Talk:Gödel's incompleteness theorems/Arguments/Archive 2

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A non-mathematical understanding of a mathematical crisis

There are roughly speaking two different ways of creating a form out of an amount of solid material, let’s say marble. The first way is to start with one piece of marble (the principle block) and keep on gluing other pieces onto it until the desired form is achieved. The second way is to start with a block of marble (the principle block) and to carve out a form. The principle block in the gluing method serves as a foundation for the whole structure (unless pieces are glued to the bottom). With respect to the carving method it would be misleading to call the principle block the foundation of the carved form.

For mathematicians it is unclear how a formal system that reflects parts of mathematics comes into being. It is therefore very important to keep an open mind if one is searching for ‘the principles of mathematics’ by means of formal systems. David Hilbert set out a program for uncovering the principles of mathematics in order to solve the foundational crisis. His strategy was based upon the idea that the principles of mathematics would appear as if they were like foundations. Gödel’s incompleteness theorem however obstructed Hilbert’s pathway and in order to encounter the principles of mathematics one is forced into a different strategy. This means that one has to master a new understanding of how a formal system that describes parts of mathematics comes into being. The understanding that a formal system is carved out of a collection of mathematical principles would provide a new pathway. In this case the strategy would be to look for the bits and pieces that have been chiseled of and glue this “stuff” back onto the formal system in order to reconstruct the shape of the uncarved block, that is, the principles of mathematics. How this stuff looks like, if it exists, is unclear but it can be recognized because it consists of the following characteristics:

1- A piece of stuff is neither a provable theorem nor a disprovable theorem of the formal system, that is, it appears as meaningless for formal system (that is why it has been chiseled of).

2- A piece of stuff must be sophisticated enough to carry the notion of truth or falsehood from the point of view of the mathematical logician, that is, it is still reasonable stuff.

3- ‘Gluing’ a piece of stuff back onto the formal system will result in some kind of simplification and is capable of reconstructing a mathematical principle.

A possible example of this stuff with respect to the formal system of arithmetic (the art of counting) is mentioned on the talk page under the title “Gödel’s theorem versus Hilbert’s program”. The argument is that Gödel’s theorem will no longer create an obstacle for this new pathway of which a clarification is given here. Sarahcroft (talk) 17:23, 20 July 2010 (UTC)

It is not clear to me what kind of "stuff" you have in mind which, when added to the axiomatic framework, would potentially save Hilbert's program. If one adds the epsilon_0 "stuff", this had been done already in the 30s, as you pointed out there. What alternative "stuff" do you have in mind? Is it "finitistic" in nature? Sorry if I am going off in a wrong direction. Tkuvho (talk) 18:22, 20 July 2010 (UTC)

Well, to be frankly honest, I was hoping that ‘zero is the immediate successor of not a number’ would be capable of doing something, at least demonstrate that the term ‘zero’ is the result of an unnecessary multiplication of mathematical entities. Obviously I’m not an expert if exactly this ‘statement’ (it is not a statement because it does not obey the rules of formation) already has been added, though I am not talking about an extension but a sophistication of the formal system. To be clear: “Gödel described a formalized calculus within in which all of the customary arithmetical notations can be expressed and familiar arithmetical relations established.” (Nagel and Newman 1958, p53) Let’s call the familiar arithmetical expressions the data of the first kind. But how about the unfamiliar arithmetical expressions that do not obey the formation rules, that is, the data of the second kind (i.e., stuff)? “But is Hilbert’s finitistic method powerful enough to prove the consistency of a system such as principia, whose vocabulary and logical apparatus are adequate to express the whole of arithmetic and not merely a fragment.” (p44) Here is where my doubts lie. Is principia sophisticated enough to express the whole of arithmetic, that is, both the data of the first kind and the data of the second kind? Leonardo da Vinci said: “simplicity is the ultimate sophistication”. Principia Mathematica is indeed very sophisticated, but is it the ultimate sophistication with respect to arithmetic which is necessary to uncover the principles of mathematics (simplicity)? My doubts lie in the quality of Principia Mathematica. Is it truly profound enough? Can it express the data of the second kind; the stuff of which I for the moment have no alternative though it must be finitistic? Sarahcroft (talk) 16:11, 22 July 2010 (UTC)

The distinction between "familiar arithmetic expressions" and "unfamiliar" ones is not really well-defined, and not merely in a pedantic sense. There are substantially different ways of drawing the line here. Thus, working in primitive recursive arithmetic imposses more stringent limits on power of expression than framework involving first-order logic. generally speaking, the failure of a particular (naive) interpretation of Hilbert's program is no reason to panic; the program is alive and well in proof theory. I suggest the illuminating article by Avigad and Reck as must reading. Tkuvho (talk) 17:14, 22 July 2010 (UTC)

Thanks, but I’m still wondering in the dark. Is there really a mathematical distinction between familiar arithmetical expressions and unfamiliar ones? For example, one could be interested in calculating the surface of a square(2-dimensional) or the volume of a cube (3-dimensional). Hence, there is familiarity with a number being raised to the power two or three. There is little familiarity though with a five-dimensional world and unfamiliarity with a two and a half, a negative or an imaginary dimensional world. Within mathematics though a number may be raised to the power 5, 2 ½, -3 or i without any fear of unfamiliarity with the related dimensional worlds. Mathematics makes no distinction and the line that is being drawn between familiar and unfamiliar arithmetical expressions appears merely as an illusion. Similarly, the Peano axioms are based upon an understanding of arithmetic which incorporates the terms; ‘the immediate successor of’, ‘a number’ and ‘zero’. These three terms originate from a human experience with arithmetic which serves as a temporarily basis and dividing line in order to create familiarity. After familiarity has been achieved one can take away this illusionary dividing line and experience the full monty. The statement ‘zero is the immediate successor of not a number’, or, ‘0=s(~s)’ does the trick because every number now can be experienced as an immediate successor. So, the three terms can be restated as ‘the immediate successor of‘, ‘an immediate successor’ and ‘the immediate successor of not an immediate successor’, that is, all three terms originate from the same stem (simplicity). Again mathematics appears profound enough to dissolve the distinction between familiar and unfamiliar arithmetical expressions. In short, Gödel’s incompleteness theorem demonstrates that the system developed in principia mathematica is not profound enough to fathom all of arithmetic and extending this system in quantity (thickness) will still not change this fact. Possible solution: one simply needs a more profound system in order to fathom all of arithmetic. This may be achieved by dissolving the dividing line between familiar and unfamiliar arithmetical expressions. Result: It is possible to fathom the arithmetical principle ‘the immediate successor of anything is an immediate successor’ (see talk page). Am I going off in a wrong direction? Sarahcroft (talk) 12:57, 25 July 2010 (UTC)

Thanks for your comment. In the end I am not sure it would be possible to gain a non-mathematical understanding of a mathematical crisis. I would make the following specific suggestions: (1) a concrete implementation of Goedel's insight may be found at Goodstein's theorem; if you study this page, you may gain a more detailed understanding of what Goedel meant; (2) to begin to understand how a distinction between "elementary stuff" and more "unfamiliar stuff" is implemented in a mathematical context, you could first focus on the difference between first order logic (quantification over elements only) and higher order logics (quantification over sets, etc); here an important example is the property of completeness. Tkuvho (talk) 13:39, 25 July 2010 (UTC)

Fair enough, but before I start climbing this tree of logic, first low then high, I would like to know if its roots are strong enough to prevent it from tipping over. What is the principle of logic? Sarahcroft (talk) 16:12, 25 July 2010 (UTC)

That's a tough one. It's not even my field. Let me try this: some people think that mathematical logic is a way of making things "really rigorous" in mathematics. I think this perception is incorrect. At least the part of logic that interests me is a way of making progress in mathematics, not striving after illusory absolute rigor. To give an example, a good understanding of principles underlying first order logic has led Hewitt (1948) and Los (1955) to a major advance, namely creating a framework where historical infinitesimals can finally be justified. This was all put together by Robinson in 1961. Tkuvho (talk) 19:13, 25 July 2010 (UTC)

Thanks for the suggestions, your point of view is slowly becoming more clear. Concerning the root of logic, wiki.answers.com gave me the following answer: “the principle behind logic is that there are relationships between two things”. Accordingly, the principle of logic is based upon a certain familiarity with reality, that is, the plurality of things. Hence, from a logical point of view there will appear a distinction between the familiar aspects of reality and the unfamiliar aspects of reality. The idea of drawing a line between these two aspects of reality is merely based upon the assumption that there is absolute rigor with respect to the laws of logic, that is, the laws of logic are to be interpreted from some kind of absolute point of view which assumes the existence of an absolute framework in which logic is embedded. The idea of an absolute framework needs further contemplation.

Consider the following situation with respect to rational thinking: the three classic laws of thought (i.e., everything that is is; nothing can both be and not be; everything must either be or not be) touch a mind in which only nothingness is reflected. Nothingness reflected in an empty mind appears as nothing but because it appears it must therefore also be something. According to the law of the excluded middle ‘nothing’ also must either be or not be. It is more common to say that ‘nothing is’ and according to the law of identity ‘nothing is nothing’. So, ‘nothing’ can be seen as the first element and all the other things are joined to this element. This is roughly the framework of common logic. This structure is, for example, found in the Peano axioms by substituting ‘nothing’ for the term ‘zero’. According to these axioms ‘zero’ is a number (being) and also an elementary term. The other numbers are joined, though in a special linear way, with the term ‘zero’ by means of the successor function. The obvious question that arises with this understanding is: does this absolute framework exist independent of the mind? Now consider an alternative route. Nothingness reflected in the empty mind appears as nothing and according to the law of non-contradiction ‘nothing’ has the aspect that it can both be and not be. The being aspect of ‘nothing’ contains all being (like space contains all matter) and within this being plurality is found. There is familiarity with plurality of being and from here common logic continues. Besides the fact that this structure looks very awkward it also allows a seemingly ambiguous (mindless) interpretation of the law of non-contradiction, that is, this law is considered to be without the necessity of an absolute framework in order to be understood. This is the structure I used earlier on with respect to the term ‘zero’ in arithmetic, a structure that apparently is unacceptable. The rejection of this structure, however, is based upon the existence of a certain framework in which logic is supposed to be embedded. The existence of this framework results in that the whole tree of pure logic is rigorously pruned into a certain shape by trimming back all the branches of unfamiliarity because the absolute framework tells us that there is a certain distinction between familiarity and unfamiliarity. This shape appears to represent the structure of common logic (first order logic, higher order logic, etc.) and by using common logic an understanding of ‘zero’ with respect to arithmetic is discovered which like a fractal supplies a distinction between familiarity and unfamiliarity. First put the rabbit in the hat and then take the rabbit out the hat.

The history of physics has already demonstrated that the idea of an absolute frame of reference with respect to classical mechanics is illusory. The structure of classical mechanics was replaced by the theory of invariance which sadly enough carries the name ‘the theory of relativity’. This theory broke through the superficial structure of familiar understanding on which classical mechanics was based in order to reveal unfamiliar but profound aspects of reality. Bearing this in mind one should at least maintain an open attitude towards the possibility of pure logic without an absolute framework, that is, a certain invariance of interpretation with respect to the laws of rational thinking. The law of non-contradiction as demonstrated earlier appears to consist of a certain invariance of interpretation also because the term ‘be’ and the term ‘not be’ can be interchanged without changing this law (invariance with respect to being). The same counts for the law of the excluded middle. The law of identity however is off-centered because it only uses the term ‘is’ and does not consist of any possible invariance of interpretation. This law clearly reflects a human being’s point of view which consists of a certain familiarity with being and an unfamiliarity with not being. Hence, this viewpoint is biased towards being and this law should be replaced by a more profound law in order to expresses pure logic. Once this is done the structure that occurs will be in harmony with the alternative route earlier mentioned. Of course, you could rightfully claim that all that is being done here is no more than non-sense which is being put in a hat, and then later the same fascinating non-sense is being pulled out of the hat. If it is true that you are not striving after illusory absolute rigor then at least you will be capable of keeping an open mind to the suggested alternative route which leads to an alternative understanding of arithmetic. Maybe William of Occam can act as an independent judge in order to decide which pathway is better? Occam’s razor can in this context be defined as: if there are two theories of arithmetic that make exactly the same observable predictions then the simplest one is closer to the principles (foundations) of mathematics. Sarahcroft (talk) 14:44, 5 August 2010 (UTC)

Occam's razor is fine to a certain extent (though there have been philosophers, such as Burgess, that have described it as a medieval superstition). The question is, what use do you put it to. I have the impression you are still traumatized by the idea that there isn't a unique mathematical structure corresponding to the naive counting numbers. I happen to think, on the contrary, that variety is best. For instance, there is no unique "continuum" either. One could distinguish four types of continua, according to a pair of binary parameters: one parameter is whether the underlying logic is classical or intuitionistic; the other parameter is whether the continuum is Archimedean (no infinitesimals) or Bernoullian (containing infinitesimals). All four are present in the literature, and all four are useful depending on the task at hand. This is, of course, even before you get to set-theoretic delicacies such as the continuum hypothesis, yet another binary variable. Tkuvho (talk) 16:24, 5 August 2010 (UTC)

A meek version of Occam’s razor was used here because the word ‘better’ does not occur in an absolute sense; “the simplest one [theory] is closer to the principles (foundations) of mathematics”. Hence, it has been put in use here as no more than accepting that a justified simplification of an understanding of mathematics is one step closer towards the principles of mathematics. Actually, Occam’s razor states that “entities should not be multiplied unnecessarily” which is difficult to disagree with due to the openness of the word ‘unnecessarily’. It is probably true to some extend that I’m traumatized though not by the lack of a unique mathematical structure corresponding to the naive counting numbers but by the mind being framed by linear order due to rational thinking. The awareness of zero arises from the awareness of the positive natural numbers. Zero is observed from only one direction because the negative integers provide only a mirror image of this observation. Awareness of the infinite arises from the fact that there is no biggest number. The infinite is also only observed from one side. Within mathematics zero and infinite cannot be the same because of the linear order. It is like when somebody is looking true a very powerful telescope and observing the back of a person’s head looking true a telescope. If the universe is flat (linear) then it must be somebody else if not it could be the same person. In this case if there was no head obstructing the view then one could look also through that telescope again and in the next one, etc., observing just... nothing (the infinite of infinite). The linear order, though, is not present within pure logic and merely imposed upon pure logic by mathematics, hence, mathematical logic. The idea that zero and infinite are different observations of the same entity is quite possible from a pure logical point of view, but even so, from a mathematical point of view the possible multiplication of this entity would still appear to be necessarily. The incompleteness theorem directs the mathematician to the infinite yet it still might be the case that the origin of incompleteness is simply found in zero. The idea of essential incompleteness is based upon the idea that an extension always increases a system. If it is possible to extend a system, and then simplify it, then map it onto another system and then give it a twist in order to return to the same system before the extension then this loop would mean that essential incompleteness does not automatically mean an infinity of extensions. The infinite inexhaustibility of mathematics is also based upon a linear way of thinking. Therefore, I do not question the correctness of the incompleteness theorem and neither its implicit results. I do however question the rational interpretation of these implicit results because rational thinking has the tendency of a linear approach due to its dualistic nature. This all revolves around one question: is there an absolute framework in which pure logic and therefore mathematical logic can be embedded or is pure logic without a framework? Sarahcroft (talk) 08:19, 6 August 2010 (UTC)

Upon mature reflection, Hermann Weyl concluded in an essay in the 1950s that, to his dismay, neither mathematics nor logic have such an absolute framework. As far as iterated infinite telescopes, you might be interested in reading section 3 of Lorenzo Magnani and Riccardo Dossena, Perceiving the infinite and the infinitesimal world: unveiling and optical diagrams in mathematics. Foundations of Science 10 (2005) 7--23. Let me know what you think of it. Tkuvho (talk) 09:10, 6 August 2010 (UTC)

Magnani and Dossena’s work is appropriate and Weyl’s work profound, yet my argument that the basis of logic does not have any solid ground for some strange reason doesn’t seem to get off the ground. Before I dissolve myself just some final contemplations. Consider the following statement similar to Berry’s paradox: ‘what cannot be defined has just been defined’. This, of course, is a contradiction. Now take the following more sophisticated statement: ‘what cannot be defined has just in principle been defined’. There is no contradiction here and the converse also holds: ‘what has been defined can in principle not have been defined’. Now contemplate the following: ‘[anything that is] being is in principle not being’. This is the law of non-identity which is the first law of pure logic, the converse also holds (invariance of interpretation). The law of non-identity reveals the relationship between being and not being. This law together with the law of non-contradiction and the law of non-third (excluded middle) are the three non-laws that form pure logic which are not embedded in an absolute framework. The three classic laws of thought, which appear to be embedded in an absolute framework (rational understanding) are derived from the laws of pure logic.

And finally this: ‘I think, therefore I am’ is a reflection. Hence it should be: ‘I think “I think, therefore I am” ’. The first part is therefore ‘I think “I think” ’ which is a form of quining. This construction has just like the equation f(x)=f '(x) two types of solutions for ‘I’. First, try for ‘I’ any Self that fits the structure, if so, then this Self supplies a correct solution, that is, a self-evident solution. For example, f(x)=e raised to the power x is a correct self-evident solution because f '(x)=e raised to the power x and ‘I equals the self-evident self’ is also a self-evident solution. Second, the trivial solution which corresponds to f(x)=0 is ‘I equals the empty self’. The first solution reflects René Descartes’ conclusion ‘I am’ because the self-evident self is. The second solution ‘I am empty’ does not contradict Descartes’ conclusion but takes the essence away from the self-evident self on which it is based, in other words, it takes away the dividing line (i.e., the idea that self-evident reveals absolute truth) between the familiar self (i.e., I am myself) and the unfamiliar self (i.e., I am in principle not myself). Sarahcroft (talk) 15:15, 7 August 2010 (UTC)

I don't know what "pure logic" means. If what you mean by this is the kind of logic that's so universal that it would underlie any attempted mathematics, philosophy, or mathematical logic, then I would have to disagree with you on certain details, such as the characterisation of the law of excluded middle as part of such "pure logic". Namely, this law has been contested by both philosophers and mathematicians, and alternative systems have been developed where the law does not hold. Tkuvho (talk) 21:16, 7 August 2010 (UTC)

Pure logic has indeed not been defined. Your description of pure logic uses the word "underlie" which reveals a mental image in which pure logic appears as a foundation. Are you aware of the persistence of this mental image with respect to rational thinking? I would prefer the following indirect description: the classic laws of thought are like a reflection of pure logic projected upon a certain surface (rational under-standing). If there is no surface (no rational understanding) then there is no reflection either (no classic laws of thought) and only pure logic remains. It appears to me that any attempted mathematics or mathematical logic is a combination of reflections of pure logic and reflections of reflections of pure logic projected upon a surface. The rational understanding of the law of the excluded middle is a reflection of pure logic, yet not a part of pure logic. Your disagreement with me is therefore based upon a certain mental image with respect to pure logic. How would you stand without this mental image? Sarahcroft (talk) 10:03, 10 August 2010 (UTC)

I am not sure, but even in terms of projection onto the surface of rational understanding, I have doubts about the law of excluded middle being part of pure logic, to the extent that it does not hold in some rather meaningful systems. If it were part of pure logic, how could it be meaningfully violated? Tkuvho (talk) 12:08, 10 August 2010 (UTC)

Apparently you didn’t take in “the law of the excluded middle is a reflection of pure logic, yet not a part of pure logic”, so let’s try it this way. In order for the law of the excluded middle to be violated it has to be untrue from a certain point of view. However, the whole idea of the law of the excluded middle either being true or untrue is based upon a framework that uses the law of the excluded middle. Hence, the violation of the law of the excluded middle is in principle not violating the law of the excluded middle. Now consider this: the law of the excluded middle is neither true nor untrue. Does this violate the law of the excluded middle? Sarahcroft (talk) 19:21, 11 August 2010 (UTC)

I am beginning to lose the thread of this discussion. What I tried to point out is that, first, I have seen no convincing reasons to disagree with Weyl, who holds that neither mathematics nor mathematical logic seem to have an absolute foundation; and second, that to do the philosophy of mathematics, it may be a good idea to learn some mathematics first. Pure speculation about pure logic may be a good way to do philosophy--I don't know enough about that field to be able to judge competently--but it is not a good way of doing the philosophy of mathematics. Tkuvho (talk) 20:21, 11 August 2010 (UTC)

Bertrand Russell’s “Definition of Number” explains how “it is simpler logically to find out whether two collections have the same number of terms than it is to define what number that is” by means of a relation called “one-one”. He ends the chapter with “A number is anything which is the number of some class. Such definition has a verbal appearance of being circular, but in fact it is not. We define “the number of a given class” without using the notion of number in general; therefore we may define numbers in general in terms of “the number of a given class” without committing any logical error.” This might be true for the numbers 2,3,4,... but for the number 1 there still appears to be some kind of circular reasoning because the reasoning involved uses a one-one relationship, that is, the concept one is used in order to define the number one. The essence that supports the idea of circularity is reflected within mathematics by the existence of the complex numbers. The familiarity of the real part of the one-one relation is within the awareness of the human being but the unfamiliarity of the imaginary part of the one-one relation does not appear to be within human awareness and is sieved out by the circular reasoning mentioned above. Most of the things said in this thread simply revolve around this issue. This thread, however, is getting very long and although you have given me enough opportunity to express myself concerning this issue I seem to have been unconvincing on all levels. Thanks for your suggestions. Sarahcroft (talk) 09:03, 14 August 2010 (UTC)

Is this material reflected in any of the wiki pages? I would like to understand Russell's viewpoint better. The idea that there should be circularity in an attempt to find some kind of absolute foundation to the concept of oneness does not surprise, as I think of it as being pre-mathematical, and I think we both already expressed the sentiment that mathematics arguably does not have "absolute" foundations. However, connecting this to the existence of imaginary numbers strikes me as odd. However, I would like to understand this better. Tkuvho (talk) 18:37, 14 August 2010 (UTC)

The use of imaginary numbers with respect to the one-one relationship is only a way of showing that mathematical insight could be a usable tool to reflect that understanding has besides a real (familiar, observable) component also an imaginary (unfamiliar, unobservable) component which both are part of a whole (awareness). The possibility of drawing a line between these two components should not imply that these components need to be considered separately. Another mathematical analogy of what I tried to portray above is that mathematical understanding is like a map (plane) of the world of mathematics (surface of a sphere). A plane can never completely reflect the surface of a sphere because its spatial properties are different, that is, the surface of a sphere has, roughly speaking, one point extra compared to a plane. Similarly, all arithmetical awareness cannot be reflected upon arithmetical understanding —this extra point becomes reflected by the Gödel sentence. For example, zero equals itself (0=0) is rationally understandable but the relationship between zero and itself is not rationally understandable, that is, zero divided by zero is not definable —the one-one relationship between zero and itself is only one relationship out of a manifold of possibilities. The ‘self’ is imposed upon zero in order to make it appear upon the plane of rational understanding in which self-evident is considered truth. Self-evident, however, is only a reflection (or approximation) of truth. The arithmetical understanding of ‘zero’ gives an incomplete picture of what ‘zero’ is because the topology of understanding differs from the topology of awareness. Gödel’s incompleteness theorem detects this difference in topology. The supposed inexhaustibility of mathematics due to Gödel’s incompleteness theorem is therefore only a relative inexhaustibility because mathematics only appears to be inexhaustible from the point of view of rational understanding. The arithmetical world, though with a limited surface, cannot be reflected completely and consistently upon the plane of rational understanding. The idea that Gödel’s incompleteness theorem leads to a negative answer of Hilbert’s second problem is based upon the idea that rational understanding is the ultimate sophistication. The wikipage “Hilbert’s second problem” clearly portrays a more meek and open minded view: “In the 1930s, Kurt Gödel and Gerhard Gentzen proved results that cast new light on the problem [Hilbert’s second problem]. Some feel that these results resolved the problem, while others feel that the problem is still open.” This is in harmony with Nagel and Newman’s concluding reflections: “The possibility of constructing a finitistic absolute proof of consistency for arithmetic is not excluded by Gödel’s results. Gödel showed that no such proof is possible that can be represented within arithmetic. His argument does not eliminate the possibility of strictly finitistic proofs that cannot be represented within arithmetic. But no one today appears to have a clear idea of what a finitistic proof would be like that is not capable of formulation within arithmetic." Here ‘formulation within arithmetic’ relates to the plane of arithmetical understanding. Hence, a finitistic absolute proof of consistency for arithmetic needs to be constructed within a mind space that consists of different spatial properties. Sarahcroft (talk) 20:14, 27 August 2010 (UTC)

Tide has changed on Wittgenstein versus Gödel

The recent literature has turned from outright hostility against Wittgenstein to active controversy. See the following publications,

• Carl Hewitt. "Common sense for concurrency and inconsistency tolerance using Direct Logic^TM and the Actor Model" ArXiv 0812.4852.
• Francesco Berto. "The Gödel Paradox and Wittgenstein's Reasons" Philosophia Mathematica (III) 17. 2009.
• Francesco Berto. There's Something about Gödel: The Complete Guide to the Incompleteness Theorem John Wiley and Sons. 2010.

The classicists don't seem to have any good arguments in response.17.244.69.240 (talk) 00:25, 28 July 2010 (UTC)

I actually don't see hostility against Wittgenstein; just against his work in this field. Most of the people commenting on Wittgenstein seem to have said he was a fool, rather than a knave. — Arthur Rubin (talk) 08:00, 28 July 2010 (UTC)

Do you mean to say that they thought he had insufficient training in mathematical logic? Tkuvho (talk) 11:17, 28 July 2010 (UTC)

There is a good introduction to this at the SEP entry. Modern philosophers who study this do not argue that Wittgenstein actually found a flaw in the incompleteness theorems. Instead, they argue that what Wittgensteins comments can be interpreted in sensible ways, if one reads between the lines and interprets them within Wittgenstein's broader philosophy of mathematics. That broader philosophy includes other tenets that are not widely accepted, such as strict finitism and the identification of truth with provability.

Modern authors such as Berto also note that Wittgenstein's comments can be interpreted from the perspective of paraconsistent logic. This is an interpretation only, of course: Wittgenstein's writings foreshadowed but strictly predated rigorous developments of paraconsistent logic, so he could not have referred to it specifically.

As a side note, the first published proof of the consistency of Principia Mathematica that I am aware of was Fitch (1938) [1]. Since Wittgenstein wrote about the incompleteness theorems in 1937–1938, it is not obvious whether he was aware of such consistency proofs. But from our modern perspective, we can recognize that "Russell's system" in Principia is consistent after all. — Carl (CBM · talk) 11:50, 28 July 2010 (UTC)

Gödel’s theorem versus Hilbert’s program

Did David Hilbert ever think of extending (extrapolating) the conception of arithmetic over the non-arithmetical plane? For example, the statement ‘zero is the immediate successor of not a number’ is an extension of the conception of the arithmetical term ‘immediate successor of’ over the non-arithmetical plane without leaving the field of mathematical logic, that is, the statement only uses concepts from mathematical logic. This statement transforms (rule of substitution) the Peano axiom ‘zero is a number’ into ‘the immediate successor of not a number is a number’. Now add to this the Peano axiom ‘the immediate successor of a number is a number’ and by using the law of the excluded middle one can derive the statement ‘the immediate successor of anything is a number’. In the light of Hilbert’s program this statement would be called a weak mathematical principle. One can also go reverse, that is, start from this weak mathematical principle and proof the two mentioned Peano axioms. This proof, though, cannot be represented within arithmetic because it incorporates a non-arithmetical conception, that is, the conception that ‘zero is the immediate successor of not a number’. Hence, by extending the formal conception of an arithmetical term over the non-arithmetical plane it becomes possible to construct a deductive system based upon a weak mathematical principle strong enough to incorporate elementary arithmetic such that within this deductive system it still could be possible to construct an absolute finitistic proof of consistency of this deductive system (and hence of arithmetic) because this proof would not be capable of formulation within arithmetic and therefore be immune to Gödel’s incompleteness theorem. Sarahcroft (talk) 14:21, 10 July 2010 (UTC)

I'm afraid I have a little trouble following your remarks step-by-step, but I'll respond to what I glean from them taken as a whole. I'm guessing a little bit, so if I guess wrong, you'll need to clarify what you're talking about.

It is not necessary for Goedel's argument that the objects of discourse of the theory under discussion (the object theory) be natural numbers. You just have to have a way to interpret some of them as natural numbers, and then be able to prove some basic facts about the naturals, thus interpreted, from the object theory. For example, the objects of discourse of ZFC are not natural numbers, but sets. But there's a standard way of interpreting some sets as natural numbers, and then proving quite a bit about the natural numbers from ZFC.

In your case, you seem to be imagining a theory that would have, as objects of discourse, the literal natural numbers (your "successors of not-a-number") in addition to some other stuff (your "non-arithmetical plane"). That seems to be parallel to the ZFC case, where you have some sets that are (reinterpreted as) natural numbers, and some that are not. The Goedel theorems definitely apply here. --Trovatore (talk) 23:27, 10 July 2010 (UTC)

Actually, incompletess is "absolute" in the sense that strictly speaking it does not really depend on the domain of discourse of a theory. Incompletess depends on the principle of "roundtripping" (see Incompleteness Theorems). 67.169.144.115 (talk) 05:50, 11 July 2010 (UTC)

"Incompleteness" depends on the principle of "roundtripping"? Only in Hewitt's imagination. — Arthur Rubin (talk) 07:47, 11 July 2010 (UTC)

Actually, roundtripping is the principle used by Gödel to prove incompleteness using Gödel numbering (see Gödel on Wittgenstein ). 67.169.144.115 (talk) 15:02, 11 July 2010 (UTC)

You're demonstrating my point. That is Hewitt's imagination. It may be the case that Hewitt calls the principle actually used by Gödel "roundtripping", but that wouldn't mean that Hewitt's definition of "roundtripping" resembles the principles Gödel and other mathematicians use(d). — Arthur Rubin (talk) 15:15, 11 July 2010 (UTC)

In Gödel on Wittgenstein , Professor Hewitt formalized the proof of incompleteness in the theory Peano+Roundtripping. It looks like Wittgenstein was correct: the theory is inconsistent. 171.66.105.162 (talk) 17:03, 14 July 2010 (UTC)

I haven't checked that paper, but, if that is correct (Peano+Roundtripping is inconsistent), then "roundtripping" is not relevant to the incompleteness theorems in conventional logic, and shouldn't possibly be mentioned here. — Arthur Rubin (talk) 18:32, 14 July 2010 (UTC)

Hewitt's proof sketch of the inconsistency of the system "Russell" comes down to this line:

Since Russell aimed to be the foundation of all of mathematics, a theorem to the effect that Russell is incomplete should be provable in Russell. And as Wittgenstein noted, self-provable incompleteness of Russell would mean that Russell is inconsistent.

In other news, because I aim to be a Formula 1 driver, I should now be able to perform any driving feat that a professional driver can perform. — Carl (CBM · talk) 21:33, 14 July 2010 (UTC)

Using "This proposition is not provable", Wittgenstein did to Russell something very analogous to what Russell had previously done to Frege using "the set of all sets thare are not members of themselves." 17.244.69.68 (talk) 22:24, 14 July 2010 (UTC)

Roundtripping is the following principle:

⊢_Peano ( ⌊⌈s⌉⌋ ⇔ s)

where ⌈s⌉ is the Gödel number of s and ⌊n⌋ is the sentence whose Gödel number is n. Therefore, Roundtripping is a theorem of Peano. 17.244.69.68 (talk) 22:43, 14 July 2010 (UTC)

Moving comments on Wittgenstein to arguments subpage

I have moved a comment about Wittgenstein by an IP editor to the Arguments subpage of this talk page. This is unfortunate, so I want to explain myself.

I had originally hoped that discussion here about Wittgenstein might lead to some improvement in the article. However, multiple lengthy threads have not led to any concrete changes. Moreover, the comments I moved today simply rephrased arguments that have already been recently discussed.

Wikipedia talk pages are not a suitable venue for pure discussion of the value of Wittgenstein's opinions or Carl Hewitt's opinions. The format of the discussions here, and the anonymity of our editors, makes it difficult to have a proper scholarly debate. There are many other forums where mathematics is actually bred and grown, such as papers, books, seminars, conferences, and coffee shops. The intended role of Wikipedia talk pages is only to discuss the writing of the article itself.

Everyone has done his or her best to discuss the issue of Wittgenstein here, and the discussions have consistently failed to come to any resolution. So, unless new comments bring substantial new material that is directly related to editing this article, I think it is appropriate to move future threads about Wittgenstein to the arguments subpage as well. I would appreciate opinions from other editors about this. — Carl (CBM · talk) 01:16, 28 July 2010 (UTC)

My opinion is that this should have been done long ago. Hewitt himself is theoretically banned from this page and he's indefinitely blocked for sockpuppetry, and the material being added here is precisely the sort of material that /Arguments pages are meant for. If the field of mathematical logic comes to agree with the proposition being advanced by the Wittgenstein/"round-tripping"/etc. argumentation, then we can report on it. We should not be devoting further effort to one person's opinion that it is important, any more than we should devote any effort to one person's claim that they can trisect an angle, or find a general formula for the roots of a fifth-degree polynomial, or any other such thing. — Gavia immer (talk) 01:46, 28 July 2010 (UTC)

In general, Wikipedia has a very hard time with paradigm shifts. Editors tend to take the received wisdom as dogma and reject new published results out of hand. Others demonize the innovators. In this case, it is interesting that someone new to demonize has appeared on the scene:

• Francesco Berto. "The Gödel Paradox and Wittgenstein's Reasons" Philosophia Mathematica (III) 17. 2009.
• Francesco Berto. There's Something about Gödel: The Complete Guide to the Incompleteness Theorem John Wiley and Sons. 2010.

So now editors have someone else to complain about besides Hewitt :-) 171.66.87.180 (talk) 03:42, 28 July 2010 (UTC)

Ad hominem attacks on Professors Berto and Hewitt are not a viable long-term strategy. If their arguments hold water, then they will win.63.249.108.250 (talk) 05:18, 28 July 2010 (UTC)

There haven't been any ad hominem attacks against Berto, unless one of you IPs is Berto, rather than Hewitt.

And Wikipedia is not supposed to report "paradigm shifts" until they actually occur. In this case, one would normally expect any shift due to a 2009 paper to occur no earlier than 2024. In any case, there is at most two persons prior to Hewitt who would have introduced paradigm shifts in distinct fields; in this case, (asynchronus) computability theory and mathematical logic. — Arthur Rubin (talk) 07:56, 28 July 2010 (UTC)

From a history of science perspective, it's fascinating that Wittgenstein conceived the paradigm shift even before Gödel did the incompleteness theorem preciptating the crisis.171.66.48.190 (talk) 21:06, 28 July 2010 (UTC)

If it were a paradigm shift, or if Wittgenstein conceived a valid refutation of Gödel before Gödel published his work, that would be notable. Neither has much weight (yet, to give the floating IP credit), and, per WP:CRYSTAL and WP:NOR, we could not note it until it does have some weight. — Arthur Rubin (talk) 08:11, 29 July 2010 (UTC)

Wittgenstein initially proposed inconsistency tolerant logic in "The Blue and Brown Books" When Gödel came up with "This proposition is unprovable in Russell's system" Wittgenstein immediately realized that Russell's system was thereby rendered inconsistent and commented "So what?" 171.66.90.247 (talk) 22:11, 29 July 2010 (UTC)

Wittgenstein (may have) proposed inconsistency-tolerant logic, but he had no proposals how to go about non-trivial inconsistency-tolerant logic. (By a trivial inconsistency-tolerant logic, I mean one which includes $\bot \vdash X$ for any sentence X.) That's not a "refutation" of Gödel, but it may be a valid note that Russell's system (with Wittgenstien's identification of "true" and "provable") is inconsistent, and hence trivial. — Arthur Rubin (talk) 22:23, 29 July 2010 (UTC)

Wittgenstein proposed that inconsistencies be isolated where they would do no harm although he did not publish a specific set of rules to accomplish this. There is no mathematically well-defined notion of "truth" for systems as powerful as Russell's. Wittgenstein focused on the the notation of provability in deriving an inconsistency in Russell's system from the incompleteness theorem. On the other hand, Gödel went out on a limb and spoke of "truth" in Russell's system using a alleged "metatheory."17.226.35.236 (talk) 23:34, 29 July 2010 (UTC)

There are several consistency proofs of Russell's system of type theory. The earliest I have seen was was published by Fitch, 1938 [2]. A more recent consistency proof was published by Mares 2007 [3]. If Wittgenstein believed the system of Principia was inconsistent, he was apparently wrong. Some of the consistency proofs (Mares' included) do give well-defined semantics ("notions of truth") for Russell's system. — Carl (CBM · talk) 01:16, 30 July 2010 (UTC)

Russell intended for his system to encompass the full power of Zermelo–Fraenkel set theory (ZF). When the incompleteness theorem was proved, Wittgenstein and Gödel were both willing to grant the point. So for the purposes of discussing the incompleteness theorem, Russell's system was taken to be equivalent to the full power of ZF. However, in full ZF:

1. "truth" is not mathematically well defined
2. we have no consistency proof of ZF

Wittgenstein focused on the the notation of provability in deriving an inconsistency in Russell's system (i.e. ZF) from the incompleteness theorem. On the other hand, Gödel went out on a limb and spoke of "truth" in Russell's system (i.e. ZF) using a alleged "metatheory."

It is true that there are various "Mickey Mouse" variants of Russell's system that do not have the power of ZF. But their existence was irrelevant to the controversy between Wittgenstein and Gödel. 171.66.105.149 (talk) 02:07, 31 July 2010 (UTC)

On consistency of Russell's system

The article quotes Gödel to the effect that:

"It is clear from the passages you cite that Wittgenstein did 'not' understand it [1st incompleteness theorem] (or pretended not to understand it). He interpreted it as a kind of logical paradox, while in fact is just the opposite, namely a mathematical theorem within an absolutely uncontroversial part of mathematics (finitary number theory or combinatorics)."

However, Hewitt (see above) pointed out:

Of course, Gödel had made an (unannounced) shift in ground because Wittgenstein was writing about incompleteness of Russell's theory. Since Russell's theory aimed to be the foundation of all of mathematics, a theorem to the effect that it is incomplete should be provable in itself. And as Wittgenstein noted, self-provable incompleteness would mean that Russell's theory is inconsistent.

64.134.239.148 (talk) 21:11, 14 August 2010 (UTC)

Hewitt's comment is misleading, if not just wrong. First, a minor point: Goedel's original paper already shows that "self-provable incompleteness" makes a system inconsistent. No comment of Wittgenstein is needed for that.

Second, the key point: Russell's system is, as far as anyone knows, consistent. There is no published claim of its inconsistency, and if Hewitt has such a proof he should certainly consider publishing it, particularly because of the several published proofs that Russell's system is consistent. If there was a trivial proof that Russell's system is inconsistent based on the incompleteness theorems, it would be well known.

The flaw in Hewitt's reasoning becomes more clear if you replace "Russell's system" with "ZFC". ZFC is also intended to be a foundation of all mathematics, but this does not mean that we can prove the consistency of ZFC within ZFC, even though we can give heuristic "proofs" of its consistency via the cumulative hierarchy.

Apart from that, any argument about Wittgenstein's remarks that hinges on using "Russell's system" specifically, rather than ZFC, is of historical interest only, because Russell's system is of historical interest only. To be relevant to the contemporary philosophy of mathematics, rather than just to Wittgenstein scholarship, any argument would need to work for contemporary foundational systems such as ZFC and topos theory. — Carl (CBM · talk) 01:15, 15 August 2010 (UTC)

Recent addition re Wittgenstein vs Goedel

This is well-written and fair and interesting (given we can corroborate the facts). Some of it I know to be fact, but it needs more-precise references (page numbers, source of criticism of Bernays etc). So I stuck some citation-flags where I think they're needed. I'll look to see if I can find any of this in my volumes of Goedel's nachlass. Bill Wvbailey (talk) 14:00, 14 August 2010 (UTC)

It's an improvement but still has a way to go to incorporate the following:

• Francesco Berto. "The Gödel Paradox and Wittgenstein's Reasons" Philosophia Mathematica (III) 17. 2009.
• Francesco Berto. There's Something about Gödel: The Complete Guide to the Incompleteness Theorem John Wiley and Sons. 2010.
• Carl Hewitt. "Incompleteness Theorems: The Logical Necessity of Inconsistency" Knol

12.197.88.252 (talk) 16:23, 14 August 2010 (UTC)

(I have moved an anonymous argument about Hewitt's reasoning to the arguments page. — Carl (CBM · talk) 01:15, 15 August 2010 (UTC))

[The first part of the following is re the "anonymous argument" referred to immediately above, now on the /Arguments page: "I am interested only in documented-source information. I know where the first quote comes from (see my history stuff with the link at the top right-hand side of this page.) But what published-in-print written source (and this does not include unvetted, self-published internet screeds) does this second quote come from?"] I haven't read any of what Agent 12.197 has proposed as sources (they look very recent, which makes them even more interesting...), so I can't comment. I encourage Agent 12.197 to add some proposed text here or on the article page so we can see what they've been reading . . .. BillWvbailey (talk) 00:36, 15 August 2010 (UTC) Bill Wvbailey (talk) 02:37, 15 August 2010 (UTC)

The argument seems to have been first published in "Common sense for concurrency and inconsistency tolerance using Direct Logic and the Actor Model" although similar comments had been previously prublished by Priest, et. al. It's an interesting argument against the quote by Gödel that is currently in the article to the effect that:

"It is clear from the passages you cite that Wittgenstein did 'not' understand it [1st incompleteness theorem] (or pretended not to understand it). He interpreted it as a kind of logical paradox, while in fact is just the opposite, namely a mathematical theorem within an absolutely uncontroversial part of mathematics (finitary number theory or combinatorics)."

Hewitt's published counterargument was

Of course, Gödel made an (unannounced) shift in ground because Wittgenstein was writing about incompleteness of Russell's theory. Since Russell's theory aimed to be the foundation of all of mathematics, a theorem to the effect that it is incomplete should be provable in itself. And as Wittgenstein noted, self-provable incompleteness would mean that Russell's theory is inconsistent.

Although not definitive, the above counterargument is not obviously entirely wrong. Also Priest, Hewitt, etc. have published additional arguments on the controversy.

We can expect more publications soon since writing about the Wittgenstein vs. Gödel controversy seems to have become something of a cottage industry :-) 12.197.88.252 (talk) 18:13, 15 August 2010 (UTC)

The source of the fact-marked passage can be found in the history:

"A google search using the phrase [with quote-marks] "Godel's comments on Wittgenstein's" coughs up only one source: Hao Wang's 1996 A Logical Journey, From Goedel to Philosophy in particular pages 179ff -- here Wang is discussing W. with Goedel after Goedel has had a chance to read the 1967 Remarks on the Foundations of Mathematics [RFM] that Carl Menger had sent him (wanting some comments for RFM) and after some information Wang has sent him in advance:

"Goedel's main comments on Wittgenstein were made on 5 April 1972 . . . his habitual calmness was absent in his comments:

"5.5.4 Has Wittgenstein lost his mind? Does he mean it seriously? He intentionally utters trivially nonsensical statements. What he says about the set of all cardinal numbers reveals a perfectly naïve view....He has to take a position when he has no business to do so. For example, “you can’t derive everything from a contradiction.” He should try to develop a system of logic in which that is true. It’s amazing that Turing could get anything out of discussions with somebody like Wittgenstein.

"5.5.5a He has given up the objective goal of making concepts and proofs precise. . . .

Goedel's 20 April 1972 written response to Menger's letter includes this:

"5.5.5b It is clear from the passages you cite [RFM: 117-123, 385-389] that Wittgenstein did not understand it [1st incompleteness theorem] (or pretended not to understand it). He interpreted it as a kind of logical paradox, while in fact is just the opposite, namely a mathematical theorem within an absolutely uncontroversial part of mathematics (finitary number theory or combinatorics). Incidentally, the whole passage you cite seems nonsense to me ."

I've not actually seen the book in its printed form (the book costs $39.15) but I double-checked this source by going to Amazon.com, finding Hao Wang's book, searching in it for the word "Wittgenstein", and eventually arriving at the relevant passages on page 179 (as noted above). It appears to be chock full of good stuff re Wittgenstein et. al. Bill Wvbailey (talk) 20:36, 21 April 2010 (UTC)

Goedel's letter to Menger is in his collected works, so it can be cited from there. It is also quoted by Floyd and Putnam [4] but I am not confident in the publication details of that paper yet. — Carl (CBM · talk) 01:17, 15 August 2010 (UTC)

Personally I think the Collected Works would be a "better" source than Wang, because they are only one step away from the original, whatever that is in this particular case . . . . there's Wang's story in his 1996 about how he discussed with G the Menger letter + the relevant Wittgenstein. As noted in the qusotes immediately above, Wang numbers the paragraphs that he quotes from, but he doesn't source them, so I am confused about where they actually appear (i.e. is he quoting directly from G's Nachlass? or from the Collected Works? or his hand-written notes?) Unfortunately I have only the first three volumes of Collected Works but not the last two with all the letters, I'll have to leave it to you to change the ref in the article if you the Complete Works is the best source. BillWvbailey (talk) 02:23, 15 August 2010 (UTC)

It doesn't seem fair to quote Gödel's attacks on Wittgenstein without comment that they have been the subject of great controversy in the literature.64.134.231.58 (talk) 22:18, 15 August 2010 (UTC)

Regarding Berto's 2010 book, I read through the section on Wittgenstein there. It summarizes the same work already summarized in this article, without really adding any new argument. Berto's 2009 paper is already cited in this article. — Carl (CBM · talk) 01:21, 15 August 2010 (UTC)

The following quotations by Wittgenstein seem relevant:

"There can’t in any fundamental sense be such a thing as meta-mathematics."

"'True in Russell’s system' means, as we have said, proved in Russell's system; and 'false in Russell's system' means that the opposite has been proved in Russell's system."

12.197.88.252 (talk) 22:54, 15 August 2010 (UTC)

They're not really of much use outside of Wittgenstein studies. They both represent idiosyncratic ideas of Wittgenstein that would be inappropriate to discuss in any depth here, but would be very appropriate to discuss in a book about Wittgenstein. — Carl (CBM · talk) 23:38, 15 August 2010 (UTC)

I am the editor who added the quote by Wittgenstein - I created the Remarks on the Foundations of Mathematics article, and looked here to see what discussion existed of Wittgenstein's unusual reaction to Gödel. Sorry if I had you all running around looking for the sources. The Rhymesmith (talk) 15:40, 17 August 2010 (UTC)

truth

I looked through some of the early versions of the article, and noticed that they do not contain the description of the theorems (attributed to Kleene in the current version) as asserting the existence of statements that are "true" but unprovable. This would appear to take a reductive Platonist view of Goedel's results. I also looked through some of the archives but have not yet found the relevant discussion. Can someone point me to such if it exists? Tkuvho (talk) 04:50, 15 August 2010 (UTC)

The article has mentioned the truth of the Goedel sentence since at least 2003 [5]. The current structure of the article dates back to around 2005 [6] (that is not the earliest revision). I doubt it was discussed back then.

The sentence you added seems to me to duplicate the footnote we already have on the word "true".

In any case, the most common interpretation of the Goedel sentence of a consistent theory is that it is true but unprovable. I'm not sure what you mean by "reductive". But since virtually every source describes Goedel sentences as true, we should do so as well. It's the most common viewpoint by far in the logic community.

By the way, any doubts about Platonism can be assuaged by realizing that the claim that "the sentence is true" is just a rephrasing of the claim "the sentence is unprovable" (the only difference is whether you talk about proofs or about proofs that have been encoded as natural numbers). So if it requires some sort of Platonism to recognize the sentence as true, it requires the same form of Platonism to recognize the sentence as unprovable. — Carl (CBM · talk) 10:44, 15 August 2010 (UTC)

Is it correct to say that the sentence is independent of, say, Peano axioms? Tkuvho (talk) 12:57, 15 August 2010 (UTC)

This is correct. The sentence is independent as both the sentence and its negation can be added to the theory without making it inconsistent. AmirOnWiki (talk) 14:25, 15 August 2010 (UTC)

I am currently puzzling over how a statement can be described as being "true" even though its negation can be added to the system without making it inconsistent. I guess this hinges on the difference between inconsistency and omega-inconsistency, see my question there :) Tkuvho (talk) 16:10, 15 August 2010 (UTC)

Back up a step. I'm sure you're comfortable with what it means to say a formula is not provable in a theory - it just means that no formal proof exists of that formula from the axioms of the theory. Now the Goedel sentence simply says, in parallel: there does not exist a natural number the codes a formal proof of that kind. So the Goedel sentence is "true" in exactly the same sense that "no such formal proof exists" is true. There are several names for this kind of truth. It can be called "disquotational truth" or it can be called "truth in the standard model". They amount to the same thing.

There is a different name for "the negation of φ cannot be added without making the theory inconsistent" - this is the same as φ itself being provable in the theory. — Carl (CBM · talk) 20:15, 15 August 2010 (UTC)

What is the "standard model"? The naive counting integers? If so, does an infinite ordinal exist for the naive counting integers? Otherwise how are we to know whether or not Goodstein's theorem is true or its negation is true for the naive counting integers? Or are you saying that there are two types of undecidable statements: the Goedel type, which really are true (but we can't prove it), and others such as Goodstein, which we don't know if they are true for the naive counting integers? Tkuvho (talk) 21:11, 15 August 2010 (UTC)

I think we should move this discussion to the reference desk, since it's getting away from discussing the actual article, and since more people will answer there. Your questions are reasonable, but (as you might have guessed) they have already been deeply explored by logicians. That's not to say there is complete unanimity about the answers, but at least the subtleties in the questions are understood.

The "standard model" consists of the naive counting integers. If you're worried about whether that is well defined, ask yourself whether you think the notion of "formal proof" is well defined, because the length of every formal proof (in lines) is a naive counting number and every naive counting number is the length of some formal proof. — Carl (CBM · talk) 23:34, 15 August 2010 (UTC)

The issue has to do with the strength of the language that is used for the Peano induction axiom (see Induction schema for Second Order Arithmetic.). Second Order Arithmetic characterizes the integers up to isomorphism and thus defines the Standard Model of Arithmetic.12.197.88.252 (talk) 23:06, 15 August 2010 (UTC)

A language on its own doesn't have a strength; the language of second-order arithmetic is trivially interpretable in first-order ZFC, and even theories of first-order arithmetic can give non-trivial interpretations of the language of second-order arithmetic. The strength of second-order logic comes from using full second-order semantics. These do lead to a characterization of the standard model, but really they are just another way of saying "disquotational truth". — Carl (CBM · talk) 23:34, 15 August 2010 (UTC)

First-order ZFC and first-order arithmetic have models of many cardinalities. It is a rigorous mathematical theorem that second-order logic with full semantics characterizes the integers up to isomorphism. By comparison "disquotational truth" is a deep mystery. 12.197.88.252 (talk) 23:54, 15 August 2010 (UTC)

The T-schema for full second-order semantics is exactly disquotational. In particular, for a domain of discourse D,

$\exists X\,\Phi (X)$ holds if and only if there is a set $Z\subseteq D$ such that $\Phi (Z)$ holds.

$\forall X\,\Phi (X)$ holds if and only if $\Phi (Z)$ holds for every set $Z\subseteq D$.

The huge strength of second-order logic comes exactly from these statements that rely on the metatheory to specify which sets exist. — Carl (CBM · talk) 01:14, 16 August 2010 (UTC)

There's no need to see "disquotational truth" as a "deep mystery"; it's a fancy name for something very ordinary. The canonical example is, what does it mean to say that the sentence "the apple is red" is true? Well, it means nothing more or less than that the apple is red. In other words, you strip off the quotation marks ("disquotation"), and you're done. --Trovatore (talk) 01:20, 16 August 2010 (UTC)

That's exactly how I read the disquotational truth page, and that's why I did not feel it answered my question as to what "truth" means here. I have not followed all the details above, but I still don't see how they answer the question about the truth of Goodstein's theorem. If one assumes the existence of an infinite ordinal for the naive counting integers, then the disquotational truth of the theorem follows. If, on the other hand, one does not assume such, then one can assume that the theorem is disquotationally false. Is Goodstein's theorem of a different kind as compared to Goedel's sentence? I am sorry if I am being slow, but I was not able to follow how the above discussion answers this. Certainly if one assumes further delicacies such as sufficient second order power, or the full ZFC background for that matter, then one can be satisfied that Goodstein is true, quotationally, disquotationally, or in any other sense. Tkuvho (talk) 04:52, 16 August 2010 (UTC)

I don't see how it's different, no. Goodstein's theorem is true. It doesn't matter what you assume; it's just true, period. Goodstein's theorem says that there is no natural number such that, when you do this funny thing to it, the sequence continues forever, and in fact there is no such natural number.

Now, it is different in one way — it has a more complicated logical form, $\Pi _{2}^{0}$ (whoops, actually $\Pi _{3}^{0}$) rather than $\Pi _{1}^{0}$. That means that the part after there is no natural number such that is not something that's a priori checkable by an algorithm guaranteed to terminate. But that doesn't affect the meaning of the claim that it's true. --Trovatore (talk) 05:09, 16 August 2010 (UTC)

I have no real problem with this position so long as we recognize it for what it is, namely a Platonist position of faith in certain second-order or set-theoretic material. That's where I came in to begin with: the page should acknowledge that the claimed "truth" of such sentences reflects a Platonist viewpoint. Tkuvho (talk) 06:16, 16 August 2010 (UTC)

It is completely standard to state mathematical results in realist language, whether one is actually a realist or not. For some reason, some people suddenly notice this when it comes to the Goedel theorems, but it's a completely general phenomenon. --Trovatore (talk) 06:53, 16 August 2010 (UTC)

As the footnote in the article says, the theorem is provable in PRA, which is a very weak form of arithmetic. So it isn't necessary to have any faith in set theory for the incompleteness theorem. You're right that Goodstein's theorem has a much higher consistency strength. Trovatore is right that the use of the word "true" is standard in the study of formal arithmetic, and does not necessarily connote a realist position. In the case of the incompleteness theorems, the truth of the Goedel sentence is a central aspect of the theorem. — Carl (CBM · talk) 10:45, 16 August 2010 (UTC)

Let me expand on the last sentence. My point here is that it is not necessary to take a realist position on mathematics in order to use the word "true".

The philosophical theory of "mathematical realism" holds that mathematical objects have an objective existence independent of people. "Platonism" is a form of realism holding that mathematical objects exist in a non-spatiotemporal sense. There are other forms of realism that hold that mathematical objects exist in a genuinely physical sense.

Two of the more important non-realist viewpoints are:

• "Structuralism" holds that the concept of "structure" is primitive. Individual mathematical objects may or may not exist - there does not have to be any object that is "the number 1", or any specific "standard model" of arithmetic - but the "structure" of the standard model exists, and can be identified by its structural relations
• "Fictionalism" says that mathematical objects are simply fictional entities that, like characters in a novel, are forced to follow the rules of the story that they appear in.

Both structuralists and fictionalists would agree that the Goedel sentence of a consistent theory is "true" in their sense. To a structuralist, this would mean that a certain type of structure is impossible. To a fictionalist, the Goedel sentence is true in the same sense that "Harry Potter is a wizard" is true. — Carl (CBM · talk) 11:22, 16 August 2010 (UTC)

Thanks very much for these clarifications (incidentally, where would "nominalism" fit? Under "structuralism"?). I would like to address a minor point, namely the remark that Goedel's theorem is "for some reason singled out for a special treatment", i.e. elsewhere in mathematics people assume the realist position without any special note, but here for some reason they are reminded of the philosophical background. The point I would like to make is that there IS a difference. Elsewhere in mathematics, ZFC is the standard background assumption (except in logic, of course). In the context of Goedel's theorem, if we assume ZFC, don't Goedel's sentences become decidable? Tkuvho (talk) 12:39, 16 August 2010 (UTC)

See related discussion here. Tkuvho (talk) 16:07, 16 August 2010 (UTC)

Concerning the remark that "the use of the word "true" is standard in the study of formal arithmetic": are you referring to the technical literature in the foundations of arithmetic? It may very well be that part of their professional convention is to use the term in this sense. It does not necessarily mean that it should be used on this page, which is addressed to a much wider audience, to whom the term does not necessarily have the same meaning. Tkuvho (talk) 16:22, 16 August 2010 (UTC)

Note that footnote 1 on disquotationalism refers to texts by Smoryński 1977 and also Franzén. But neither of these texts is mentioned at disquotationalism! I think the context of the "truth" claim should be explained better. In particular, the introduction makes a blanket claim about "truth" without even qualifying it by a reference to disquotationalism. A number of people complained about this in 2007 and there seemed to be a consensus that something should be done about it. Tkuvho (talk) 16:28, 16 August 2010 (UTC)

Some structuralists are nominalists (rejecting abstract objects) but (apparently) not all. One key point is whether mathematical "structure" (e.g. the natural numbers) is itself an abstract object, or whether it just exists "in" phisical instantiations of the structure.

Regarding the use of the word truth, since pretty much every reference on Goedel's theorem explains why the Goedel sentence is "true", including introductory textbooks, it would be irresponsible for us to try to avoid using the term here. It's a key point about the theorem.

I haven't looked at our article on disquotationalism and I have no idea what it contains. I have previously argued that we could add an entire paragraph to this article explaining how the Goedel sentence is true, rather than a footnote. — Carl (CBM · talk) 00:48, 17 August 2010 (UTC)

I see, I was not aware of the situation with the "introductory textbooks". One has to be very determined to go against what they say, short of writing one's own :) Writing a paragraph would be helpful. Incidentally, do the textbooks describe Goodstein's theorem as "true", as well, regardless of one's hypotheses concerning infinite ordinals? Notice that the footnote currently contains three explanations of the "truth" of Goedel's sentence: the first is a link to disquotationalism, which is unconvincing because the page does not really say anything relevant; and the second, a claim that the sentence is true because it indeed "cannot be proved as it says", etc. The second explanation certainly does not clarify the situation with Goodstein. The third explanation is in terms of PRA and relies on "consistency of T". I personally would appreciate a clarification of the relation between the said consistency and existence of infinite ordinals. Tkuvho (talk) 02:11, 17 August 2010 (UTC)

I think you're confusing epistemology with semantics. Your "hypotheses concerning infinite ordinals" have nothing to do with whether Goodstein's theorem is true, but only with how much confidence you might have that it's true. The meaning of the claim that Goodstein's theorem is true (which is the same as the meaning of Goodstein's theorem itself) refers only to the natural numbers, and not to any infinite ordinals.

To agree that this meaning is well-specified, you don't have to be a realist about ordinals, or about a completed collection of all natural numbers, but only about the naturals themselves. That is, it suffices to accept that the predicate "is a natural number" is well-specified (and to accept excluded middle as self-evident, without needing to refer to a completed collection). --Trovatore (talk) 02:27, 17 August 2010 (UTC)

Thanks for your comments. As this discussion shows, the page could be more helpful to someone who confuses epistemology with semantics. The issue of the meaning of the term "truth" as it is used here should not be relegated to a footnote. Tkuvho (talk) 04:25, 17 August 2010 (UTC)

Most introductory logic books discuss the incompletness theorems; fewer discuss Goodstein's theorem. But of course the ones that do discuss Goodstein's theorem point out it's true because, well, it is. (There's not much interest in false sentences of arithmetic that are unprovable in PA.) Goodstein's theorem is not only true, it's provable in ZFC. Mathematicians choose systems such as PA and ZFC to work with because of the general recognition among mathematicians that theorems in these systems are true.

A key point point of the footnote is the references. You should read them (or at least some of them) if you haven't already. I'd recommend Franzen's book in particular, because it's well-regarded, aimed at a general audience, and has a long exposition on the truth of the Goedel sentence. — Carl (CBM · talk) 02:33, 17 August 2010 (UTC)

Thanks, I will try to look it up. Could this page perhaps merit to have a summary of that exposition on the truth? Tkuvho (talk) 04:25, 17 August 2010 (UTC)

The mathematical consensus is that "True for arithmetic" means exactly the same as "Provable in Second-order Arithmetic (including the Axiom of Choice)"

The mathematical consensus since Peano used his (and Dedekind's) Second-order Logic axioms to prove that all models of arithmetic are isomorphic has been the following:

"True for arithmetic" means exactly the same as "provable in Second-order Arithmetic"

However, "true for Second-order Arithmetic" is controversial.

Of course, using roundtripping, it is still possible to construct the Gödelian "This proposition is not provable in Second-order Arithmetic" and thus prove incompleteness of Second-order Arithmetic. However, the truthiness of this proposition is open to question.63.249.108.250 (talk) 22:53, 10 September 2010 (UTC)

The deductive systems for second-order logic are also incomplete, and so there are still true statements about the natural numbers that cannot be proved in these deductive systems. It is true that you can prove the natural numbers are categorical in second-order logic, but this does not mean that you can prove every sentence about the natural numbers or its negation. So "true" does not mean "provable in second order arithmetic." — Carl (CBM · talk) 00:06, 11 September 2010 (UTC)

The consensus is that provablity in Second-order Arithmetic defines "truth for arithmetic" because there is no other acceptable definition. Therefore, there are propostions that are neither "true for arithmetic" nor "false for arithmetic" such as the following: 'This proposition is not provable in Second-order Arithmetic'.

Nevertheless the following proposition is provable in Second-order Arithmetic:

'This proposition is not provable in Second-order Arithmetic' ∨ ¬'This proposition is not provable in Second-order Arithmetic'

Consequently the above proposition is "true for arithmetic".

63.249.108.250 (talk) 20:53, 12 September 2010 (UTC)

Given that there is no computable complete deductive system for second-order arithmetic, it's unclear to me what "provable in second-order arithmetic" is supposed to mean. It's true that truth in arithmetic is the same as logical implication in second-order arithmetic (in the sense of full second-order logic). I guess if you like, you could take "provable" to mean "logically implied" in this context. --Trovatore (talk) 21:02, 12 September 2010 (UTC)

Provable in the theory of Second-order Arithmetic is perfectly well defined by its axioms and rules of inference. However, the theory is incomplete because the following propostion is neither provable nor disprovable:

This proposition is not provable in Second-order Arithmetic.

It is important not to confuse provablity with "logical implication." 63.112.0.74 (talk) 21:43, 12 September 2010 (UTC)

Well, but here I think you're talking about second-order arithmetic using first-order logic. There is AFAIK no standard set of "rules of inference" for second-order logic. --Trovatore (talk) 21:46, 12 September 2010 (UTC)

There are systems of rules of inference for second-order logic, but they are not complete for full second-order semantics. Indeed no effective system of inference rules for second-order logic can ever be complete for those semantics. Which means there are plenty of things that are true in the standard model but not provable in these deductive systems. So truth in the standard model is not the same as provability in second-order arithmetic. — Carl (CBM · talk) 22:48, 12 September 2010 (UTC)

There are such systems, but no canonical such system if I'm not mistaken. There would be a hierarchy of stronger and stronger ones. There could be said to be a complete one in some sense, though as you say it would not be effective.

It's not clear that a deductive system has to be effective in order for its inferences to be called "proofs". For example Ω-logic is said to have "proofs", which are universally Baire sets of reals. Is there some similar sort of "certificate" that captures full second-order logical implication? It's a vague question of course, but not an uninteresting one. --Trovatore (talk) 22:53, 12 September 2010 (UTC)

Our edits overlapped. The system I described below is "canonical enough" that is has a name ($Z_{2}$), while stronger systems are defined as $Z_{2}$ plus some new axiom or axiom scheme. Much like Peano arithmetic for first-order arithmetic. — Carl (CBM · talk) 23:21, 12 September 2010 (UTC)

If I understand you correctly, you've described only axioms, not rules of inference, is that correct? In that case this seems to me to be a pure first-order theory; I don't see that second-order logic is being used at all. --Trovatore (talk) 02:03, 13 September 2010 (UTC)

Yes, that's the crux of the matter. Any set of true axioms for second-order logic works equally well for first-order-ish Henkin semantics and for full second order semantics. The same is true for rules of inference, although if you use infinitary rules you'll need to replace first-order logic with some infinitary logic. Full second-order semantics are so strong because they manage to transcend syntactic description by directly referring to truth in the metatheory. As I pointed out below, either $\Sigma _{1}^{1}$ determinacy or its negation is a logical consequence of "there are infinitely many elements in the domain" in full second order semantics. We know from experience in set theory the sort of consequences that would have for any concrete set of deduction rules. — Carl (CBM · talk) 02:24, 13 September 2010 (UTC)

I don't think it's quite true that all rules of inference work equally well for both sorts of semantics. You could have a rule of inference that allows A |- B even though there are Henkin models for A^¬B. As long as those Henkin models necessarily failed to have the full powerset of the collection of type-0 objects, that would still be a sound rule. --Trovatore (talk) 03:38, 13 September 2010 (UTC)

Yes - for rules of inference, you have to narrow the Henkin semantics to only consider the models for which the inference rule is sound. But this is true in first order logic too. For example, if you stated induction in Peano arithmetic as a rule of inference $\Phi (0),\forall n[\Phi (n)\to \Psi (S(n))]\vdash \forall n\Phi (n)$ instead of an axiom, then you'd have to narrow the semantics, or else the rule would be unsound. And whenever you add a rule of inference you have to check whether things like the deduction theorem still hold. But the incompleteness theorems at least apply to theories with additional finitary rules of inference, so you won't manage to get a complete axiomatization of arithmetic by adding those. — Carl (CBM · talk) 04:09, 13 September 2010 (UTC)

OK, I think we're going a bit far afield here. The question was about a canonical deductive system for second-order arithmetic with second-order logic. The one you mention is what I would call second-order arithmetic with first-order logic. --Trovatore (talk) 04:14, 13 September 2010 (UTC)

PS. If someone asked me what the standard axiom system for second-order arithmetic was, I would say: the basic axioms PA⁻, the full induction scheme, and the full comprehension scheme. But there are stronger axioms, for example certain dependent choice principles, which are not provable in this system although they are true (and provably true in ZFC). Also, there are many set-theoretic facts that can be expressed in second-order arithmetic but which cannot even be proved in ZFC, for example determinacy for ${\boldsymbol {\Sigma }}_{1}^{1}$ sets. — Carl (CBM · talk) 22:59, 12 September 2010 (UTC)

You omitted the Axiom of Choice, which was implicitly included by Peano. Given that the Axiom of Choice is included, what truths of arithmetic are not provable in Second-order Arithmetic? 67.169.144.115 (talk) 12:19, 13 September 2010 (UTC)

You should look at the section on choice schemes in Simpson's Subsystems of second-order arithmetic for a detailed treatment of the strengths of various ways of stating the axiom of choice in that setting.

However, no amount of axiom of choice is going to prove either $\Sigma _{1}^{1}$ determinacy or its negation (and one or the other is a truth of second order arithmetic). This principle is equivalent in set theory to the existence of 0^#, which is independent of ZFC. All the choice principles for arithmetic are provable in ZFC. — Carl (CBM · talk) 12:55, 13 September 2010 (UTC)

It is questionable whether $\Sigma _{1}^{1}$ determinacy or the existence of 0^# should be called "truths of arithmetic" 67.169.144.115 (talk) 18:53, 13 September 2010 (UTC)

Well, $\Sigma _{1}^{1}$ determinacy can be expressed by a single formula Φ of second-order arithmetic. Either that formula is true in the standard model, or its negation is true there. So one of them is a truth of second-order arithmetic.

If you don't like the fact that that sentence is second-order, you can take a $\Pi _{1}^{0}$ Goedel sentence for your favorite consistent axiomatization of second-order arithmetic. This will also be true in the standard model but not provable in your axiomatization. — Carl (CBM · talk) 19:12, 13 September 2010 (UTC)

So the question is whether the following proposition is "a truth of arithmetic":

This proposition is not provable in Second-order Arithmetic.

67.169.144.115 (talk) 20:49, 13 September 2010 (UTC)

Yes, assuming that you pick any effective, consistent set of axioms and deduction rules, the Goedel sentence of the theory will not be provable, and therefore will be true in the standard model. That's kinda the point here. — Carl (CBM · talk) 22:16, 13 September 2010 (UTC)

So thesis is that the following propostion is a "truth of arithmetic":

Second-order Arithmetic is consistent. ⇒ This proposition is not provable in Second-order Arithmetic.

But is the above proposition a theorem of Second-order Arithmetic?71.198.220.76 (talk) 00:09, 14 September 2010 (UTC)

Sure. Assuming, as I've been emphasizing, that you're using "second-order arithmetic" to mean "second-order arithmetic with first-order logic". You (assuming it was you) made the start of the discussion a bit confusing by talking about second-order logic.

(Actually, the proposition listed is a theorem of much weaker theories; some weak fragment of first-order Peano arithmetic will suffice.) --Trovatore (talk) 00:15, 14 September 2010 (UTC)

Oh, wait a minute. Of course it depends on what you mean by "this proposition". "This proposition" refers to the bit that comes after the ⇒, of course, not to the whole thing. --Trovatore (talk) 00:18, 14 September 2010 (UTC)

And another wait-a-minute (I have to specify all the caveats with you). Of course the part starting with "this proposition" is to be interpreted as shorthand for what you get out of the standard process of translating such things into statements of arithmetic. It is not to be understood as incorporating any explicit self-reference into the language or anything like that. --Trovatore (talk) 00:21, 14 September 2010 (UTC)

So it looks like the mathematical consensus really is that "True for arithmetic" means exactly the same as "Provable in Second-order Arithmetic (including the Axiom of Choice)." 67.169.144.115 (talk) 07:27, 14 September 2010 (UTC)

No, it doesn't look like that at all. At least if you mean second-order arithmetic using first-order logic. --Trovatore (talk) 07:42, 14 September 2010 (UTC)

Of course, "second-order arithmetic using first-order logic" is complete nonsense.207.47.36.238 (talk) 16:47, 17 September 2010 (UTC)

You're aware that second-order arithmetic is usually studied as a theory in first-order logic, right? The point here is that changing to semantics does not change the deductive system, and so even though the theory become categorical in those semantics, this does not mean the theory is complete. — Carl (CBM · talk) 17:54, 17 September 2010 (UTC)

More accurate would be "second-order axioms for arithmetic using classical logic."207.47.36.238 (talk) 19:02, 17 September 2010 (UTC)

Second-order logic is a classical logic, it's just stronger than first-order logic. ω-logic is another classical logic. — Carl (CBM · talk) 19:49, 17 September 2010 (UTC)

What can be proved about arithmetic using ω-order axioms that be proved using 2nd-order axioms?207.47.36.238 (talk) 20:13, 17 September 2010 (UTC)

As has frequently been the case in these discussions, your refusal to use standard language makes it difficult to respond directly, because it isn't clear what you mean.

An example of a true proposition of arithmetic that cannot be proved in the theory called second-order arithmetic (understood as a theory of first-order logic) is Con(ZFC). --Trovatore (talk) 20:18, 17 September 2010 (UTC)

I was referring to ω-logic, a classical infinitary logic. — Carl (CBM · talk) 23:28, 17 September 2010 (UTC)

First-order logic is unsatisfactory for ordinary mathematics because of the Löwenheim–Skolem theorem. So mathematicians moved on to the superstructure of sets over N (the natural numbers) where they successfully characterized N, the real numbers and everything else they needed. Then Gödelization came along with roundtripping which was used to proved.

Ordinary Mathematics is consistent. ⇒ This proposition is not provable in Ordinary Mathematics.

However, if the above proposition is a theorem of Ordinary Mathematics, then Ordinary Mathematics is inconsistent. The obvious way out of this problem is to reject roundtripping (or to restrict it as is done in Common sense for concurrency and inconsistency tolerance using Direct Logic and the Actor Model).216.131.196.2 (talk) 20:53, 20 September 2010 (UTC)

That is not even wrong. You've redefined the terms of mathematical logic to make your point. — Arthur Rubin (talk) 00:56, 21 September 2010 (UTC)

They probably meant say something like the following:

Let G be the proposition This proposition is not provable in Ordinary Mathematics.

Using roundtripping, Gödel informally proved the following propositions in Ordinary Mathematics (where ⊢ means provable in Ordinary Mathematics):

1. Ordinary Mathematics is consistent. ⇒ ⊬G
2. Ordinary Mathematics is consistent. ⇒ ⊬¬G

209.133.114.32 (talk) 18:57, 21 September 2010 (UTC)

"Ordinary mathematics" is not a formal theory of any sort, and therefore the techniques used to form Goedel sentences do not apply to it. The lesson to take away from the Goedel theorems is that mathematics transcends complete formalization. --Trovatore (talk) 19:39, 21 September 2010 (UTC)

If there were' such a thing as "ordinary mathematics", it wouldn't include "roundtripping". — Arthur Rubin (talk) 19:57, 21 September 2010 (UTC)

I didn't say there's no such thing. Of course there's such a thing. But it can't be completely formalized. --Trovatore (talk) 20:04, 21 September 2010 (UTC)

It still doesn't include "roundtripping", except according to Hewitt. 20:18, 21 September 2010 (UTC)

According to Universe of Mathematics, Ordinary Mathematics is set theory with the domain SN (the superstructure over N) together with induction and choice.171.66.80.49 (talk) 23:16, 21 September 2010 (UTC)

Our article doesn't say that. The domain is SN, but it doesn't specify the axioms. Is it Z+induction/replacemnt+AC? Z+Induction+AC?

• Induction/replacement means that if φ is a formula such that $(\forall X)(\exists !Y)\phi (X,Y)$ then there is a function object G on N such that $G(0)=\emptyset$ and $(\forall n\in \mathbb {N} )\phi (G(n),G(n+1))$.)
• Induction would be $\phi (0)\wedge (\forall n\in \mathbb {N} )(\phi (n)\rightarrow \phi (n+1))\rightarrow (\forall n\in \mathbb {N} )\phi (n)$ — Arthur Rubin (talk) 05:01, 22 September 2010 (UTC)

Truth in arithmetic = model-theoretic truth

In his 1980 paper, “The Mathematics of Non-Monotonic Reasoning. Journal of Artificial Intelligence”, Martin Davis mentions in passing that the simple recursive definitions of addition and multiplication, which are embedded in all axiomatizations of arithmetic, have an unique minimal model, which is the standard model of arithmetic.

These definitions do not include induction. But induction is true in this minimal model, where truth is defined in the standard Tarskian manner.

The source of incompleteness is the fact that an universally quantified sentence is true in the standard model iff its infinitely many instances are true in the standard model. The incompleteness theorem shows that there exist universally quantified sentences whose truth cannot be proved by finite means.86.169.162.255 (talk) 07:36, 11 September 2010 (UTC)

While I'm not sure I'd put it exactly that way, you haven't said anything obviously silly. But what's your point exactly? Is there some change you're proposing to the article on the basis of the above? Talk pages are not intended for general discussion of the article's subject matter. In case you are new around here, you might want to take a look at WP:TALK, which explains in more detail. --Trovatore (talk) 07:43, 11 September 2010 (UTC)

Inconsistency Robustness

I was unable to find the book

Carl Hewitt (2014). "Inconsistency robustness in foundations: Mathematics self proves its own consistency and other matters". Inconsistency Robustness. College Publications. ISBN 978-1-84890-159-9. {{cite book}}: Unknown parameter |editors= ignored (|editor= suggested) (help)

at the College Publication site http://collegepublications.co.uk/about/ or anywhere else. John Woods is a widely published author many of whose articles are reviewed at MathSciNet. It should be possible to mention this at the talkpage. Tkuvho (talk) 10:08, 15 April 2015 (UTC)

The book is available on Amazon:

Carl Hewitt and John Woods (editors), 2015. Inconsistency Robustness College Publications. ISBN 978-1848901599.

The book has several articles that bear on the incompleteness theorems. — Preceding unsigned comment added by 50.242.100.195 (talk) 23:38, 17 August 2015 (UTC)

The editors, and possibly the authors, are not experts in the field of this article. That being said, it might be noted, in this article, as a notable Comment on the theorems, if not an expert opinion. That is, provided that Hewitt doesn't make the edit, per his ban from Wikipedia. — Arthur Rubin (talk) 02:26, 18 August 2015 (UTC)

Arthur Rubin has been feuding with Professor Hewitt and his colleagues at Stanford for years. — Preceding unsigned comment added by 50.247.81.99 (talk) 04:55, 2 September 2015 (UTC)

In the book, professors Hewitt and Woods explain how using types, Computer Scientists developed a completely different take on Goedel's writings. — Preceding unsigned comment added by 4.15.127.211 (talk) 22:45, 20 August 2015 (UTC)

Please add the following references to the article from the book Carl Hewitt and John Woods (editors), 2015. Inconsistency Robustness Studies in Logic. Vol. 52. College Publications. ISBN 978-1848901599 :

• "Inconsistency robustness in foundations: Mathematics self proves its own consistency and other matters" Carl Hewitt
• "Inconsistency: Its present impacts and future prospects" John Woods

I had argued before for a sentence referencing Hewitt's work. I don't know when that disappeared from the article, but it should be included somewhere. Re the book, if there is any specific paper in it that is worth referencing here, that would be great, but it makes no sense to cite an entire book of papers, few of which are likely to be about Goedel's incompleteness theorem. The book itself is a collection of papers, not a monograph. It is also worth remembering that this article is not about incompleteness and inconsistency in general, and is not a catch-all for such topics. This article is about Goedel's theorems in particular. — Carl (CBM · talk) 11:06, 18 August 2015 (UTC)

Looks like amateurism versus the professors arguing over Goedel's incompleteness theorem with the amateurs denying the existence of the professors' book. 50.242.100.195 (talk) 00:05, 4 September 2015 (UTC)

Gödel's writings on incompleteness play a major role in the following articles:

• "Inconsistency robustness in foundations: Mathematics self proves its own consistency and other matters" Carl Hewitt
• "Inconsistency: Its present impacts and future prospects" John Woods

However, Professor Hewitt's article is from the viewpoint of Computer Science and is completely at variance with the common view that Gödel proved that Mathematics does not prove its own consistency. — Preceding unsigned comment added by 50.247.81.99 (talk) 05:15, 19 August 2015 (UTC)

I have resolved an edit request added by banned user Carl Hewitt. — Carl (CBM · talk) 02:03, 4 September 2015 (UTC)

Edit request

Please add the following references to the article:

Carl Hewitt (2014). "Inconsistency robustness in foundations: Mathematics self proves its own consistency and other matters". Inconsistency Robustness. College Publications. ISBN 978-1-84890-159-9. {{cite book}}: Unknown parameter |editors= ignored (|editor= suggested) (help)

John Woods (2014). "Inconsistency: Its present impacts and future prospects". Inconsistency Robustness. College Publications. ISBN 978-1-84890-159-9. {{cite book}}: Unknown parameter |editors= ignored (|editor= suggested) (help)

Thomas Foster's SEP essay (2006), "Quine's New Foundations", shows that it is wise to careful about statements on consistency in mathematics. "the status of NF does seem to be the oldest outstanding consistency question", where NF is Quine's New Foundations. --Ancheta Wis   (talk | contribs) 03:59, 2 September 2015 (UTC)

I have resolved an edit request added by banned user Carl Hewitt, editing as IP 107.85.105.128 (the edit history of the page is hard to follow). — Carl (CBM · talk) 02:06, 4 September 2015 (UTC)