W J Eckerslyke

Richard's Principle

Richard's Principle asserts that no mathematical “proof” is valid which postulates the existence of a collection of all entities of a specific kind and then purports to define on the basis of that collection a new entity which must by definition be part of that collection but clearly is not.

Richard's Paradox (updated version)

Consider that set S of real numbers between 0 and 1 which can be defined in English. Require only that each definition be a finite string of characters from a finite alphabet. It is then trivially easy to represent each definition as an integer. Therefore S is clearly countable. It does not matter that each definable real has an infinite number of possible definitions, nor that most integers do not correspond to valid definitions.

Take it for granted that each of the numbers in S has a decimal expansion. Now define a new real number between 0 and 1 such that the Nth digit after the decimal point in its decimal expansion has the value:

• 4 if the integer N does not correspond to a valid definition
• 5 if the Nth digit in the decimal expansion of the real number defined by N is not 5
• 6 if the Nth digit in the decimal expansion of the real number defined by N is 5.

Clearly this new number differs from each of the numbers previously defined, and so is not in S. But it is also a real number defined in English, and therefore should by definition be in S.

Analysis

There is clearly an unacceptable contradiction. What has gone wrong?

The definition used in the above “paradox” of the new real number clearly contains a finite string of characters, and hence clearly maps readily into an integer K as asserted. And that integer K corresponds with the definition of no other real number. So far, so good.

But if that definition does indeed define a real number Q then it states clearly that Q differs from itself in the Kth decimal position. Which is blatant and unarguable nonsense. Such definitions cannot be allowed.

Conclusion

The conclusion has to be that circular logic of that kind is never right. The general rule (Richard's Principle) is that it is never valid to postulate the existence of a collection of all entities of a specific kind and then to define on the basis of that collection a new entity which must by definition be part of that collection but clearly is not.

Implications

This implies, in particular, that all mathematical reasoning based on any form of the “Cantor diagonal argument”, of which the paradox above is a clear instance, is inherently and irredeemably invalid.

On the other hand

It may be argued that until it has been defined the number Q cannot be taken to be in the set S. Obviously this allows an infinite sequence of additions to S, each new Q being defined on the basis of S and all preceding Qs. But there is no contradiction in this quite reasonable position, which seems to completely resolve (i.e. avoid) the paradox.

On the other hand, this approach would seem to be based on the view that the “all” used in the definition of S can only mean “all defined so far”. Which again seems quite reasonable. But if we deny the right to use “all” to mean “absolutely all”, including things not yet defined, things definable only on the basis of the “all”, and even things constitutionally undefinable, then all arguments which rely on that right also fail. And that possibly includes rather more than just things based on “diagonalisation”.

In other word this alternative resolution of the paradox strengthens rather than weakens the conclusions.

Wanted

This almost certainly needs some cleaning up before being fit for human consumption. But it all seems so obvious that I am sure it is not new. Can some kind reader please give me enough “sources” for it to warrant posting it to WikiPedia. Otherwise I fear it may have to languish in Wikinfo.

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ZFC

A critique of and/or some questions about ZFC. Can some kind reader please help me understand what I have misunderstood.

Chickens and eggs

ZFC “is the standard form of axiomatic set theory and as such is the most common foundation of mathematics”. Yet its “universe of discourse” is claimed to comprise “all mathematical objects”. How can this be? How can ZFC claim to be based on objects which are supposedly based on ZFC? At the very least some clarification is needed.

What about non-mathematical objects?

Why should a set theory be confined to discussion of mathematical objects only? Surely a set theory, even an axiomatic one, could usefully be applied to sets of books, laws, or foxes?

Individuals

Is it really necessary, or even desirable, to assume that “all individuals ... are sets”? This assumption certainly offers some potential simplification, but it also causes a serious problem which ZFC does not seem to address. The question is whether for all individuals x it is true that {x}=x. If this is not true then all individuals have the same members, i.e. none, and are thus indistinguishable from each other and from the empty set. But if it is true then neither the Axiom of Regularity nor the Axiom of Infinity bears examination!

Domain of discourse

Are there any individuals (other than the empty set)? “Many authors require a nonempty domain of discourse as part of the semantics of the first-order logic in which ZFC is formalized.” I cannot see why this should be optional. If there is no given 'domain of discourse' then the whole thing just enables the construction of confections of the empty set.

Active sets?

“The axioms of ZFC govern how sets behave and interact.” Strange wording! Sets are essentially passive objects, and do not behave or interact at all. It seems to me the axioms do no more than specify what sets are deemed to “exist”.

Consistency

There seems to be some doubt that ZFC is in fact consistent. Doubtless there is no actual proof of its consistency, but at first glance there really does not seem to be enough in it to allow the possibility of inconsistency. However, its blithe use of reference to “any property” (Axiom 3) and to “any formula” (Axiom 6), without any attempt to limit the scope of the “any”, clearly leaves infinite scope for things to go wrong, and consistency is therefore likely to depend on just what scope is chosen for properties and formulas. It also seems to be left open whether or not “the background logic includes equality” (Axiom 3). The conclusion might perhaps be that ZFC may or may not have any consistent realisations, but it is very likely to have some inconsistent ones.

Infinite sets

“The minimal set X satisfying the axiom of infinity is the von Neumann ordinal ω.” Indeed, it would seem that ω is the only set whose existence this axiom guarantees, even if there is a non-empty domain of discourse. So perhaps this should be called the Axiom of ω?

Power set

There is what seems to be a dangerous imprecision in the presentation of the axiom of power set. The text says “The power set of x is the class whose members are every possible subset of x.” This “possible” may be interpreted as meaning “conceivable”, whatever that means. But that would be wrong. In this context “possible” can only mean “possibly existing in accordance with these axioms”. This is perhaps clearer in the formal definition: $\forall x\exists y\forall z[z\subseteq x\Rightarrow z\in y],$ in which the scope of the $\forall$ is more obviously limited to sets allowed under ZFC.

Definability

“Much research has sought to characterize the definability (or lack thereof) of certain sets whose existence AC asserts.” Is that a good use of effort? Would it not be more sensible instead to develop a set theory based on a set definition capability, and to accept the existence of all sets which can be defined by it and no others?

Another opinion

You raise a number of doubts, but I prefer to discuss first only one of them, the most principal one. Boris Tsirelson (talk) 16:30, 26 January 2009 (UTC)

Thanks greatly for your most eloquent responses (below). I do hope that you have some other use for this splendid prose (or should I say poetry?), and that it is not wasted on just me. It is highly persuasive, and I shall definitely have to think again about my aversion to ineffable entities, and perhaps refrain from further comment until I have found out more about the irreplaceable properties you ascribe to them.

However, the main thrust of my comments was not against ineffable entities as such, nor even against their claimed uncountability, but rather against the validity of alleged proofs of their uncountability. I am happy to accept that such things are probably in fact always uncountable, and indeed it would feel very strange if somebody were able to produce a set of ineffable entities which were demonstrably countable. Do you know of any such?

As for the ZFC stuff, I am minded to transfer it to the Discussion page of ZFC. But not today.

W J Eckerslyke (talk) 16:45, 28 January 2009 (UTC)

Thanks for the compliments. When I want to clam credit, I write to arXiv or a mathematical journal. Here in Wikipedia I express myself, hopefully also for a benefit of others. I am really glad if it is interesting to you. Feel free to use my text for any (non-commercial) goal, as any other text in Wikipedia.

Against the validity of uncountability proofs? I am puzzled. Surely you understand now that (Cantor's) diagonal argument is designed for a man that accepts the continuum of (ineffable or not) points. Then, why against? Boris Tsirelson (talk) 19:53, 28 January 2009 (UTC)

I am "against" (Cantor's) diagonal argument because I believe it to be demonstrably invalid. I do not think that for whom it is designed affects its logical validity. I think I shall have to make my comments directly to the Cantor's diagonal argument discussion page. W J Eckerslyke (talk) 05:58, 31 January 2009 (UTC)

So, we fail to understand each other. I still do not see any problem with the diagonal argument; you still prefer incomplete counterparts of Euclidean spaces and other continua. Well... I did my best. Now join the Talk:Cantor's diagonal argument/Arguments page. Good luck. Boris Tsirelson (talk) 18:18, 31 January 2009 (UTC)

About "a set of ineffable entities which were demonstrably countable". I have no exact definition for "ineffable", but here is some thought. A point on the Euclidean plane, is it "effable"? (Sorry for my way of using English, I do not know how to say it better.) A spot of chalk on a blackboard is not a point on the Euclidean plane but only a hint toward it. The abstract notion of Euclidean plane does not stipulate coordinates, the origin etc.; all its points have equal rights... I cannot single out one of them; in this sense each of them in ineffable. (Do you agree?) Of course, they are a continuum; however, we can consider in the same spirit a plane over the field of rational (rather than real) numbers; it is countable. Or even a finite geometry! Boris Tsirelson (talk) 07:25, 30 January 2009 (UTC)

I shall have to think more on this, and possibly be less carefree in my use of "ineffable". But essentially I see points in Euclidean geometry as being relative, in the sense that the whole thing is independent of all shifts, and rotations and changes of scale. Clearly Euclid depends on continuity, but only in the sense that between any two points there is another, but that does not commit us to anything beyond rational numbers. However, given only Pythagoras we clearly could not base our geometry on just rational distances between points. On the other hand, there seems no advantage in allowing Euclidean distances to be undefinable reals. W J Eckerslyke (talk) 05:58, 31 January 2009 (UTC)

Alternatively, your words can be interpreted as the following conjecture: "every countable definable set has a definable enumeration". I do not know, whether it is true or not. This is a good question to experts in set theory and logic; I am only a probabilist. Boris Tsirelson (talk) 09:18, 30 January 2009 (UTC)

Philosopher, mathematician, physicist

Often, a mathematician finds arguments of a physicist to be very non-rigorous, but the physicist is not bothered. For him, a mathematical proof is not decisive anyway; rather, a physical experiment is decisive. The mathematician says: you physicist prefer usefulness to reliability! I guess that, similarly, a philosopher often finds arguments of a mathematician to be very unconvincing, but the mathematician is not bothered. For him, a formal proof is decisive. The philosopher says: you mathematician prefer usefulness to reliability!

I look at it somewhat differently. The mathematician develops formal structures. In these proof is indeed decisive, subject of course to it being found acceptable to other mathematicians. These structures do not have to have any intended, or even possible, application to anything real. The physicist (for example) applies mathematical structures to create models of reality. For him the physical experiment is indeed decisive, but only in determining the appropriateness of his model. But experiment can never prove it right, and it never is. In particular, there is always a limit on the scope of is applicability. Newtonian physics is still right enough, but not if things get too big or too small or too fast. And the application of integer arithmetic (re your comment below) is similarly limited, and works only with objects which are stable enough, identifiable/separable enough, and not moving about too quickly. W J Eckerslyke (talk) 05:58, 31 January 2009 (UTC)

PS It is possibly only a probabalist who could think that the arguments of a philosopher might be more rigorous than those of a mathematician. :-) W J Eckerslyke (talk) 09:36, 31 January 2009 (UTC)

The risky mathematics

Arithmetics of natural numbers from 1 to 1000 is very reliable, and sometimes useful (say, for a trader). It is more useful and less boring to consider all natural numbers, but it is more risky. We are not sure that arithmetics is consistent. Also, axiom of induction is challenged by continuum fallacy. Large numbers, such as 10^1000, probably have no interpretation in the physical universe. Thus, arithmetics is a fiction. But quite useful.

Geometry of Euclid, with its zero-size points, is also a risky fiction, but very useful and elegant. Compare it with a well-known joke: the main theorem of applied geometry states that any three points lie on a straight line, provided that the points are thick enough...

Theory of finite sets is rather reliable, and surely useful (especially in combinatorics and computer science). However, we take the higher risk by assuming existence of infinite sets. Many objections were voiced a century ago: infinity should be treated as potential only, not as actual!

Still more exciting but risky, we assume existence of the set of ALL subsets of a given set (say, of natural numbers), no matter what means are used in constructing these subsets. You may wonder, why it did not yet lead to a contradiction. This is an important fact: it did not. For now, of course.

Continuum is useful

We like to describe points of a geometric continuum by coordinates, – real numbers; and the real numbers – by digits, say, binary digits. Sequences of binary digits correspond evidently to subsets of the naturals. Thus, the geometric continuum is mapped to the set-theoretic continuum.

Some real numbers are rational, others are irrational. However, a geometer does not want to classify the points of the geometric continuum into "first class citizens" and "second class citizens". Your approach leads inevitably to complicated hierarchies of real numbers according to degrees of (non)constructivity. ("Obviously this allows an infinite sequence of additions to S, each new Q being defined on the basis of S and all preceding Qs"... — only a tip of the iceberg.) These are investigated, indeed. However, they are of no interest to the geometer. For him it is much more natural to assume that all points of the continuum exist irrespective of any human effort/ability to single them out by individual definitions/constructions. They exist just in space! Yes, it is an exaggeration; they exist in Euclidean space, not in our physical space. Still, this is the idealization used. Otherwise geometry and analysis will turn into a nightmare. This was tried by several scools of intuitionists/constructivists.

The real line, as we know it today, is connected and locally compact. If you delete even a single point, these nice and useful properties disappear. And of course, a dense countable set never has such properties.

Being a probabilist I appreciate very much the idea of a point chosen at random from the interval [0,1], according to the uniform distribution. For now, this is well-defined. Try it for rational numbers only. You must give some probability to 1/2, some probability to 1/3, to 2/3, to 1/4, and so on, according to your favorite series that converges to 1. But you never get anything that may be called the uniform distribution! The same trouble persists for any other countable set instead of the continuum.

A quote

A philosopher once said, "It is necessary for the very existence of science that the same conditions always produce the same result." Well, they don't! ... What is necessary for 'the very existence of science,' and what the characteristics of nature are, are not to be determined by pompous preconditions, they are determined always by the material with which we work, by nature herself. We look, and we see what we find, and we cannot say ahead of time successfully what it is going to look like. ... It is necessary for the very existence of science that minds exist which do not allow that nature must satisfy some preconceived conditions.

-- Richard Feynman, The Character of Physical Law.

Vacuous sets

The empty set is a vacuous set. All other vacuous sets are sets which contain only vacuous sets. For example, in most set theories {{},{{},{{}}},{{{}},{{{}},{{{}}}}}} would be an allowable vacuous set.

Vacuous sets are of interest because they are the only sets whose existence is guranteed in ZFC. And, since ZFC is widely accepted as the foundation of mathematics, mathematics may be regarded as being essentially the study of vacuous sets.

An established term for these is pure set. — Carl (CBM · talk) 16:47, 3 February 2009 (UTC)

Thanks greatly for that. A lovely name for it. Much better than "hereditary set", and, being somewhat euphemistic, distinctly less contentious than "vacuous set". I shall use that term in future. W J Eckerslyke (talk) 10:31, 4 February 2009 (UTC)

But "vacuous set" sounds as if it means the same thing as "empty set". Michael Hardy (talk) 23:13, 4 February 2009 (UTC)

It does a bit, doesn't it. But I interpreted it to mean "lacking in substance", being free of all urelements. OTOH, "pure" set sounds as if it means free from corruption! Which is what urelements would be regarded as, I guess. W J Eckerslyke (talk) 10:06, 5 February 2009 (UTC)

I have always read pure as "containing nothing else", as in "pure sodium". — Carl (CBM · talk) 18:06, 5 February 2009 (UTC)

Richard's principle

Someone has proposed deleting Richard's principle on the grounds that it is "original research", and thus forbidden by WP:OR. I can see how someone reading your "Sources" section could get that impression from what you wrote there. Generally the pronoun "I" is not appropriate in such a context in a Wikipedia article, since it is always contemplated that such an article will probably become a joint work—nobody "owns" it. Michael Hardy (talk) 23:11, 4 February 2009 (UTC)

Thanks. Point taken. Hardly original! So one reference now included. W J Eckerslyke (talk) 09:59, 5 February 2009 (UTC)

I haven't thought through the possible connections between the topic of that article and that of impredicativity, but they seem somewhat similar. Michael Hardy (talk) 23:11, 4 February 2009 (UTC)

Similar, but not identical, I feel. Impredicativity is defining a set in terms of itself, whereas Richard's principle rules out defining an entity as a member of a set it must belong to but cannot, which ultimately means it must differ from itself. That would be not so much a circular definition as a self-contradictory one.

OTOH, the second form of impredicativity described has to be wrong. Whereas defining a set in terms of itself is clearly dangerous, defining an entity in terms of a set it belongs to is not. Indeed, if it were the ZFC Axiom of specification would have to be declared illegitimate! W J Eckerslyke (talk) 09:59, 5 February 2009 (UTC)

An impredicative definition defines a new set by reference to a totality of sets that includes the new set being defined. It need not literally define the set in terms of itself. Thus, for example, any definition of a real number that includes quantification over the set of all real numbers is impredicative. — Carl (CBM · talk) 18:09, 5 February 2009 (UTC)

In which case I have no problem with impredicative definitions provided they include an "if any": you always have to allow for the possibility that nothing in the given set satisfies the definition of the "new" set. If I define x to be the smallest even number which is greater than 2 and less than 4 you will think me a bit silly. But if I define x to be a real number which differs (e.g. in one decimal position) from all real numbers, including presumably itself, you will apparently raise no objection. I am puzzled. W J Eckerslyke (talk) 12:34, 7 February 2009 (UTC) modified W J Eckerslyke (talk) 02:08, 8 February 2009 (UTC)

I read through Richard's letter and he does not directly advance any principle. I have therefore proposed on the talk page of that article that it should be merged with the article on Richard's paradox. — Carl (CBM · talk) 18:09, 5 February 2009 (UTC)

I have proposed that it be deleted, on the grounds that changes have rendered it useless. But I have no problem if you wish to salvage some of it for use in the article on Richard's paradox. W J Eckerslyke (talk) 12:34, 7 February 2009 (UTC)

1. Over time I did come to think that that issue you were raising there was about predicativity. There has been a lot of interest in predicative mathematics, actually, and it's an important level in the hierarchy of foundation systems for mathematics.

Even in predicative systems, it is possible to prove the following:

For every sequence of real numbers <r_n> there is a real number r such that r is different from r_n for all n.

Proof. Given the sequence, for each i let the ith decimal place of r be 2 if the ith decimal place of r_i is not 2. Let the ith decimal place of r be 3 otherwise. Then verify that r is different from each r_i.

As you can see, the definition of r here is not circular in any way. In involves no quantifiers over real numbers at all.

I am happy with that. It avoids the self-contradictary definition which (it still seems to me) invalidates the traditional argument. But it does not achieve the desired result, I think, because Richard's "paradox" still seems to show why the fact that you can add one to any sequence of real numbers does not necessarily prove that the real numbers are uncountable. W J Eckerslyke (talk) 05:17, 17 February 2009 (UTC)

I have not yet seen the Hodges paper you recommended, which I assume is the one entitled "An Editor Recalls Some Hopeless Papers", but I have seen one which refers to it. I shall keep looking, in the hope that he includes some justification for the scorn he undoubtedly feels for ignorant amateurs. W J Eckerslyke (talk) 05:17, 17 February 2009 (UTC)

2. Regarding the Richard's principle article, I felt somewhat bad about rewriting it all, and did leave the first paragraph as you had it, but I felt that this part in particular needed to either be attributed to a specific reference, or removed:

"This implies, in particular, that all mathematical reasoning based on any form of the “Cantor diagonal argument”, of which the paradox above is a clear instance, is inherently and irredeemably invalid."

In his letter, Richard does not argue (in my reading) that the diagonalization procedure is invalid. Rather, he foreshadows the idea that there is a hierarchy of definitions, and that the definition in his paper of the new real number is not at the same level as the definitions over which the diagonalization is performed.

3. I would be happy to continue discussing this stuff with you. Your questions about ZFC were very relevant and brought out several of the subtle nuances that are often glossed over in texts aimed at general audiences. Unfortunately there is a significant part of the justificational aspect of mathematics that is passed on orally during graduate school, which can make it difficult for people who are trying to come in from the outside. I can expand on many of the things that I said, or find references that explain them in more detail, if you have access to a library with a good mathematics collection. — Carl (CBM · talk) 13:48, 8 February 2009 (UTC)

I appreciate your kind offer. Unfortunately I do not have ready access to a good mathematics library. But, as you will have noticed, I do have Internet access most of the time. W J Eckerslyke (talk) 05:17, 17 February 2009 (UTC)

I do still have two concerns about the ZFC article.

(a) I feel that an encyclopaedia should not be a mathematical textbook so much as an enlightening introduction for a lay reader. In which case the ZFC article really should emphasise, and spell out, that ZFC is not a general theory of sets but a theory of pure sets only, and that, despite this, mathematicians have found it useful in constructing, or examining, the foundations of mathematics.

(b) I remain convinced that in an axiomatic system you can have only what the axioms give you. So if they give you only the "constructible universe" then that is all you can rely on. I believe that if you wish to have a theory based on the "von Neumann universe" combined with ZFC then you should call it that, and if you wish to make it an axiomatic system then you need to add enough axioms to ZFC to guarantee the existence of the "von Neumann universe". But clearly there is something fundamental I have missed. What?

W J Eckerslyke (talk) 05:17, 17 February 2009 (UTC)

I did edit the ZFC article some days ago to say,

"It has a single primitive ontological notion, that of a hereditary well-founded set, and a single ontological assumption, namely that all individuals in the universe of discourse are such sets."

Do you think the article reads like a textbook?

About (b), it's normal for axioms to only describe some of the properties of the objects that they study. For example, in studying the arithmetic of the real numbers, I might use the axioms for a type of field; all you need to know to follow the rest of this paragraph is that the field axioms describe the addition and multiplication operations of the real numbers. Now the only numbers that can be proven to exist by these particular axioms are the rational numbers. But this does not prevent me from using the axioms to study the real numbers. In particular, if I can prove something about the real numbers using only the field axioms, I know that I have not used any other insights or properties except the ones in the axioms I chose.

ZFC is the same way. Its axioms include things like, "for any set that happens to be in the universe, the powerset of this set is also a set in the universe". The range of the "for any" quantifier is determined by the specific model of set theory under consideration, but the general principle of the axiom is valid regardless of which model is being considered. — Carl (CBM · talk) 23:24, 17 February 2009 (UTC)

AfD nomination of Richard's principle

An article that you have been involved in editing, Richard's principle, has been listed for deletion. If you are interested in the deletion discussion, please participate by adding your comments at Wikipedia:Articles for deletion/Richard's principle. Thank you. Trovatore (talk) 20:05, 5 February 2009 (UTC)