Talk:Axiom of limitation of size

Added section "Zermelo's models and the axiom of limitation of size"

I added a new section "Zermelo's models and the axiom of limitation of size" because of my personal experience with the axiom. When I first heard about the axiom, my reaction was: "Why can't there be classes of different sizes? After all, the ordinals form a 'thin' class and the universal class is 'much thicker'." When I learned of Zermelo's models, it became clear why the axiom can be true.

So I've added a model-theoretic section to complement the current article (which concentrates on the implications of the axiom and its relationship to other axioms). My approach was to write an exposition of Zermelo's work in 3 parts: The first gives the reader an overall view and emphasizes two statements Zermelo proved (which in my exposition are Theorems 1 and 2). A more interested reader may like to see why these theorems are true for a simple model — this is covered in the subsection "The model V_ω." This subsection naturally leads to the subsection "The models V_κ where κ is a strongly inaccessible cardinal." Zermelo handled the cases κ = ω and κ is a strongly inaccessible cardinal at the same time. However, Zermelo was writing a research article. I'm thinking of Wikipedia readers, and I realized that by dividing the exposition into 3 parts, each a bit harder than the previous, I'm catering to readers of different math levels and different interest levels. --RJGray (talk) 19:45, 3 February 2013 (UTC)

Rewrite

I have rewritten the lead following the guidelines in WP:LEAD and have added the sections "Formal statement" and "Implications" that use and handle much of what the old lead covered. The beginning of the lead was inspired by the French article on the axiom (see fr:Axiome de limitation de taille), which starts off by connecting the axiom to limitation of size principle and the paradoxes. I also borrowed two phrases from the French article.

The formal expression of the axiom of limitation of size has been rewritten a little. It reads better on two lines and doesn't shrink the font size of the entire article when printing it (which I found annoying). Also, by specifying that lowercase variables denote sets, "$F$ is onto" can be expressed with "$\forall y\exists x\left(x\in A\land (x,y)\in F\right)$" rather than "$\forall y{\bigl [}\exists B\left(y\in B\right)\Rightarrow \exists x\left(x\in A\land (x,y)\in F\right){\bigr ]}$" because the lowercase $y$ ranges over sets.

I also added some proofs and a new section titled "Limitation of size doctrine." I removed the "See Also" section because all its links appear in the article and WP:ALSO states: "As a general rule, the 'See also' section should not repeat links that appear in the article's body …"

I have given Hallett's arguments that the limitation of size doctrine does not justify the power set axiom. I also mention that he has analyzed the arguments given by Fraenkel and Levy, and has found what he considers flaws in their arguments. I tried to find authors who have found flaws in Hallett's arguments, but could not. After doing searches on Google, in Google Books, and at www.jstor.org, I found four who comment on Hallett's arguments and only one of them makes an (inaccurate) negative comment:

• Robert Bunn states: "The usual limitation of size arguments relating to the "motivation" or justification of the axioms … are subjected to a devastating critique [by Hallett] in the first section of chapter 5. They are circular. The power set axiom is particularly problematic. In summary form the reason is that “There is no reasonable internal criterion of 'extent' or 'size', so the limitation of size conception has to use a notion of relative size. And there is no good reason to suppose that a power set P(a) always remains small with respect to any measure not involving P(x) …" (Bunn, Robert (1988), "Michael Hallett's Cantorian Set Theory and Limitation of Size", Philosophy of Science, 55: 469.)
• Peter Clark states: "If we ask however whether the axioms of say ZFC (Zermelo-Fraenkel theory with choice) satisfy the conception of limitation of size, it is clear that as such they do not. The principle reason is the power set axiom (when combined with the axiom of infinity). … Despite the failure of ZFC to implement the limitation of size idea (or to make it a precise part of formal set theory) Hallett sees it as having an important heuristic role, for if it is assumed that the axioms of infinity, power set and union permit only legitimate sets, then it can be shown that restricted comprehension … and any number of reiterations of it cannot lead out of the domain of legitimate sets." (Clark, Peter (1986), "Cantorian Set Theory and Limitation of Size. By Michael Hallett.", Mind, 95: 527.)
• Penelope Maddy states: "Advocates of limitation of size suggest that the power set of a given set will not be large because all its members must be subsets of something small. Hallett casts some well-deserved doubt on this last form of justification for the Power Set Axiom …" (Maddy, Penelope (1988), "Believing the Axioms, I", Journal of Symbolic Logic, 53: 486–487, doi:10.2307/2274520.)
• Adam Rieger states: "Hallett also makes much (in Chapter 5) of the technical result that we have very little idea of the size of the power set of ω, arguing that this refutes ZF's claim to embody a 'limitation of size' conception. This, however, seems to depend on thinking of 'limitation of size' in the style of Russell, as 'no sets allowed that are bigger than such-and-such a cardinal'; rather, as I have been trying to convey, the point is that however big, it is P(ω) is still a set, and therefore not as large as the universe." However, as pointed out in this Wikipedia article, Hallett not only deals with cardinal size, but also considers arguments that treat size in terms of "comprehensiveness" and "extendability." Rieger does not try to justify the power set axiom in terms of limitation of size. Instead, he analyzes how the argument based on the iterative conception is used to support ZF and concludes that this argument is not convincing. Hallett argues that the iterative conception (which he says is a descendant of limitation of size) goes wrong for the same reason that the limitation of size argument goes wrong—namely, in its explanation of the power set axiom. So both Rieger and Hallett find the argument based on the related iterative conception unconvincing. (Rieger, Adam (2011), "Paradox, ZF, and the Axiom of Foundation", in Devidi, David; Hallett, Michael; Clark, Peter (eds.) (eds.), Logic, Mathematics, Philosophy, Vintage Enthusiasms: Essays in Honour of John L. Bell, Springer, pp. 172, 180–184, ISBN 9789400702134 {{citation}}: |editor-first3= has generic name (help). Also, Hallett 1984, pp. 214, 221.)

I learned some interesting math and philosophy of set theory (limitation of size doctrine) by doing research for this rewrite. I hope that readers will finding the rewrite informative and interesting. --RJGray (talk) 17:52, 13 January 2017 (UTC)

What was wrong with my original formal statement?

Hi JRSpriggs,

I kept your change, just reformatted it a bit and used x and y as is usual when talking about functions. However, I liked my original statement because it was shorter, it gets readers to appreciate the usefulness of the convention that lowercase variables range over sets, and I thought it was equivalent. Please help me understand what was wrong with my original statement. Here's what I wrote above in the Rewrite section:

The formal expression of the axiom of limitation of size has been rewritten a little. It reads better on two lines and doesn't shrink the font size of the entire article when printing it (which I found annoying). Also, by specifying that lowercase variables denote sets, "$F$ is onto" can be expressed with "$\forall y\exists x\left(x\in A\land (x,y)\in F\right)$" rather than "$\forall y{\bigl [}\exists B\left(y\in B\right)\Rightarrow \exists x\left(x\in A\land (x,y)\in F\right){\bigr ]}$" because the lowercase $y$ ranges over sets.

Thank you, --RJGray (talk) 16:41, 14 January 2017 (UTC)

We should be able to use the same version of the axiom in any theory where it applies. So it should work in (one-typed) class theories (like MK) where all variables refer to classes. In effect, your version hides the restriction of lower case variables to the class of sets by using a special language which may not be understood by everyone.

I also preferred my choice of variable names for the following reasons: "C" because it is short for "possibly-proper class" while "A" has no mnemonic value; "W" suggests "V" the class of all sets while "B" means what?; "s" for "a set in C"; and avoid using subscripts which make variable names harder to read.

JRSpriggs (talk) 22:37, 14 January 2017 (UTC)

I always enjoy conversing with you since you explain your thinking so well. I agree that the same version of the axiom should work in all applicable theories. (By the way, Gödel's set theory is also one-sorted. He does introduce "set" and "class" as primitives, but his first axiom states that every set is a class. So everything in his system is a class. Later expositions, like Mendelson's just start with classes and define sets.)

So I've kept the old form of the axiom. Also, I've changed "A" and "B" back to "C" and "W". I like "C" for "class". The "W" did not suggest "V" to me, but "W" is as good as any other letter. I've removed the subscripts—they did make it harder to read. However, I prefer "x, y, z" because "F" is a function and these letters are commonly used with functions. Personally, I find it easier to read with "x, y, z". Also, I never guessed that "s" stood for set, and I was puzzled why "s" was there with "x" and "y" (which, by the way, are also sets).

After thinking about my previous statement: "I liked my original statement because it was shorter, it gets readers to appreciate the usefulness of the convention that lowercase variables range over sets …", I realized that just having the shorter statement won't get readers to appreciate anything—they need a comparison. So I explain Gödel's convention and restate the axiom using it. It adds some redundancy to the section, but I do reference Gödel's monograph several times in this article so it will be helpful to readers who look up his monograph. Also, it's a simple example of formal language in action and the interested reader can see how Gödel's convention converts the first statement of the axiom into the second. --RJGray (talk) 22:49, 15 January 2017 (UTC)

I like your new rewrite which gives both versions and explains why they are the same. JRSpriggs (talk) 00:38, 16 January 2017 (UTC)

I'm glad you like the new rewrite. Thanks for helping me improve the article. --RJGray (talk) 19:59, 16 January 2017 (UTC)