Talk:Scott's trick

Wikipedia is not well-founded

It seems to me that these math articles in Wikipedia are not well-founded. This article defines an ordinal number based on the cumulative hierarchy. Cumulative hierarchy redirect to Von Neumann universe, which in turn defines the rank based on ordinal numbers! This seems to strongly violate the axiom of Foundation for articles! :-) Albmont (talk) 14:24, 15 May 2009 (UTC)

Not directly addressing this confusion (which is about the relationship between "naive" ordinals and their representatives), I rewrote the section to be about cardinality, which I think it less opaque. — Carl (CBM · talk) 20:33, 14 July 2009 (UTC)

Where and when?

If Scott published this, it would be nice to have a reference. — Preceding unsigned comment added by 213.122.6.196 (talk) 20:24, 13 October 2011 (UTC)

Again about the introduction

The introduction still seems unclear. First of all, a method for choosing sets of representatives for equivalence classes without using the axiom of choice (AC) suggests that in the presence of AC Scott's trick is unnecessary. Yes, we can do with AC in order to define cardinals, but the trick is necessary to take an ultrapower of the universe. I feel that it is necessary also for defining order types rigorously. (PS: No: order types can be introduced without using Scott's trick if you allow the Axiom of Choice: once you have a representative for each cardinal, then there is only a set of linear orders over that representative, hence you have a set of representatives for all order types, say, of cardinality $<\lambda$ - of course, all order types form a class. See, e.g., the footnote on p. 239 of Sierpinsky, Cardinal and Ordinal numbers, second edition, 1965. So the only thing for which Scott's trick is credited to be indispensable is to take an ultrapower of the universe)

(When we use AC in order to define cardinals, we do not use it to choose directly some representative: we use it to well-order a set X. The well order is (order)-isomorphic to an ordinal, then we take the initial ordinal as the cardinality of X. If we had to use directly some form of choice, for defining cardinals, we should choose an element from a class of classes; this cannot even be stated in most set theories. Perhaps Penelope Maddy in "Proper classes" (J. Symbolic Logic 48 (1983), no. 1, 113--139) talks about this, and perhaps this can be even done in some conservative way, but it is more involved than Scott's trick anyway).

For the rest, I confess that I do not understan Forster: to perform Scott's trick you need the ordinals, so you cannot define the ordinals by using the trick. I agree with Albmont that this seems circular. (There are many alternative ways to introduce the ordinals, apart from the presently "trendy" Von Neumann's trick (the alternative ways can be found, for example, in Bachman, Transfinite Zahlen, especially if you now german better than I do ;). Paolo Lipparini (talk) 12:28, 2 April 2015 (UTC)

PS: According to Tarski (same link as Scott's original abstract) "The theory of ordinals can be developed within the theory of ranks", that is, one can introduce a rank function without first defining what an ordinal is. I still do not see why this is useful, since you can introduce ordinals without choice and without regularity (see Jech's book). Paolo Lipparini (talk) 17:09, 21 April 2015 (UTC)

"it is possible that some sets do not have the same cardinality as any ordinal number"

Section Application to cardinalities said:

In Zermelo–Fraenkel set theory with the axiom of choice, one way of assigning representatives to cardinal numbers is to associate each cardinal number with the least ordinal number of the same cardinality. But if the axiom of choice is not assumed, it is possible that some sets do not have the same cardinality as any ordinal number, and thus the cardinal numbers of those sets have no ordinal number as representatives.

I changed the part "it is possible that some sets do not have the same cardinality as any ordinal number" to "for some cardinal numbers it may not be possible to find such an ordinal number".

Rationale: I'm not sure whether the original statement is true, but even if it is true, I think it is not a sufficient explanation for why without the axiom of choice it may be impossible to associate each cardinal number with the least ordinal number of the same cardinality. Even if there are some ordinal numbers that have the same cardinality as a certain cardinal number, it may not be possible to find the least of these ordinal numbers without the axiom of choice.

If I understand correctly, the section Ordinal number#Initial ordinal of a cardinal confirms my understanding:

In theories with the axiom of choice, the cardinal number of any set has an initial ordinal, and one may employ the Von Neumann cardinal assignment as the cardinal's representation. In set theories without the axiom of choice, a cardinal may be represented by the set of sets with that cardinality having minimal rank.

But I'm far from being an expert in set theory, so I may be mistaken. -- Chrisahn (talk) 00:22, 28 December 2018 (UTC)

The rewording is fine, but this is not a problem; you can always find the least ordinal from a nonempty set (even a proper class) of ordinals.

The confusing point (I admit I have been quite confusing in the above comment, too) is that there is no unique possible definition of "ordinal". Intuitively, an ordinal is an equivalence class of isomorphic well-ordered sets. Unfortunately, the above "definition" is not admissible in many axiomatizations of set theory, since such an equivalence class is generally a proper class (not a set), hence it cannot be a member of some set, so you could not have "sets of ordinals". If you want to use the intuitive definition, you should deal with a theory in which it is possible to talk of "classes of classes" and this is unusual.

To circumvent the problem, one usually chooses some representative, for each isomorphism class. One possibility is using Scott's trick, as explained in the page with emphasis on the notion of cardinality, but there is no essential difference when treating ordinals. The more frequent definition uses a trick due to Von Neumann, but here another confusing point arises, since, according to the actual definition you choose, you might need to assume Foundation. However, if you define an ordinal as in Jech's book, you need not assume Foundation (there still another delicate issue, namely, you need Replacement in order to show that Von Neumann definition corresponds to the intuitive notion, that is, that every well-ordered set is isomorphic to some ordinal. Hystorically, it seems that the issue tricked both Zermelo and Von Neumann himself!)

In any case, whatever the definition, if an ordinal corresponds in some way to some isomorphism type of well-ordered sets, you always have the minimum of a nonempty set of ordinals; this is a property of well-ordered sets. You need no special axiom to see that any two well-ordered sets are comparable. Then, consider a nonempty family F of well-ordered sets and choose one, say W in F. If W is the smallest member of the family, you are done. Otherwise, each element of the family is either larger than W, or isomorphic to W, or isomorphic to some proper initial segment of W (since any two well-ordered sets are comparable). But each initial segment of W is determined by some element w of W. Since W is well-ordered, you have the minimum such w, which gives the order-type of the minimal element of the family. Paolo Lipparini (talk) 22:17, 11 October 2019 (UTC)