Talk:Implementation of mathematics in set theory

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It is (really!) not the aim of this article to support a polemic for ZFC or NFU. I do notice that NFU gets credit for allowing certain natural abstractions to be first-class objects which are often missed in ZFC (the universe, the Frege natural numbers). However, it must also be noted that constructions in NFU often get rather baroque (especially where large sets need to be taken into account) and it is pretty clear that the world of NFU contains nonstandard objects (in particular, nonstandard ordinals). Easy access to the "nonstandard" phenomena coded by the external endomorphism of the ordinals (the T operation) allows strong axioms of infinity to be stated in appealingly (perhaps deceptively) simple forms (see the New Foundations article for some of these). I actually think that ZFC on the whole is better (because easier) but not that much better (despite the fact that I study NF and related systems, I'm not a partisan of some wholesale revolution in set theoretical practice!) But I think it is good for those who study foundations to be aware of alternatives. Randall Holmes 03:19, 22 December 2005 (UTC)

More will be coming. Randall Holmes 03:19, 22 December 2005 (UTC)

The handling of indexed families is a little different from that in my book: I save a level of typing by allowing only index sets of singletons. It is not completely general but is a little easier -- but I still made some mistakes setting it up! Randall Holmes 06:26, 22 December 2005 (UTC)

[edit] terminology

I find it a little problematic to refer to doing these things "in" ZFC or NFU. ZFC and NFU only prove things; they don't construct or define anything. I've had quite a job explaining this at articles like definable number (people want to talk about things like "real numbers definable in ZFC", which is nonsense).

What the article seems to be about is how to define various concepts in the language of set theory (not ZFC or NFU) in such a way that ZFC (resp. NFU) proves that they behave the way one wants them to. I think that's fine; I just would rather not see this called "doing things in ZFC or NFU". That's a reasonable shorthand when everyone understands each other, but is likely to cause or reinforce misconceptions among neophytes.

Unfortunately, that's the way I talk. But I will try to bear this in mind. Randall Holmes 08:54, 23 December 2005 (UTC) (I moved this remark to the right place in the conversation).

There is certainly a sense in which one theory (not just language) may "define" or "construct" an object which another theory with the same language may not: it may prove that there is a unique object x such that phi(x), thus establishing that "the x such that phi" exists in the world or class of worlds described by that theory, where another theory with the same language may not prove that the description is satisfied. Randall Holmes 11:05, 23 December 2005 (UTC)

Further, I don't see that the style you object to is avoidable in practice if one is to talk about this issue at all. It is problematic to talk about analogous objects in different theories (in the sciences as well as in mathematics), and here I am not only talking about analogous objects in different theories but also about abstractions from other areas of mathematics to be imported into the two set theories. A possible line of argument might be that one simply cannot safely talk about this issue at all except to experts: but the issue shows up (and is potentially visible to neophytes) at the very beginning of the project of founding mathematics in set theory (why are we using this particular set of axioms?) I do agree (I am very much aware) that one must talk about these things carefully... Randall Holmes 11:05, 23 December 2005 (UTC)

A subordinate but related point is that, of course, the implementations said to be "done in ZFC" could equally well be done in weaker or stronger theories with the same intended interpretation (say, ZC, or ZFC+"there exists a huge cardinal). So it's really the intended interpretation that controls, not the precise formal theory, at least in the "ZFC" case. For NFU it's harder to say, because I'm unaware whether or not NFU has an intended interpretation (you'd know more about that than I). --Trovatore 08:42, 23 December 2005 (UTC)

Re intended interpretation, see the model construction in the New Foundations article. The world of NFU is best understood to be an initial segment of the cumulative hierarchy with an external automorphism moving a rank (which is then used to tweak the membership relation used). It actually presents some of the same difficulties you raised in your discussion of the intended interpretation of KM, with the additional feature that some elements of the NFU universe are clearly in some sense "nonstandard" (large ordinals moved by the T operation, for example). Another way of looking at NFU is to note that it is motivated more by the idea that a set is an abstraction from a predicate than by the idea of a set as a generalization (to the transfinite) of the everyday notion of set (a finite collection) [the latter being what some say is going on in ZFC; but I don't see that this helps with getting a picture of what the world of NFU is like]. Randall Holmes 08:54, 23 December 2005 (UTC)

I do think about the issue of weaker theories. In fact, Mac Lane set theory (bounded Zermelo set theory) is really the Zermelo-style theory I am most often thinking of: it has the same strength as the base theory NFU + Infinity + Choice, while there is actually no natural extension of NFU with the same strength as ZFC: there is an overshoot to the level of n-Mahlo cardinals for each n. In my mind (and more explicitly in various places in my publications) I am comparing Zermelo-style set theory and Quine-style set theory -- the general views of the world, not the specific axiom sets. Randall Holmes 09:35, 23 December 2005 (UTC)

Well, that was kind of my point; the article as written (and perhaps even the title) is misleading for that exact reason. --Trovatore 09:12, 23 December 2005 (UTC)

I'll see if I can put in something about that. I'm not sure the title is particularly misleading: "set theory" as a general subject includes different general approaches as well as different concrete theories. But I should perhaps say that I am comparing two general approaches to set theory rather than two particular theories. Randall Holmes 09:15, 23 December 2005 (UTC)

I added some language about this (second paragraph). Randall Holmes 09:34, 23 December 2005 (UTC)

If you look at the end of the New Foundations article, you can see a summary of the stronger extensions of NFU and the levels of strength in terms of the usual approach to set theory that they represent. Randall Holmes 09:15, 23 December 2005 (UTC)

[edit] new section

I added the new "preliminaries" section to clarify issues of what I mean by "working in a theory" or "defining" or "constructing" objects there. Randall Holmes 18:41, 24 December 2005 (UTC)