Talk:Implementation of mathematics in set theory

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)[reply]

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

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)[reply]

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.

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)[reply]

Unfortunately, that's the way I talk. But I will try to bear this in mind. Randall Holmes 08:54, 23 December 2005 (UTC)[reply]

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 (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 of the everyday notion of set (a finite collection) [the latter being more what 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)[reply]