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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.
: 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. [[User:Randall Holmes|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... [[User:Randall Holmes|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). --[[User:Trovatore|Trovatore]] 08:42, 23 December 2005 (UTC)
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