Content deleted Content added
Line 20:
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. [[User:Randall Holmes|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. [[User:Randall Holmes|Randall Holmes]] 11:05, 23 December 2005 (UTC)
Line 27 ⟶ 29:
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)
▲: Unfortunately, that's the way I talk. But I will try to bear this in mind. [[User:Randall Holmes|Randall Holmes]] 08:54, 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]. [[User:Randall Holmes|Randall Holmes]] 08:54, 23 December 2005 (UTC)
| |||