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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. 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); 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)
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