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