Talk:Zermelo set theory

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original text should be in wikisource -MarSch 14:37, 18 Apr 2005 (UTC)

[edit] ZC and categorical logic

IIRC ARD Mathias says that MacLane's weakened set theory (ZBQC), which is close to topos theory, is essentially the same as ZC (try different senses of essentially and maybe this recollection comes out). The reference might be The Strength of MacLane's set theory (PDF). If the claim does come out, then it justifies viewing Zermelo set theory as of more than historical interest. --- Charles Stewart 02:22, 31 December 2005 (UTC)

this is not true: Mac Lane set theory is essentially weaker than Zermelo and you can find this out in the article of Mathias that you cite. Randall Holmes 05:12, 31 December 2005 (UTC)

Details: Mac Lane set theory is Zermelo with separation restricted to bounded (Δ₀) formulas. This gives the same strength as the simple theory of types or NFU + Infinity; this is essentially weaker than full Zermelo (which can prove the consistency of Mac Lane). As is almost always the case, adding Choice makes no difference at all with respect to consistency strength. I wouldn't describe Mac Lane set theory as especially close to topos theory; topos theory interprets intuitionistic type theory, which is at the same level of strength as Mac Lane but is not especially similar to it on a formal level. Randall Holmes 16:59, 31 December 2005 (UTC)

Ah, lazyweb works. Many thanks, all these comments are interesting. By same level of strength, do you mean there is connection along the lines of that which Aczel describes between constructive set theories and Kripke-Platek style set theories? --- Charles Stewart 21:54, 31 December 2005 (UTC)

By same strength I mean that consistency of the one theory is equivalent to consistency of the other, probably in PA; they are mutually interpretable. Randall Holmes 04:07, 1 January 2006 (UTC)

my best advice is to read the Mathias paper; it is really good and contains lots of stuff (though nothing about topos theory). Randall Holmes 04:08, 1 January 2006 (UTC)

Ok, it is clear that my idea doesn't pan out. Mathias' paper is on my pile of to-read papers. which is unfortunately very high. --- Charles Stewart 17:36, 1 January 2006 (UTC)

[edit] ZC and mathematics

Harvey Friedman claims, in a post on Predicative foundations, that:

Mathematicians will not adhere to any explicit formalism. Currently they (the community as a whole) implicitly adheres to ZC, which is ZFC without replacement.

This claim makes ZC of more than purely historical interest. --- Charles Stewart^(talk) 21:52, 8 February 2006 (UTC)