Talk:Tarski–Grothendieck set theory

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[edit] Questions

Is TG simply ZF minus Infinity, augmented by Tarski's axiom? Does that axiom also ensure the existence of infinite sets? What is known about the metamathematics of Tarski's axiom? Why is Grothendieck's name associated with TG? Has anyone written on TG outside of the Journal of Formalized Mathematics?132.181.160.42 03:38, 10 August 2006 (UTC)

This theory looks like very sloppy work to me, if this article correctly represents it. JRSpriggs 03:00, 21 August 2006 (UTC)

ad existence of infinite sets: yes, for any ordinal, Tarski's axiom gives you a limit ordinal containing it (http://mmlquery.mizar.org/mml/current/ordinal1.html#T51); the smallest containing the empty set is omega (http://mmlquery.mizar.org/mml/current/ordinal1.html#D12) JosefUrban 19:00, 8 June 2007 (UTC)

ad usage by Grothendieck: http://modular.fas.harvard.edu/sga/sga/4-1/4-1t_185.html;

and as for "sloppiness", I do not know how to measure this, but provided that this is used by two top-level mathematicians of 20. century, I'd be a bit cautious with such words (and if used at all, I'd certainly try to justify them) JosefUrban 19:09, 8 June 2007 (UTC)

I can't read Mizar, and I don't see how Tarski's axiom implies the axiom of infinity. It looks to me that V_ω, which is a Grothendieck universe, is a model of these axioms, since all of its subsets are either hereditarily finite or have cardinality ω. — Charles Stewart (talk) 12:26, 24 June 2009 (UTC)

Tarski's axiom basically says that every set is a member of a set which is a Grothendieck universe. V_ω is not a model of this axiom, because no nonempty element of V_ω is a Grothendieck universe. More generally, every nonempty Grothendieck universe is infinite, which is why Tarski's axiom implies the axiom of infinity. — Emil J. 12:42, 24 June 2009 (UTC)

Ah, yes, of course, how silly of me. Thanks. So ZF+Tarski's axiom has the same theory as ZFC+the inaccessible cardinal axiom. — Charles Stewart (talk) 13:24, 24 June 2009 (UTC)

[edit] Implies axiom of choice?

"Tarski's axiom implies the Axiom of Choice"

why/how?does anyone have a proof of this?

16:25, 28 March 2007 (UTC)

yes, a verified one: http://mizar.uwb.edu.pl/JFM/Vol1/wellord2.html (or in full detail: http://mmlquery.mizar.org/mml/current/wellord2.html#T26, http://mmlquery.mizar.org/mml/current/wellord2.html#T28). JosefUrban 18:15, 8 June 2007 (UTC)