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Does TG (Tarski-Grothendieck Set Theory) prove that ZFC2 (second-order ZFC) has a model? Does it at least prove the consistency of ZFC2?

Asaf Karagila
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goblin GONE
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1 Answers1

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Gathering two bits from the Wikipedia entries:

It is a non-conservative extension of Zermelo–Fraenkel set theory (ZFC) and is distinguished from other axiomatic set theories by the inclusion of Tarski's axiom which states that for each set there is a Grothendieck universe it belongs to (see below).

Tarski's axiom implies the existence of inaccessible cardinals, providing a richer ontology than that of conventional set theories such as ZFC. (Wikipedia, Tarski–Grothendieck set theory)

And in turn

Secondly, under ZFC it can be shown that $\kappa$ is inaccessible if and only if $(V_\kappa,\in)$ is a model of second order ZFC. (Wikipedia, Inaccessible cardinal)

So the answer is indeed positive.

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Asaf Karagila
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