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The wikipedia article on Tarski-Grothendieck set theory states:

"[Tarski's axiom] also implies the existence of inaccessible cardinals, thanks to which the ontology of TG is much richer than that of conventional set theories such as ZFC."

From http://en.wikipedia.org/wiki/Tarski-Grothendieck_set_theory

What is TG about and why does it provide a richer ontology? How does the richer "ontology" of category theory express itself in terms of definitions of syntax? It is said that in category theory the main objects of study are functors instead of of sets, but obviously these are interchangeable, because there is the category of sets and sets define functions as mappings of elements.

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It allows you to talk about categories of categories; and categories of categories of categories; and so on. Whereas in plain ZFC it is not the case. –  Asaf Karagila Sep 22 '12 at 10:48

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ZFC is a theory of sets, in the sense that it allows us to talk about arbitrary sets (more formally, the quantifiers $\forall x$ and $\exists x$ are intended to be interpreted as "for every set $x$" and "for at least one set $x$"), but we cannot talk directly about classes of sets that are not themselves sets (like the class of all sets or the class of all ordinal numbers), or collections of such classes, etc. Strictly speaking, the same is true for TG, but TG also offers us a second notion of set (in fact many such notions) for which we can speak about bigger collections like classes that are not sets, collections of such classes, etc.

Specifically, TG provides a lot of sets $U$, called universes, that share many of the properties that one would want for the class of all sets. More precisely, lots of facts that one knows about sets in general are also true about "sets in $U$". For example, if $X$ is a set in $U$, then so is the collection of all subsets of $X$. For another example, there is an infinite set in $U$. In particular, every axiom of ZFC remains true if one systematically replaces "set" by "set in $U$" throughout the axiom; therefore the same goes for any theorems provable in ZFC. (The details of this information about $U$ are built into the definition of what it means to be a universe.) The upshot of this is that whatever people normally do in ZFC, using arbitrary sets, can be done equally well in TG using just sets in $U$.

The advantage of TG is that, because $U$ is a set, there is no problem with talking about arbitrary subsets of $U$ (which would correspond to classes in the ZFC approach), collections of such subsets, etc. In the TG approach, it would seem that, if one wants to prove things about a particular set $X$, by analogy with ZFC arguments, one would need that $X\in U$; what if $X\notin U$ (for example, what if $X$ is $U$ itself)? Fortunately, TG gets around that problem by giving us not just one universe $U$ but arbitrarily large universes. Specifically, any set $X$ is an element of some universe $U$.

A disadvantage of TG arises from this multiplicity of universes. One often needs to check that some definition doesn't depend on which universe it's carried out in (or perhaps that it depends only in inessential ways). That checking tends to be a routine business that only distracts from the essential mathematical content that one is really trying to develop.

Another aspect of TG that might be perceived as a disadvantage is that its axiom of universes cannot be proved consistent relative to ZFC. That is, it is conceivable that ZFC is consistent but TG is not. From the point of view of contemporary set theory, though, the axiom of universes is a very mild assumption. Far stronger "large cardinal" hypotheses have been studied for decades, are quite well understood, and have led to no contradicitons. I think it's fair to say that no set theorist really worries about the possibility that TG might be inconsistent.

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When you say "One often needs to check that some definition doesn't depend on which universe it's carried out in", do you mean often if you're a set theorist doing foundations or do you mean in general? If I'm only interested in sets for doing topology and propability theory, will this named disadvantage be really one? –  NikolajK Mar 27 '13 at 18:46
    
I now asked this elsewhere too, and much more. –  NikolajK Mar 28 '13 at 0:33

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