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My belief is that no result in algebraic geometry that does not explicitly engage the universe concept will fully require the use of universes. Indeed, I shall advance an argument that no such results …

A Grothendieck universe is known in set theory as the set Vκ for a (strongly) inaccessible cardinal κ. They are exactly the same thing. Thus, the existence of a Grothendieck universe is exactly equiv …

The usual relations to consider in the large cardinal hierarchy are Direct implication: every A cardinal is also a B cardinal Consistency strength implication: if ZFC + there is an A cardinal is con …

François has excellently addressed your question 1; allow me to address question 2. I understand the question to be: what will be the mathematical effects if someone were to show that there are no (we …

Two weeks ago a conference was held on precisely the topic of your question, the Workshop on Set Theory and the Philosophy of Mathematics at the University of Pennsylvania in Philadelphia. The confere …

My view is that the large cardinal hierarchy already has all the principal features of the project you are proposing. Each of the large cardinal concepts can be regarded as corresponding to a certain …

There are numerous examples of such statements. Let me organize some of them into several categories. First, there is the hierarchy of large cardinal axioms that are relatively consistent with V=L. Se …

Let me try to answer as a set theorist, rather than as a category theorist, since I think that your question concerns at bottom a matter often considered in set theory. Namely, the essence of your qu …

I like the question very much. First, let me mention briefly that the question has a flaw in the quantifier order, since you have first fixed the theory $\Gamma$ and then ask for a cardinal $\kappa$ …

The line of reasoning you mention at the end of your post, firmly in support of large cardinals, was first argued forcefully in W. N. Reinhardt, “Remarks on reflection principles, large cardinals, …

With regard to the title question, I believe that the main argument people would provide would be that it is the actual existence of the large cardinals that explains the consistency assertions that o …

Because of Goedel's Incompleteness Theorems, we know that we cannot describe a complete axiomatization of mathematics. Any proposed axiomatization $T$, if consistent, will be unable to prove the princ …

Update. My new article grows out of and extends my 2010 answer to this question. The new part is the conservativity result, showing that the Vopěnka principle has the same first-order consequences as …

Cantor's Attic, which I founded with Victoria Gitman, is a compendium of information on the consistency strength hierarchy in set theory, spanning the range from ZFC (which in this context is viewed a …

Beyond the infinite Ramsey's theorem on N, there is, of course, a kind of super-infinite extension of it to the concept of Ramsey cardinals, one of many large cardinal concepts. Most of the large ca …