What is the relationship between a requirement for consistency of a theory and what it can prove?

This is a question that I always had in mind.

When it is said that the consistency of a theory $$T$$ requires the assumption of existence of some specified cardinal $$\kappa$$. Is that taken to mean that $$T$$ can prove the existence of all cardinals strictly smaller than $$\kappa$$?

A clarifying example, suppose its said that a theory $$T$$ requires existence of $$0^\sharp$$, one might get the impression that $$T$$ can interpret each theory ZFC+ $$\kappa$$, where $$\kappa$$ is a cardinal property such that ZFC + $$\kappa$$ is interpretable in ZFC+$$0^\sharp$$. So viewing the list of large cardinal properties, then one would for example expect that $$T$$ can interpret ZFC+ Mahlo cardinal exist. Is that impression correct?

This is a situation where there is a fair amount of abuse of language, unfortunately.

Generally statements like this are shorthand for a consistency strength claim over the base theory ZFC; e.g. when people say "X requires a measurable cardinal" what is meant is that ZFC+"There is a measurable cardinal" is consistent relative to ZFC+X (even more formally, that PRA proves "If ZFC+X is consistent, then ZFC+"There is a measurable cardinal" is consistent).

Sometimes a different base theory on one or the other side is meant; e.g. "AD requires a measurable" means that ZF+AD is consistent only if ZFC+"There is a measurable" is, and this is clear from context since ZFC+AD is outright inconsistent.

Note that this is all strictly weaker than saying that the large cardinal actually needs to exist; when we say that X requires a measurable cardinal we generally do not mean that ZFC proves "If X, then there is a measurable cardinal." In particular, there's a huge difference between "There is a measurable cardinal" and "There is an inner model with a measurable cardinal."

(Incidentally, it's not even clear to me what "$$T$$ can prove the existence of all cardinals strictly smaller than $$\kappa$$?" would mean in the first place.)

• I've added a clarifying example addressing the last two lines of your answer. – Zuhair Apr 13 at 18:17
• I think the implications in your answer are the converse. I mean if I say that X requires measurable cardinal then its taken to mean that PRA proves that if ZFC+measurable cardinal is consistent, then ZFC+X is consistent. – Zuhair Apr 13 at 18:25
• @Zuhair Absolutely not - under that reading, "$2+2=4$" would require a measurable! "Requires" is analogous to "only if." – Noah Schweber Apr 13 at 18:33
• I wouldn't call it abuse of language. It is standard jargon. – Andrés E. Caicedo Apr 13 at 18:47
• @NoahSchweber, Oh yes of course, I was confused, I thought its the least cardinal in that direction. Anyhow i think its equivalent to the direction you've stated. – Zuhair Apr 13 at 18:56