Here it is mentioned that someone claims to have proven that there are no weakly inaccessibles in ZF.
Question 1: What reasons are there to believe that weakly inaccessibles exist?
Question(s) 2: Since all large cardinals are weakly inaccessible, this would have a profound effect on set theory. What are some of the most significant results whose only known proof assumes the existence of weakly inaccessibles? Might any of the arguments go through without their existence? For example, I've heard that the original proof of Fermat's Last Theorem (FLT) assumed (something equivalen to) a large cardinal, but it was then shown that the argument went through without such an assumption.
Edit. I just added the phrase "whose only known proof" to Question 2 above, which is what I intended originally. The point of course, is that I want to know what which results, if any, would be "lost" if weakly inaccessibles were lost. FLT is not an example of that, but would have been before it was known that weakly inaccessibles are not necessary in its proof.