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5 added 10 characters in body

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.

4 Clarified Question 2; added 6 characters in body

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 that assume 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 results 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.

Post Reopened by Georges Elencwajg, Daniel Litt, quid, Mark Sapir, David Roberts
3 Removed reference to M. Gardner, removed Question 2
Post Closed as "subjective and argumentative" by Will Jagy, David Roberts, Noah Snyder, Andreas Blass, Qiaochu Yuan
2 Added second sentence "By...scrutinized." to Question 2
1