Thread: Is the continuum hypothesis a hopeless situation?

I've recently heard that Kurt Gödel and another guy proved that the continuum hypothesis can neither be proven nor disprove under the axioms of set theory, with or without the axiom of choice. Does this mean that the situation is completely hopeless? Is it possible that we can prove/disprove it with new axioms, or a completely different approach altogether? I'm starting to get worried. Perhaps a lot of the unsolved problems in mathematics cannot be solved under assuming whatever is self-evident? I understand that a lot of things are true but unprovable (Gödel's first incompleteness theorem), but most of these statements are self-evident and can be immediately accepted by nearly anybody without proof. The universe, I think, has completely screwed us over with axioms that are true but cannot be proven.

Originally Posted by anticorncob28

I've recently heard that Kurt Gödel and another guy proved that the continuum hypothesis can neither be proven nor disprove under the axioms of set theory, with or without the axiom of choice. Does this mean that the situation is completely hopeless? Is it possible that we can prove/disprove it with new axioms, or a completely different approach altogether? I'm starting to get worried. Perhaps a lot of the unsolved problems in mathematics cannot be solved under assuming whatever is self-evident? I understand that a lot of things are true but unprovable (Gödel's first incompleteness theorem), but most of these statements are self-evident and can be immediately accepted by nearly anybody without proof.

Modern set theorists have done a lot of work on the problem. Here is one link summarizing some approaches from the site Mathoverflow.net. It's a math discussion forum for professional mathematicians so much of this is not accessible. But it gives the flavor.

set theory - Solutions to the Continuum Hypothesis - MathOverflow

also

http://mathoverflow.net/questions/14...uum-hypothesis

There's a set theorist names Seharon Shelah who's done a lot of work in this area.

Saharon Shelah - Wikipedia, the free encyclopedia

Originally Posted by anticorncob28

The universe, I think, has completely screwed us over with axioms that are true but cannot be proven.

The axioms of set theory don't come from the universe. They were developed by mathematicians between around 1870 and 1920 or so. They're historically contingent. People did brilliant math before there was set theory and they're doing brilliant math today that goes beyond set theory. There are already many alternative foundational approaches being developed, such as category theory, homotopy type theory, and complexity theory.

In other words if ZFC has problems -- and we've always known that it does -- then foundationalists will figure out something else. And the great majority of working mathematicians don't care one way or the other. They're studying mathematical objects, not rooting around in foundations.