How does Godel's incompleteness theorem affect the way that mathematicians understand and see mathematics as well as the world (if at all)? I'm not even close to a mathematician, but even a slight dose of the idea and theorem were enough to affect me so I suppose that I'm curious.

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This depends in part upon what you mean by "mathematicians". Ordinary mathematicians, by which I mean mathematicians who aren't particularly or specially interested in logic, have generally, as a group, been utterly uninterested in Gödel's theorems. Reactions vary from case to case, and some are based on ignorance. But we do know, generally, that a huge proportion of "ordinary mathematics" can be done in what are, by the standards of set theorists, very weak theories. So the incompleteness of these theories tends not to be an issue. We've hardly exploited the strength they have.

Another way to put this point is that, by and large, we know of very few "interesting" mathematical claims---claims that would be interesting to an "ordinary mathematician"--- that can be shown to be independent of these same, quite weak theories, let alone independent of Zermelo-Frankel set theory plus the axiom of choice, which is what most logicians would regard as sufficient to formalize the principles used in ordinary mathematical reasoning. That is: Although ZFC is incomplete, by Gödel's result, its incompleteness often seems to lie outside "ordinary mathematics". So the question tends to be, to put it bluntly: If the only questions ZFC can't decide are boring, then who cares?

One of the great logicians of our time, Harvey Friedman, has been working furiously over the last several years to answer this kind of argument, by trying to find "interesting" mathematical claims, that can be shown to be independent of ZF and even of much stronger theories. He has been making some progress, but even his best results still strike me as not entirely convincing.

Now, if the question were about how Gödel's results have affected the way philosophers and logicians see mathematics and our knowledge of it, that would be a very different matter.

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