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It seems that Godel himself believed that the incompleteness theorems seem to imply the inexhaustibility of mathematics; since he states you can simply add the consistency statement of the system as a new axiom thereby creating a logically stronger system. But my question is do these ever stronger systems ever allow us to prove anything that is of interest or fundamentally new, as in when adding such statements you can suddenly construct some object that was not possible before?

And secondly are there other possible axioms that are independent of a given system of arithmetic besides the Godel sentences and consistency statements of that formal system.

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  • $\begingroup$ Welcome to stackexchange. That said, I'm voting to close this question because it's much too vague to have a proper mathematical answer. If you have a particular more precise question about the foundation of mathematics, ask it and perhaps we can help. $\endgroup$ – Ethan Bolker Jul 6 '18 at 0:44
  • $\begingroup$ To your second question, there's been interest in finding 'natural' or 'interesting' statements that are independent of PA. See e.g. en.wikipedia.org/wiki/Paris%E2%80%93Harrington_theorem and the links at the bottom, e.g. Goodstein's theorem. $\endgroup$ – spaceisdarkgreen Jul 6 '18 at 1:06
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There are many statements that are now known to be independent of particular axiom systems. Some of them are mathematically quite interesting. See e.g. List of statements independent of ZFC

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