According to Putnam, Gödel's theorems show that the set of truths in Number Theory (i.e., true propositions involving natural numbers and their properties) is not recursively enumerable, whereas all the statements that can be proved about natural numbers are recursively enumerable.
That is, there are true propositions in Number Theory that can not be proved.
If we take classic rationalism as the view that starting from a set of axioms and by the use of analytic reason, we can both decide and know the truth or falsehood of any affirmation; does this not disprove the main starting point of rationalism?
I have read many accounts claiming that Gödel's theorems have no philosophical implications beyond the Philosophy of Mathematics, or that any application of them to bear on epistemological questions is for some reason not legitimate; but the authors do not generally give arguments for this.
The only sound argument generally asserts that Gödel's results are restricted to the ZFC axiomatic system, so that every proposition of interest can be decided in a system with additional axioms.
This argument however, seems to have two strong flaws: First, it fails to recognise that virtually every working Mathematician assumes that ZFC is a good model of the axioms that are employed when trying to solve a question in Mathematics; so that ZFC can be reasonably regarded as encompassing the natural axioms presupposed in the practice of Mathematics, as well as making the addition of other axioms arbitrary or not natural.
And second, it overlooks that this is not a flaw unique to ZFC, in that every recursively defined system of axioms will also have this flaw.
Also, it seems strange to me that such a significant result is not more widely discussed in connection to the all-too-popular contemporary schools of positivism and rationalism, although it seems to be a very reasonable argument against them in the worst case.
Am I missing something?
Thank you in advance for your input.