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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.

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    It seems like you may be using the term "rationalism" to mean something more like logicism or formalism--philosophers usually understand the term in the context of rationalism vs. empiricism where the "rationalist" position is that certain truths are know by a kind of direct intuition rather than derived discursively from some premises and rules of inference.
    – Hypnosifl
    Apr 27 at 20:39
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    If "logicism" is defined broadly as the idea that all truths of mathematics are analytical truths of the kind saying "such and such a proposition follows from this set of premises/axioms and logical rules of inference", then a version of logicism is still defensible after Gödel if you allow for non-computable rules of inference like the ω-rule, see my answer on this topic here.
    – Hypnosifl
    Apr 27 at 20:42
  • "there are true propositions in Number Theory that can not be proved in a specific formal system". Apr 28 at 6:57
  • And yes, it is about "every recursively defined system of axioms". The issue is: is the "mathematical mind" a recursive machine? We do not know. Apr 28 at 6:58
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    Not at all, logicians especially Godel are very rational people in general. It hints its limitations certainly… Apr 29 at 22:40

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Godel's incompleteness can be considered as a mathematical prove to what Spinoza said in philosophy. To build a logical system you need to accept somethings as principles. Whether these principles maybe correct or not we have to assume them. Then he asked how to find out if these principles are correct. We need an external criteria to judge. this external criteria is our assumption. Spinoza chose the compliance with the laws of nature as the criteria. And today we still believe that only those assumptions that do not violate the laws of nature should be our criteria. By the way, rationalism is a subset of this logicism, rationalism and empiricism follow the same criteria for choosing the external assumptions. But in rationalism we assume the law is responsible for matter distribution, in empiricism we assume the matter distribution makes the law.

So, rationalism is compatible with incompleteness theorem because it has an external assumption - the compliance with the laws of nature

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    This is not correct as a description of what "rationalism" means to philosophers, it's definitely not a subset of logicism, nor do rationalism and empiricism use the same criteria for assumptions--see the article on rationalism vs. empiricism I linked to in another comment.
    – Hypnosifl
    Apr 27 at 22:11
  • "Godel's incompleteness can be considered as a mathematical prove to what Spinoza said in philosophy. To build a logical system you need to accept somethings as principles. Whether these principles maybe correct or not we have to assume them." This is totally not what incompleteness is about. Incompleteness says that no matter what axioms you choose, there are always statements that the axioms can neither prove nor disprove.
    – user4894
    Apr 28 at 0:13
  • @Hypnosifl. I read that article and I see the point, philosophers generalizes the concepts of rationalism and empiricism to also include obsolete historical logics. While rationalism may be broader than what I described, one can easily disprove those ideas.
    – user58159
    Apr 28 at 5:16
  • @user4894 There are two theorems, you only read the first theorem. the combination of both theorems result into what I said.
    – user58159
    Apr 28 at 5:19
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    @user4894, yes, that's the technical point of the 1st incompletelenss theorem... but the larger point of interest comes from the 2nd. Whatever axiomatic system we use, we can never prove its consistency. So on what basis do we choose a system. We use some other kind of judgment... Why do mathematicians choose ZFC at all? They choose it based on because it hasn't been proven inconsistent yet, and there's some kind of unprovable intuition at its base. That fits perfectly with rationalism imo. Apr 28 at 5:22

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