Ok, lot to respond to here. I’m familiar with the relevant mathematics.
First, I think it’s clear from the litany of failed attempts that you can’t write down a finite set of rules that tells us what is true about the world. At least to me, it’s also intuitively clear that you can’t write down such a set of rules. That is not, without considerable auxiliary claims at least, a consequence of gödel’s incompleteness theorem, nor does gödel’s incompleteness theorem follow from it.
The essential issue here is that the incompleteness theorem deals with formal statements and formal reasoning in formal languages. There is a significant gap between the perfectly acceptable reasoning we use every day to understand the world around us, which if it can be written down at all often requires us to use informal language, and the sort of thing the incompleteness theorem addresses. There are real philosophical claims to be made and argued, which in at least implicit form go far back and have yet to be pinned down. For example, the sorites paradox can be understood as an (informal) proof that induction can fail in informal arguments. The whole thing, rather than being clarified, becomes more and more hopelessly complex the more one thinks about it.
I agree that inasmuch as objectivism pretends to formality it makes itself vulnerable to mathematical theorems, and surely would evaporate on contact with them. But the failure of the pretense to formality itself renders the issue moot.
Second, the question of what is or is not a “consequence” of this or that theorem is, given the nature of implication, a little difficult to pin down in borderline cases (are not all theorems a consequence of “T -> T”?). I’m perfectly fine with calling cantor’s theorem, the halting theorem, the incompleteness theorem, etc instances of lawvere’s fixed point theorem. But there is significant work required to take the hypotheses of some of these types of theorems and maneuver things into such a position as to apply the FPT. So I don’t think it’s a consensus opinion.
Third, if we want to describe a postmodern movement in mathematics, while I’m not sure about the dates, I get what you’re going for and it makes sense to me. But I think the description you’re putting forth here gives way too much weight to theorems. The movement toward a post-modern sensibility is imo much more marked by a movement toward guiltlessly abstract definitions and axioms. Consider the centuries of the torment that mathematicians experienced trying to justify or explain what negative numbers or complex numbers really are. In contrast, in the 20th century one has the definition of a scheme in algebraic geometry, a kind of space characterized by functions on it which are not really functions and which can be 0 at every point and yet not the zero function. What is the meaning of such a thing? Well, it is up to the individual mathematician to accept their own metaphors explaining that matter. Totally unthinkable a century prior. Examples of this sort of thing abound (for example, test functions in functional analysis). The movement toward such things was doubtless urged on by the impossibility theorems you refer to but goes far beyond them and is far more impactful imo.
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