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16
votes
3answers
665 views

What are the philosophical consequences of the undecidability of the spectral gap in quantum theory?

An article published in Nature yesterday proves that finding the spectral gap of a material based on a complete quantum level description of the material is undecidable (in the Turing sense). One of ...
5
votes
5answers
1k views

Do Gödel's incompleteness theorems have any consequences for epistemology?

Do Gödel's results have any impact on the theory of knowledge: realism vs anti-realism, the existence of the noumenal, the existence of synthetic a priori truths, etc... or are they just relevant to ...
1
vote
3answers
311 views

Was Gödel the first person to bring up that truth always exceeds the grasp of proof?

Was Gödel the first person to pose and solve this question in mathematics? In the larger philosophical debate, has this question been posed before? Say by Plato or Aristotle? One could interpret for ...
3
votes
2answers
2k views

Relation of Gödel's incompleteness theorems and Karl Popper falsification

Falsifiability is considered a positive (and often essential) quality of a hypothesis because it means that the hypothesis is testable by empirical experiment and thus conforms to the standards of ...
29
votes
4answers
12k views

What are the philosophical implications of Gödel's First Incompleteness Theorem?

Gödel's First Incompleteness Theorem states Any effectively generated theory capable of expressing elementary arithmetic cannot be both consistent and complete. In particular, for any ...