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Andrius Kulikauskas

  • m a t h 4 w i s d o m - g m a i l
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  • My work is in the Public Domain for all to share freely.

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  • 读物 书 影片 维基百科

Introduction E9F5FC

Questions FFFFC0

Software

Logic, Category theory

Set theory

https://en.wikipedia.org/wiki/Goodstein%27s_theorem

  • Shows that the provability of a statement may depend on how powerful is the notation supported by the axioms.
  • Gives clues on how the limitations of Goedel's Incompleteness theorem https://en.wikipedia.org/wiki/G%C3%B6del%27s_incompleteness_theorems might be overcome, for example, by a growing set of axioms, potentially infinite, by which the notation would become ever more powerful, thus becoming more complete.

Amelia: [The axiom of function extensionality is] inconsistent with many axioms of a more "computational" nature. For example, "formal Church's thesis" says that for any function N→N, there is a "program" (we call it a realizer) that realizes it. You can kinda see what goes wrong: this would be able to tell e.g. "λ x → x" and "λ x → x + 0" apart. You could imagine an assignment of realizers that sidesteps this, though, so to see that it's actually inconsistent takes slightly more work.

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This page was last changed on December 26, 2024, at 09:26 PM