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title
Algorithmically Incompressible Thoughts
description
... or are they?
image
site name
Algorithmically Incompressible Thoughts
author
updated
2026-02-21 09:52:39
raw text
Algorithmically Incompressible Thoughts | … or are they? Algorithmically Incompressible Thoughts … or are they? Undecidability from a syntactic perspective November 7, 2010 Robinson arithmetic, a finitely axiomatized fragment of arithmetic, is undecidable, therefore, so is classical first-order predicate logic. This is one usual way of proving the undecidability of first-order logic (with an arithmetic language). What do we mean by undecidability here? That we have no algorithm for deciding whether a formula belongs to a certain theory or not, of course, but how is this notion formalized? Via the notion of Gödel numbering, which allows us to translate formulas into numbers, and the notion of recursive sets of numbers, which in arithmetic terms defines sets membership in which can be algorithmically decided: a theory is decidable if the Gödel numbers of its formulas form a recursive set. These definitions work out very well, but one might wonder whether, in talking about the ...
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