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www.lesswrong.com | ||
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nickdrozd.github.io
Goedel's first incompleteness theorem is the claim that any sound, consistent formal system of sufficient power is incomplete; that is, there are statements in the language of the system that can neither be proved nor disproved. Traditionally the theorem is proved by exhbiting a statement g which is provably equivalent to a statement encoding its own disprovability in the system S. |
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blog.sigfpe.com
[AI summary] The article explains why defining a truth predicate for set theory (ZF) is impossible despite the success of similar definitions in propositional calculus and Peano arithmetic, based on Tarski's theorem and the limitations of transfinite recursion when applied to infinite collections of children. |
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unstableontology.com
(note: one may find the embedded LaTeX more readable on LessWrong) The Löwenheim-Skolem theorem implies, among other things, that any first-order theory whose symbols are countable, and which has an infinite model, has a countably infinite model. This means that, in attempting to refer to uncountably infinite structures (such as in set theory), one "may... |
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lawrencecpaulson.github.io
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