- Oct 2024
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Local file Local file
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Cohen (Independence of the Axiomof Choice; The Independence of the Continuum Hypothesis I, I1) completed theproof of independence for each by showing neither could be deduced from theexisting axioms (by showing the negation of each could consistently be added tothe Zermelo—Fraenkel axiom scheme). See P. J. Cohen (Set Theory and theContinuum Hypothesis) for a discussion of these results and his intuition about thecontinuum hypothesis. Another expository reference is Cohen (IndependenceResults in Set Theory).
In 1963 Paul Cohen completed the work of Gödel by proving the independence of the axiom of choice and the continuum hypothesis from the Zermelo-Fraenkel set theory axioms. He did this by showing that neither could be deduced from the existing axioms and specifically by showing that the negation of each could be added to ZF consistently.
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Godel (The Consistency of the Axiom of Choice and of the Generalized Con-tinuum Hypothesis with the Axioms of Set Theory) proved in 1940 that additionof either the axiom of choice or the continuum hypothesis to existing set theoreticaxioms would not produce a contradiction.
Gödel's breakthrough in 1940 was to prove that one could extend the axioms of set theory to include the axiom of choice or the continuum hypothesis without introducing contradictions.
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- Dec 2022
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math.stackexchange.com math.stackexchange.com
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My freely downloadable Beginning Mathematical Logic is a Study Guide, suggesting introductory readings beginning at sub-Masters level. Take a look at the main introductory suggestions on First-Order Logic, Computability, Set Theory as useful preparation. Tackling mid-level books will help develop your appreciation of mathematical approaches to logic.
This is a reference to a great book "Beginning Mathematical Logic: A Study Guide [18 Feb 2022]" by Peter Smith on "Teach Yourself Logic A Study Guide (and other Book Notes)". The document itself is called "LogicStudyGuide.pdf".
It focuses on mathematical logic and can be a gateway into understanding Gödel's incompleteness theorems.
I found this some time ago when looking for a way to grasp the difference between first-order and second-order logics. I recall enjoying his style of writing and his commentary on the books he refers to. Both recollections still remain true after rereading some of it.
It both serves as an intro to and recommended reading list for the following: - classical logics - first- & second-order - modal logics - model theory<br /> - non-classical logics - intuitionistic - relevant - free - plural - arithmetic, computability, and incompleteness - set theory (naïve and less naïve) - proof theory - algebras for logic - Boolean - Heyting/pseudo-Boolean - higher-order logics - type theory - homotopy type theory
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- Jul 2022
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bafybeicuq2jxzrw7omddwzohl5szkqv6ayjiubjy3uopjh5c3cghxq6yoe.ipfs.dweb.link bafybeicuq2jxzrw7omddwzohl5szkqv6ayjiubjy3uopjh5c3cghxq6yoe.ipfs.dweb.link
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worldview as a complex mental object makes sense onlyin the light of evolution – as the work in progress that it is; both fluid and firm at thesame time.
!- question : strange loop * I'm not sure if I agree with this claim, I'll have to read and see if he can justify it * I would claim instead that language and symbols are even more profoundly entangled, as per Nagarjuna's work * https://hyp.is/go?url=http%3A%2F%2Fdocdrop.org%2Fvideo%2FHRuOEfnqV6g%2F&group=world
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- Jun 2021
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www.fudco.com www.fudco.com
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cheap trick
Does Morningstar think that math too suffers from the same issues he finds in critical theory, or just Godel's incompleteness theorem (I'm assuming that's what Morningstar is alluding to)? Explore a deep discussion about whether Godel's incompleteness theorem is a cheap trick.
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- Jan 2021
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opentheory.net opentheory.net
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But there can also be a buildup of tension as one gathers information that is incompatible with one’s key signature, which gets progressively more difficult to maintain, and can lead to the sort of intensity of experience that drives an annealing-like process when the key signature flips.
cue [[Godel Escher Bach]]'s [[Little Harmonic Labyrinth]]: ends in a pseudo-tonic that never returns back to original [[tonic]]. results in tension dangling from continual musical modulation without resolution
similar to [[linguistic]] structures as well - unconsciously keeping track of constructions and hierarchies within sentences
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- Dec 2020
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stopa.io stopa.io
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explanation of godel's theorems
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