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Results tagged with lo.logic
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user 20597
first-order and higher-order logic, model theory, set theory, proof theory, computability theory, formal languages, definability, interplay of syntax and semantics, constructive logic, intuitionism, philosophical logic, modal logic, completeness, Gödel incompleteness, decidability, undecidability, theories of truth, truth revision, consistency.
6
votes
Reinhardt's ultimate classes
You can find Reinhardt's philosophy of set theory in
"Set existence principles of Shoenfield, Ackermann, and Powell", Fundamenta Mathematica, vol 84, pp 5-34 and
"Remarks on reflection principles, la …
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0
votes
1
answer
269
views
What is the smallest countable limit ordinal in which 'lost melodies' occur
The question is in the title. This question is in response to the following paragraph found at the end of Prof. Hamkins' answer to my MathOverflow question, Are ITTM's necessary to compute Turing's " …
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1
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2
answers
831
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Is there an analogue of the Lost Melody Theorem in ordinary recursion theory and if not, why...
In their arXiv preprint, "Infinite Time Turing Machines" (arXiv:math/9808093v1 [math.LO] 21 Aug 1998) Hamkins and Lewis state the Lost Melody Theorem for ITTM's as follows:
Lost Melody Theorem 4.9 [p …
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1
vote
0
answers
313
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Is there a second-order expression of '$\kappa$ is Reinhardt' in $NGB$, where $\kappa$ is a ...
In their paper, "Generalizations of the Kunen inconsistency", (Annals of Pure and Applied Logic 163 (2012) 1872-1890, doi:10.1016/j.apal.2012.06.001, arXiv:1106.1951), Hamkins, Kirmayer, and Perlmutte …
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0
votes
Regarding Gentzen's note regarding 'Godel-points' (i.e., "Where is the Godel-point hiding?")
In his answer to David Roberts' mathoverflow question, "$Z_2$ versus second-order $PA$" (question 97077), Prof. Ali Enayat writes (Under the subheading, "Regarding the second question)":
One way t …
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4
votes
0
answers
260
views
Proof theory and subsystems of second-order arithmetic: in particular the reverse mathematic...
While doing some research on reverse mathematics, I came across the following document under the address, http://www.andrew.cmu.edu/user/avigad/Talks/survey1.pdf:
Proof theory and Subsystems of Se …
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0
votes
3
answers
1k
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Regarding Gentzen's note regarding 'Godel-points' (i.e., "Where is the Godel-point hiding?")
Consider the following note written by Gerhard Gentzen in early 1932, on the onset of his work on a consistency proof for arithmetic:
The axioms of arithmetic are obviously correct, and the princi …
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2
votes
What is induction up to $\varepsilon_0$?
The answer to your question can be found in Maria Hameen-Anttila's paper, Nominalistic Ordinals, Recursion on Higher Types, and Finitism, Bulletin of Symbolic Logic, 25(1), 101-124 (2019) doi:10.1017/ …
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1
vote
1
answer
462
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What does "can almost be proven in PA" mean regarding Theorem 2 of Timothy Chow's expository...
In his expository article, "The Consistency of Arithmetic" (MSN), Prof. Chow has the following theorems:
Theorem 1. If $a_1, a_2, a_3,\dotsc$ is a sequence of ordinals and $a_i \ge a_j$ whenever …
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6
votes
3
answers
3k
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The Lucas argument vs the theorem-provers -- who wins and why?
In his paper, "Minds, Machines and Gödel", J.R. Lucas writes the following:
Gödel's theorem [First Incompleteness Theorem, that is—my comment] must apply to cybernetic machines, because it is of t …
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2
votes
Is V, the Universe of Sets, a fixed object?
I believe the answer to your question revolves around correcting a subtle confusion between classes and sets in the Cumulative Hierarchy. This can be shown by reference to Samuel Coskey's Senior Thes …
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0
votes
1
answer
614
views
What are the difficulties involved in proving that the Kunen inconsistency holds in $NGB$
or (contrariwise) that $NGB$ + "There exists a Reinhardt cardinal" is consistent?
The question is partially in the title. $NGB$ is used for the reasons stated in the Hamkins, Kirmayer, and Perlmutte …
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2
votes
Can we add set complements on top of ZF?
I believe that an answer to your question [1] is the system that Dana Scott developed in his paper, "Axiomatizing Set Theory" found in Proceedings of Symposia in Pure Mathematics, Volume 13, Part II, …
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3
votes
Is there a physically realizable inductive turing machine that can solve Hilbert's $10$th pr...
Consider the following paper, written by A. Steven Younger, Emmett Redd, Hava Siegelmann, and Conrad Bell:
"A Physical Machine Based on a Super-Turing Computational Model" [found under title on th …
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2
votes
0
answers
323
views
The universe and multiverse views of set theory from the perspective of $ZFC^2$
(Note: the 'Second-order $ZFC$' ($ZFC^2$) I am talking about is the theory [in the second order language of set theory consisting of a single non-logical symbol $\in$ ] consisting of the axioms Exte …
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