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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.
69
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5
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What was Hilbert's view of Gödel's Incompleteness Theorems?
According to Solomon Feferman, in his slide presentation "Three Problems for Mathematics", Hilbert wrote (in regards to Gödel's second incompleteness theorem):
...the end goal [is] to establish as …
- 6,132
21
votes
3
answers
2k
views
In What Way are Set Theorists' 'Experiences' in the CH Worlds Flawed, if Any?
This is in regards to Joel David Hamkins' new paper "IS THE DREAM SOLUTION OF THE CONTINUUM HYPOTHESIS ATTAINABLE?" (look under title in arXiv). I quote from the last paragraph of his paper:
"My chal …
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18
votes
2
answers
1k
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What sort of cardinal number is the Löwenheim–Skolem number for second-order logic?
In their paper “On Löwenheim–Skolem–Tarski numbers for extensions of first order logic”, Magidor and Väänänen make the following statement:
“For second order logic, $\mathrm{LS}(L^{2})$ [the Löwenheim …
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11
votes
Vopěnka's Principle for non-first-order logics
Since the title of your question is, "Vopenka's Principle for non-first-order logics", this, from Magidor's and Vaananen's paper "On Lowenheim-Skolem-Tarski numbers for extensions of first order logic …
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11
votes
Has decidability got something to do with primes?
You might be interested in A. Grzegorczyk's paper "Undecidability without arithmetization." (Studia Logica, 79(2): 163-230, 2005) in which he dispenses with arithmetization altogether (but does not d …
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10
votes
1
answer
831
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Are there 'finitistic' nonrecursive functions (assuming Church's Thesis is false)?
[Note: In what follows, I will be using the same type of argument Laszlo Kalmar did in his paper "An Argument Against the Plausibility of Church's Thesis" found in Constructivity in Mathematics, (Amst …
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8
votes
On independence and large cardinal strength of physical statements
Though this does not directly answer your question, here is a foundational paper that might help one derive results that might answer your question:
Marian Boykan Pour-El and Ian Richards: "Nonco …
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7
votes
Belief in consistency of extremely large cardinals
There is another path that can be used to justify the existence of very large cardinals. Consider, for example, the abstract to Magidor's paper, "On the Role of Supercompact and Extendible Cardinals …
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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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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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5
votes
Is a paraconsistent and provably non-trivial foundation for math possible?
One should note that as regards your question 1, systems of relevant arithmetic with inconsistent models such as $R{\sharp}$, $R{\sharp}{\sharp}$, and the systems $RM3^{i}$ can prove (with finitary pr …
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4
votes
Are Separation and Types the only way to avoid paradoxes?
In point of fact, there are (at least) a couple more ways to circumvent the paradoxes brought about by the use of the unrestricted comprehension axiom
$\exists$$y$$\forall$$x$($x$$\in$$y$$\leftrighta …
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4
votes
Set theoretical multiverse and truths
Since the Fundamental Theorem of Arithmetic is a theorem of $PA$, it holds for both standard and nonstandard models of $PA$. Since one can interpret $PA$ in both $ZFC$ and $GBC$ (e.g., for $ZFC$, it …
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4
votes
0
answers
260
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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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4
votes
Direct axiomatization of ordinal and cardinal numbers
You might consider taking a look at a paper by Athanassios Tzouvaras titled "Cardinality without enumeration" (look under title on the Web), especially at Definition 3.1. Since it is short, I will qu …
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