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Results tagged with lo.logic
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user 5017
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.
2
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2
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Question about Godel's 2nd Theorem
Let Con(PA) be the sentence of arithmetic which translates as "Peano Arithmetic is consistent." Then according to Godel's 2nd incompleteness theorem, assuming PA is consistent then PA can neither prov …
- 4,597
1
vote
1
answer
186
views
Can a class of arithmetical statements containing its own soundness condition be closed unde...
Given a class $C$ of arithmetical sentences,
an arithmetical theory $T$ is said to be $C$-sound if
all the theorems of $T$ which are in $C$ are true.
For instance, $T$ is $\Sigma_1$-sound if all th …
- 4,597
5
votes
2
answers
1k
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What is the state of research on Horn Angles?
The ancient Greeks struggled with the concept of a horn angle, the "angle" formed by the intersection of two curves. The only information I find in Mathworld is that horn angles are examples of non-A …
- 4,597
1
vote
4
answers
892
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Why can't an explicit well-ordering of the reals be ruled out in ZF?
The statement A = "There exists a well-ordering of the reals" is independent of ZF. My understanding is that the statement B = "There exists an explicit well-ordering of the reals" is also independen …
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16
votes
1
answer
2k
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Fulfilling Pythagoras' Dream using Nonstandard Models of Arithmetic and/or Surreal Numbers
Pythagoras and his followers believed that the Universe was made of numbers. Specifically, they thought that if you compared any magnitudes of the same kind, say the lengths of two objects, you would …
- 4,597
23
votes
1
answer
2k
views
Can we axiomatize Omnific Integers without the Surreal Number system?
Omnific integers are the counterpart in the Surreal numbers of the integers. The surreal numbers are usually defined using set theory, and then the omnific integers are defined as a particular subset …
- 4,597
8
votes
1
answer
396
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What subsystem of second-order arithmetic is needed for the recursion theorem?
In its simplest version, the recursion theorem states that for any $m\in\mathbb{N}$ and any function $g:\mathbb{N}\rightarrow\mathbb{N}$, there exists a function $f:\mathbb{N}\rightarrow\mathbb{N}$ su …
- 4,597
8
votes
1
answer
463
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What subsystem of third order arithmetic proves the real numbers are Dedekind complete?
Reverse mathematics is mainly about subsystems of second-order arithmetic, but in recent years it’s expanded to cover subsystems of third-order arithmetic as well. Now the fact that the real numbers …
- 4,597
5
votes
1
answer
943
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Does a nonlinear additive function on R imply a Hamel basis of R?
A function is additive if $f(x+y) = f(x) + f(y)$. Intuitively, it might seem that an additive function from R to R must be linear, specifically of the form $f(x) = kx$. But assuming the axiom of cho …
- 4,597
-3
votes
3
answers
828
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Can different extensions of ZF have contradictory consequences for first-order arithmetic?
My question is basically, does there exist a statement X independent of ZF such that ZF + X implies a statement P of first-order arithmetic, but ZF + not X implies not P?
Now X cannot be the axiom …
- 4,597
13
votes
0
answers
422
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Is it an open problem whether fast-growing hierarchies can be defined without fundamental se...
Googology Wiki says this, concerning the relation between fast-growing hierarchies defined for all countable ordinals, and the existence of a system of assigning a canonical fundamental sequence to ea …
- 4,597
3
votes
0
answers
352
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Understanding a part of Friedberg’s Priority Argument Paper
This is Richard Friedberg’s original 1957 proof of the Friedberg-Muchnik Theorem, the origin of the ground-breaking priority argument. (The result proven is that there are two recursively enumerable …
- 4,597
5
votes
1
answer
270
views
Does there always exist a categorical extension of $ZFC_2$ with no set models?
$ZFC_2$, i.e. second-order Zermelo-Fraenkel set theory with Choice, has only one proper class model upto isomorphism, namely $V$. But it may or may not also have set models. If $V$ has no inaccessib …
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9
votes
0
answers
960
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Has anyone pursued Frege's idea of numbers as second-order concepts?
Gottlob Frege was a pivotal figure in the history of mathematical logic. He gave an analysis of numbers that proceeded along roughly the following lines, in his books "The Foundations of Arithmetic" …
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5
votes
1
answer
862
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Why is adopting Russell's Axiom of Reducibility as strong as eliminating the Ramified Hierar...
In order to respond to concerns of impredicativity, Bertrand Russell developed a system of ramified second-order logic, which is like regular second-order logic except the comprehension schema is divi …
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