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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.
38
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4
answers
4k
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Illustrating Edward Nelson's Worldview with Nonstandard Models of Arithmetic
Mathematician Edward Nelson is known for his extreme views on the foundations of mathematics, variously described as "ultrafintism" or "strict finitism" (Nelson's preferred term), which came into the …
- 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 …
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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 …
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13
votes
0
answers
422
views
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 …
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12
votes
3
answers
644
views
Has the Ramified Theory of Types been applied to NBG?
Questions of predicativity are well-studied in the context of arithmetic. We have a base theory, first-order Peano arithmetic. Some people, like Edward Nelson (in chapter 1 of his book) and Charles …
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11
votes
1
answer
395
views
What is the Turing degree of the monadic theory of the real line?
The monadic theory of the real line is the set of all sentences in the monadic second-order language of order which are true in $\mathbb{R}$. In this 1982 paper, Gurevich and Shelah show that true fi …
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10
votes
1
answer
405
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Is $\mathbb{Z}$ universally definable in any number fields other than $\mathbb{Q}$?
In 2009, Jochen Koenigsmann showed that $\mathbb{Z}$ is universally definable in the field $\mathbb{Q}$. My question is, are there any other number fields in which $\mathbb{Z}$ is universally definab …
- 4,597
10
votes
1
answer
868
views
What is the theory of the random poset?
$\DeclareMathOperator\Th{Th}$The random poset is the Fraisse limit of the class of finite posets, just like the random graph is the Fraisse limit of the class of finite graphs? That is, the random po …
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9
votes
0
answers
960
views
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" …
- 4,597
8
votes
1
answer
396
views
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
248
views
What is the lowest complexity definition of $\mathbb{Z}$ in an infinite extension of $\mathb...
In 2009, Jochen Koenigsmann showed that $\mathbb{Z}$ is universally definable in the field $\mathbb{Q}$. And in 2012, Jennifer Park proved a result which implies that $\mathbb{Z}$ is $\exists\forall$ …
- 4,597
8
votes
3
answers
2k
views
Does the Feferman-Schutte analysis give a precise characterization of Predicative Second-Ord...
A definition is called impredicative if it involves quantification over a domain that contains the thing being defined. For instance, if you define hereditary property to be a property which applies …
- 4,597
8
votes
1
answer
463
views
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
7
votes
4
answers
2k
views
How can you formalize the metamathematics conventionally used to state Godel’s theorem?
Gödel’s incompleteness theorem states that for any sufficiently strong formal system $T$ there exists a statement $G$ such that if $T$ is consistent, then $G$ is true but not provable in $T$. But I’m …
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6
votes
1
answer
516
views
What is the Galois group of one ultrapower over another ultrapower?
Let $F$ be a field, let $E$ be a field extension of $F$, and let $U$ be an ultrafilter. Then my question is, what is the relationship between the Galois groups $Gal(\Pi_U E/\Pi_U F)$ and $Gal(E/F)$?
…
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