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Results tagged with lo.logic
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user 49
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.
30
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6
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Mathematics without the principle of unique choice
The principle of unique choice (PUC), also called the principle of function comprehension, says that if $R$ is a relation between two sets $A,B$, and for every $x\in A$ there exists a unique $y\in B$ …
- 48.8k
14
votes
1
answer
603
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Ordinal realizability vs the constructible universe
Koepke's paper Turing Computations on Ordinals defines a notion of "ordinal computability" using Turing machines with a tape the length of Ord and that can run for Ord-many steps, and shows that a set …
- 4,378
13
votes
3
answers
2k
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Intuitionistic Lowenheim-Skolem?
Is there a version of the Löwenheim-Skolem theorem in intuitionistic logic? I'm particularly interested in the "downward" form. The standard proof I know uses the Tarski-Vaught test for elementary s …
- 1,467
8
votes
1
answer
331
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Distributivity of ! over?
Has anyone studied a variant of linear logic, or of its semantic counterpart (exponential modalities on linearly distributive categories / $\ast$-autonomous categories / polycategories) for which ther …
- 1,646
15
votes
3
answers
1k
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A rigid type of structure that can be put on every set?
Call a type of structure rigid if any automorphism of such a structure is an identity. (This is a bit different from some other uses of the word, but hopefully I'll be forgiven.) For example, well-o …
- 63.9k
5
votes
1
answer
497
views
Failure of SVC in Grothendieck toposes
The axiom SVC (for "small violations of choice") asserts that there is a set $S$ such that for every set $X$ there is a choice set $A$ such that $X$ is a subquotient of (i.e. admits a surjection from …
- 42.4k
36
votes
3
answers
2k
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Internal logic of the topos of simplicial sets
I am looking for a closed statement (i.e. not depending on any parameter objects) which is true in the internal logic of the topos of simplicial sets, but is not an intuitionistic tautology. Ideally, …
- 47.7k
16
votes
2
answers
839
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Cauchy real numbers with and without modulus
In constructive mathematics there are many possible inequivalent definitions of real numbers. The greatest variety seems to be in Dedekind-style approaches: in addition to "the" Dedekind real numbers …
- 9,683
6
votes
0
answers
147
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Linear logic with storage preserving positives
Has anyone studied a version of linear logic in which the storage modality $!$ preserves the positive connectives and quantifiers $\otimes,\oplus,\exists$? That is, such that we have $!(A\otimes B) = …
- 66.8k
12
votes
1
answer
780
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Is Girard's LU just an embedding of classical and intuitionistic logic into linear logic?
This question is about Girard's system LU, presented in his paper On the unity of logic. Girard starts by giving a "modal" sequent calculus with two zones of both hypotheses and consequents, $\Gamma; …
- 1,646
21
votes
3
answers
3k
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Approximate intermediate value theorem in pure constructive mathematics
The ordinary intermediate value theorem (IVT) is not provable in constructive mathematics. To show this, one can construct a Brouwerian "weak counterexample" and also promote it to a precise counterm …
- 1,616
18
votes
1
answer
1k
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Can Vopenka's principle be violated definably?
One form of Vopenka's principle (a large cardinal axiom) states that no locally presentable category contains a full subcategory which is large (= a proper class) and discrete (= contains no nonidenti …
- 32.5k
9
votes
1
answer
979
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constructive Serre classes
A Serre class (of abelian groups) is a class of abelian groups closed under subgroups, quotients, and extensions. For instance, finitely generated groups and finite groups are both Serre classes.
Ho …
- 66.8k
12
votes
4
answers
823
views
A well-founded relation on lists
Let $A$ be a set equipped with a well-founded relation $<$, let $LA$ be the set of finite lists of elements of $A$, and define a relation $\prec$ on $LA$ such that $\ell \prec m$ if $\ell$ is obtained …
- 21.3k
9
votes
1
answer
832
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Axiom of class collection
One version of the Axiom of Collection says that any surjection $A\to B$ from a class $A$ to a set $B$ is factored through by some surjection $C\to B$ where $C$ is a set.
Note that assuming $B$ is a …
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