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Results tagged with lo.logic
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user 2672
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.
4
votes
1
answer
365
views
"Is it possible to give a restricted set-theoretical definition of addition of natural numbe...
In his paper "Restricted set-theoretical defintions in arithmetic" Raphael Robinson cites a problem posed by Tarski:
Is it possible to give a restricted set-theoretical
definition of addition of …
- 9,720
4
votes
2
answers
291
views
Goedelizability and decidability of a property of Peano formulas
Sorry for not knowing the answers to these elementary questions:
Is the property of formulas of the first-order language of Peano arithmetic of "defining a finite set of natural numbers" goedelizable …
- 9,720
1
vote
Categories of logical formulae
I found the answers to a somehow related question on Entailment and Implication @ n-Category Café very enlightening, too.
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0
votes
4
answers
310
views
Deficiency of necessary conditions
Motivation
Consider the situation: You know that
every $x$ that has property $P$ must have property $Q$. $Q$ is a
rather strong condition but not strong
enough to fulfill $P$. What is mi …
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8
votes
5
answers
809
views
Categories of logical formulae
Consider the set of formulas of a logic. If there was only one sort of "unary" deduction $\phi \Rightarrow \psi$ - like $(\forall x)\phi(x) \Rightarrow \phi(a)$ - we would immediately have a category …
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-1
votes
3
answers
1k
views
Naturally definable sets of natural numbers
(This is a follow-up question from over there: Natural models of graphs.)
(And it has a follow-up question over there: Naturally definable sets of natural numbers (2): Can the circle be broken?)
Motiv …
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-1
votes
1
answer
678
views
Naturally definable sets of natural numbers (2): Can the circle be broken?
(follow-up to: Naturally definable sets of natural numbers)
Every formula $\Psi(x)$ in the first-order language of Peano arithmetic defines a set of natural numbers. Some of these sets are finite, ot …
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1
vote
1
answer
364
views
Naturally definable sets of natural numbers (3)
[This shall be the last of a series of questions, see Naturally definable sets of natural numbers (2)]
I cannot explain why I have been so stubborn not to see the most straight-forward definition for …
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3
votes
2
answers
1k
views
Lindenbaum algebras and models
Sorry for this question out of the blue (especially if its answer should be trivial, obvious, or folklore):
(When and how) can we construct models of a consistent
first order theory $T$ from its
Lind …
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4
votes
2
answers
586
views
Formulaic definitions
In Jech's Set Theory, p. 194, I read - as a comment on the definition of ordinal-definable sets ("A set X is ordinal-definable if there is a formula such that [...]") -:
It is not immediate clear …
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24
votes
When are two proofs of the same theorem really different proofs
Maybe this might be of interest: Blass, Andreas; Dershowitz, Nachum; Gurevich, Yuri, When are two algorithms the same?, Bull. Symb. Log. 15, No. 2, 145-168 (2009). ZBL1192.03021. JSTOR link.
Here is a …
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1
vote
0
answers
203
views
A question on definable categories
One way to define a category set-theoretically might be to give four $\in$-formulas (not sets!)
$$\begin{array}{rl}
\mathsf{O}(X)&\text{(“$X$ is an object”)}\\
\mathsf{M}(X,Y,z)&\text{(“$z$ is a morp …
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-2
votes
1
answer
202
views
Natural constructions (not depending on parameters) [closed]
Consider graph clusterings as a prototypical example of (logical) constructions.
Let a clustering of a graph $(V,E)$ be any covering of $V$, i.e. a set $C$ with $\bigcup C = V$.
I am looking for a …
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3
votes
Set theories that do require the existence of urelements?
In the final end I found such a theory, it's called ZFCUA (= Zermelo Frankel set theory with the axiom of choice and unlimited atoms), see Faithful Representation in Set Theory with Atoms by Harvey Fr …
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1
vote
4
answers
1k
views
Condition of possibility = Co-Implication
Sorry, but I do not know another place to post this question.
Condition of possibility is an important philosophical concept. Naively, this concept could be formally defined this way:
$q$ is a c …
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