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Results tagged with lo.logic
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user 9896
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
votes
Accepted
Godel 's Ladder: Undecidable PI_N sentences for N =2, 3, ......
The halting set relative to an oracle deciding $\Pi_{n-1}$ sentences, provides such a sentence. This set is $m$-complete for $\Pi_n$.
- 3,475
1
vote
5
answers
2k
views
Why do we ignore non-standard finite sets in ordinary mathematics?
Here is a how a typical proof might look like in group theory--- Suppose we are given a finite group $G$. Enumerate the elements $g_1, \dots, g_n$. Now consider a formula $\phi(g_1, \dots, g_n)$ which …
- 3,475
1
vote
5
answers
345
views
"local variables" in first-order formulas
Often in logic, you want to define a formula $\phi(x)$ which "says something about $x$". For example, $\phi(x)$ may say that $x$ is a prime. In order to form $\phi$, you may need internal bound variab …
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14
votes
3
answers
3k
views
Definition of relativization of complexity class
Is there any general definition, for a class $C$ of languages, what is the relativized class $C^A$ for an oracle $A$?
Usually, these classes and their relativizations seem to be defined in an ad-hoc …
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4
votes
1
answer
725
views
Proof system with same complexity as "informal mathematics"?
The Completeness Theorem in first-order logic states that any mathematical validity is derivable from axioms. Hence, any informal mathematical proof (which is rigorous) can be translated into a formal …
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15
votes
1
answer
813
views
Undecidable theories easier than $Q$
Most proofs of undecidability for various theories (pure logic with binary relation, group theory, etc.) show that the natural numbers and Robinson's $Q$, in one form or another, can be encoded approp …
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5
votes
2
answers
825
views
Semantic definition of sentence
This is a follow-up to question Completeness vs Compactness in logic 68788. One common theme was that compactness in logic is a purely semantic notion, so should have no need of completeness.
The def …
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23
votes
10
answers
5k
views
Completeness vs Compactness in logic
One standard approach to showing compactness of first-order logic is to show completeness, of which compactness is an easy corollary. I am told this approach is deprecated nowadays, as Compactness is …
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5
votes
Are there examples of nonconstructive metaproofs?
If the proof system is recursively axiomatizable, this situation cannot occur.
If there exists a proof of $\Theta$, there exists an algorithm to find that proof. Namely, search the recursively enumer …
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1
vote
Accepted
Equivalence of monadic axioms
The comment showed that decidability of axiom equivalence is implied by decidability of pure logic. (I.e. to decide if $\Theta$ and $\Theta'$ are equivalent, it is equivalent to decide if $\vdash \The …
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16
votes
2
answers
3k
views
Can randomness add computability?
I have been looking at Church's Thesis, which asserts that all intuitively computable functions are recursive. The definition of recursion does not allow for randomness, and some people have suggested …
- 3,475
2
votes
Why is it OK to rely on the Fundamental Theorem of Arithmetic when using Gödel numbering?
Here is a way to think about it. When you use something like Fund Theorem of Arithmetic to prove Godel numbering, you are referring to the abstract set $\mathbf N$, which "exists". The Fundamental The …
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0
votes
Seemingly complex logic/set-theoretic puzzle
Based on the answers proposed to this question, I think this problem is ill-defined. The set of possible questions and assignment of truth valuations is not well defined. A fair definition of a "quest …
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42
votes
4
answers
2k
views
Inconsistent theory with long contradiction
What can one say about an inconsistent theory $T$ which has no contradictions (i.e. deductions of $P \wedge \neg P$) of length shorter than $n$, where $n$ is some huge number?
There have been some di …
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3
votes
Graph properties: definability and decidability
Almost any language I can think of (FO theories, etc) isinvariant under isomorphism, i.e. if $\cal A$ satisfies the sentence $\sigma$ then so does $h(\cal A)$ for any isomorphism $h$. Hence if a class …
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