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Results tagged with lo.logic
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user 4600
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.
3
votes
What is the name for Boolean algebra's version of $\models$ between sets of identities and i...
It seems the name for this idea is equationally complete theory, see page 30 of Walter Taylor's Equational Logic survey.
Not every theory is like that:
for example in the theory of lattices, which is …
- 24.8k
13
votes
How did the Baker-Gill-Solovay paper come to be?
Apparently, we would have gotten at least half of the BGS result without any of the three named authors and also without any of the 4 people they credit, all we needed was Dekhtiar. 😊
The Annals of t …
- 24.8k
13
votes
Examples of $\aleph_0$-categorical nonhomogeneous structures
How about: dense linear order with endpoints.
It's $\aleph_0$-categorical by the same proof as for the case without endpoints.
It's not homogeneous because of the endpoints.
- 24.8k
2
votes
Examples of statements with a high quantifier complexity
The Interchange lemma in the theory of formal languages has a $\Pi_5$ form:
$\forall L$, if $L$ is a context-free language, then
$\exists c$, $c$ is a positive integer, such that
$\forall n\ge 2$, $R …
- 24.8k
2
votes
Source on smooth equivalence relations under continuous reducibility?
Seems like this structure must be pretty complicated. For example, consider Brownian motion $\{W_t\}_{t\ge 0}$ with the equivalence relations
$$t\sim_\omega s\iff W_t(\omega)=W_s(\omega).$$
Here $\ome …
- 24.8k
20
votes
Accepted
Why is weak Kőnig's lemma weaker than Kőnig's lemma?
The issue is that for a finitely branching subtree $T$ of $\omega^{<\omega}$, the function $f$ mapping $\sigma$ to the greatest $n$ such that the concatenation $\sigma ^\frown n$ is in $T$ may not be …
- 24.8k
9
votes
Accepted
Is the equational theory of groups axiomatized by the associative law?
Yes. It suffices to show that any free semigroup embeds in a group.
For this I refer you to MO question 3235:
Let $F$ be a free semigroup (say, $2$-generated) which is embedded in a group $G$, an …
- 24.8k
4
votes
How does proof assistant organize knowledge?
Of course a hard part is to know whether two similar-looking lemmas are really related, and even more whether two superficially very different statements might have a short proof of their equivalence. …
- 24.8k
2
votes
Variously pointed closed sets
Let $\mu$ be a measure on $2^\omega$ which doesn't have a least Turing degree.
This exists by Theorem 4.2 of
Day, Adam R.; Miller, Joseph S., Randomness for non-computable measures, Trans. Am. Ma …
- 24.8k
2
votes
Downward density of w-REA sets under arithmetic reducibility?
Probably still open. James Barnes' dissertation (2018) addresses initial segments under the arithmetic reducibility, but is not specifically about $\omega$-CEA degrees.
Barnes, James S., On the deci …
- 24.8k
2
votes
Accepted
Correct Proof Of ZBC Theorem From Odifreddi? Also Extension Question
A proof of Harrington’s ZBC Lemma can be found in Theorem 2.5 of
Hinman, Peter G.; Slaman, Theodore A., Jump embeddings in the Turing degrees, J. Symb. Log. 56, No. 2, 563-591 (1991). ZBL0745.03036. …
- 24.8k
1
vote
Semantics/Syntax distinction vs. Meta/Object language distinction
Meta/object $\to$ syntax/semantics:
Once you have the distinction between a meta-language and a formal language, you certainly have the idea of a language. And the very idea of a language (as opposed …
- 24.8k
3
votes
Can we have a theory $T$ that is complete for simple sentences in the language of $T$ that a...
Partial answer:
If there is such a theory $T$, it would mean that every simple sentence that's independent of $T$, must decide Con($T$).
Note that every $\Pi^0_1$ sentence is equivalent to a simple …
- 24.8k
5
votes
Accepted
1
vote
What are the definable sets in Skolem arithmetic?
The fundamental theorem of arithmetic states that the monoid of positive integers under multiplication is a free commutative monoid on an infinite set of generators, the prime numbers.
So you get wha …
- 24.8k