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Results tagged with lo.logic
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user 8133
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
0
answers
149
views
Are the natural powers of two conservatively embedded in $\mathbb{C}$?
This is a followup to this question.
Consider $\mathbb{C}$ as a structure - in the sense of first-order logic - with the graphs of addition and multiplication. Let $\mathcal{X}$ be the substructure wi …
- 20.7k
9
votes
0
answers
211
views
Can $\mathbb{C}$ have a "doppelganger" in $L(\mathbb{R})$ with countable automorphism group?
Working in $\mathsf{ZFC}$ + large cardinals (a proper class of Woodins, to be precise), is there a field $F\in L(\mathbb{R})$ such that $V\models F\cong\mathbb{C}$ and $L(\mathbb{R})\models\vert\mathi …
- 20.7k
12
votes
0
answers
241
views
Is there a decidable theory of arithmetic with a non-collapsing quantifier hierarchy?
This question is very close to this old MSE question of mine, which is still unanswered.
Is there an (ideally reasonably-natural!) expansion of the structure $(\mathbb{N};+)$ in a finite language whos …
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8
votes
1
answer
176
views
Weakly compact cardinals in $L$: how long do branches take to appear?
Throughout, we work in $\mathsf{ZFC+V=L+}$ "There is a weakly compact cardinal," $\kappa$ is the first weakly compact cardinal and "tree" means "subtree of $2^{<\kappa}$ of height $\kappa$"
Despite t …
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5
votes
1
answer
252
views
What oracles make finding isomorphism (of finite structures) easy?
Below, all structures are finite, in a finite language, with underlying set an initial segment of the natural numbers. This has been edited to fix errors pointed out by Emil Jerabek in his answer belo …
- 20.7k
6
votes
1
answer
207
views
Can we computably escape infinitely many functions (allowing partiality)?
Let $(p_i)_{i\in\omega}$ be a uniformly computable sequence of partial functions (i.e. the partial function $q(i,x)=p_i(x)$ is computable) such that infinitely many $p_i$s are total. Must there be a …
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3
votes
0
answers
89
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Can the differential field of d.c.e. reals be nicely construed as a field of functions?
This question is basically a special case of this older question of mine, which is still unanswered.
Let $\mathcal{D}$ be the field of d.c.e. reals; these turn out to be exactly the reals $\alpha$ for …
- 20.7k
9
votes
1
answer
364
views
Can the canonical Eudoxus-real representatives be defined easily?
(See e.g. here for background on the Eudoxus reals, which motivates this question.)
Let $\mathcal{Z}=(\mathbb{Z};+,<)$. Say that a Eudoxus function is an $f:\mathbb{Z}\rightarrow\mathbb{Z}$ such that …
- 20.7k
4
votes
0
answers
254
views
Läuchli's "intermediate thing"
On page 230 of An abstract notion of realizability ..., Läuchli writes the following:
If we drop the restrictions put on $\Theta$, then we get classical logic in one case and an intermediate thing in …
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6
votes
1
answer
181
views
$L(\mathbb{R})$-absoluteness from a proper class of Woodins: source?
For a paper I'm writing I need to use (as a blackbox) the following theorem: if there is a proper class of Woodin cardinals and $G$ is set-generic, then $L(\mathbb{R})$ and $L(\mathbb{R})^{V[G]}$ are …
- 20.7k
14
votes
2
answers
719
views
Are there any non-conjugation "extendible automorphisms" in the category of finite groups?
Let $\mathbf{Grp}$ be the category of groups. Given a subcategory $\mathscr{G}$ of $\mathbf{Grp}$ and $G\in\mathit{Ob}(\mathscr{G})$, a $\mathscr{G}$-extendible map on $G$ will here mean an assignment …
- 20.7k
6
votes
0
answers
217
views
Decidably clarifying ordinals
For a computable$^*$ ordinal $\alpha$ (viewed as an $\{<\}$-structure), say that a first-order sentence $\sigma$ in a relational language $\Sigma\supseteq \{<\}$ decidably clarifies $\alpha$ iff $\alp …
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5
votes
0
answers
80
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Which pseudovarieties have (Eilenberg/Schutzenberger) minimal descriptions?
Suppose $\mathscr{V}$ is a pseudovariety in a countable language $\Sigma$. Say that a minimal description of $\mathscr{V}$ (in the sense of Eilenberg/Schutzenberger rather than Reiterman) is a pair $( …
- 20.7k
3
votes
0
answers
98
views
Counting Eilenberg/Schutzenberger-type definitions of pseudovarieties
See Eilenberg/Schutzenberger, On pseudovarieties for background on pseudovarieties. I've phrased things in terms of pairs-of-sets to avoid some annoying language about multisets. Also, I'm aware that …
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4
votes
1
answer
147
views
Comparing semiring of formulas and Lindenbaum algebra
This is motivationally related to an earlier question of mine.
Given a first-order theory $T$, let $\widehat{D}(T)$ be the semiring defined as follows:
Elements of $\widehat{D}(T)$ are equivalence c …
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