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Results tagged with lo.logic
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user 2362
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
204
views
Can there be a segment of regular cardinals with the tree property capped by an almost-stron...
Recall that a cardinal $\kappa$ is $(\lambda,\infty)$-almost-strongly-compact if every $\kappa$-complete filter can be refined to a $\lambda$-complete ultrafilter. A cardinal $\mu$ has the tree proper …
- 64k
2
votes
1
answer
131
views
Is the Lascar group over $A$ trivial when $T=T^{eq}$ and $A = acl(A)$?
Let $T$ be a first-order theory which eliminates imaginaries, and let $A$ be an algebraically closed set in a model of $T$. Let $Gal_L(T[A])$ be the Lascar group of the theory $T[A]$, which is $T$ wit …
- 64k
7
votes
2
answers
527
views
What is the status of the assertion "There are arbitrarily large cardinals with the tree pro...
Of course, if you want your cardinals with the tree property to be strongly inaccessible, then you're asking about weakly compact cardinals. But what if you don't want them to be strongly inaccessible …
- 64k
7
votes
0
answers
212
views
Automorphisms that preserve every algebraically closed set
Let $T$ be a theory, and $A$ be an algebraically closed set. Let $L$ be the lattice of algebraically closed subsets of $A$. Can one understand the kernel of the natural map $\chi: Aut(A) \to Aut(L)$? …
- 64k
7
votes
1
answer
540
views
Vopenka's principle is equivalent to the existence of a strong compactness cardinal for any ...
According to Cantor's attic, Vopenka's principle is equivalent to the existence of a strong compactness cardinal for any "logic". But I can't find a definition of what a "logic" is either there or in …
- 64k
8
votes
1
answer
729
views
Stationarity and Fodor's lemma for a (nice) poset?
The notion of a stationary set is peculiar in that it applies to subsets of certain very particular posets -- ordinals or powersets. At least to a non-set-theorist, the situation seems to beg for the …
- 64k
12
votes
1
answer
466
views
Stone duality for the algebra of Boolean functions such that $f(\top,\dots,\top) = \top$, or...
$\newcommand\FinSet{\mathit{FinSet}}\newcommand\FinBool{\mathit{FinBool}}\newcommand\FreeFinBool{\mathit{FreeFinBool}}\newcommand\Set{\mathit{Set}}\newcommand\Psh{\mathit{Psh}}$It's well-known that th …
- 64k
7
votes
2
answers
345
views
How many elementary embeddings can there be?
If $T$ is a complete first-order theory and $\kappa$ is a cardinal, let $\mathrm{Mod}_\kappa(T)$ be (a skeleton of) the category of $\kappa$-small models of $T$ (i.e. of cardinality $<\kappa$), with e …
- 64k
2
votes
0
answers
219
views
Countable Fodor's Lemma?
Does Fodor's lemma fail for countable ordinals?
For the usual statement of Fodor's lemma to make sense, one needs well-behaved notions of club and stationary sets, which fail for countable ordinals, s …
- 64k
2
votes
1
answer
230
views
Is any real closed extension of $\mathbb R$ characterized up to isomorphism by its ladder?
Let $R$ be a real closed field. Recall that the ladder of $R$ is the divisible, ordered abelian group obtained by quotienting $R$ by a certain equivalence relation.
Note that $R$ has trivial ladder if …
- 64k
10
votes
0
answers
505
views
Sunflower / $\Delta$-system lemma in a more general poset?
The sunflower lemma (or $\Delta$-system lemma) may be viewed as a statement about the poset $P_\omega(\omega_1)$, and the generalized sunflower lemma may be viewed as a statement about the poset $P_\l …
- 64k
10
votes
1
answer
592
views
Is Vopenka's Principle + "ORD has the tree property" consistent?
Vopenka's principle implies the existence of weakly compact cardinals (a proper class of them, I believe). My question is whether Vopenka's principle is consistent with the assertion that the universe …
- 64k
4
votes
1
answer
299
views
Can a Vopenka cardinal be supercompact?
Can a Vopenka cardinal be supercompact?
I asked a weaker question on here before. Unfortunately, I don't know enough set theory to see whether the positive answer there generalizes to a positive answe …
- 64k
12
votes
1
answer
678
views
Can there be a small complete category in ZF?
It's a ZFC theorem of Freyd, any small complete category is a preorder. Freyd's theorem continues to hold in any Grothendieck topos. But Hyland showed it fails in some elementary toposes.
I don't actu …
- 64k
11
votes
2
answers
1k
views
Does ZF+AD have any unusual arithmetic consequences?
Motivation:
This question is motivated by wondering to what extent "natural" theories are linearly ordered (or at least ordered in a directed manner) by their (first-order) arithmetic consequences, i …
- 64k