All Questions
Tagged with model-theory higher-order-logics
33
questions
5
votes
0
answers
174
views
Whence compactness of automorphism quantifiers?
The following question arose while trying to read Shelah's papers Models with second-order properties I-V. For simplicity, I'm assuming a much stronger theory here than Shelah: throughout, we work in $...
- 20.7k
3
votes
0
answers
121
views
Comparing two fragments of SOL with the downward Lowenheim-Skolem property
For $S$ a set of (parameter-free) second-order formulas and $\mathfrak{A},\mathfrak{B}$ structures, write $\mathfrak{A}\trianglelefteq^S\mathfrak{B}$ iff $\mathfrak{A}$ is a substructure of $\mathfrak{...
- 20.7k
5
votes
1
answer
534
views
The "first-order theory of the second-order theory of $\mathrm{ZFC}$"
$\newcommand\ZFC{\mathrm{ZFC}}\DeclareMathOperator\Con{Con}$It is often interesting to look at the theory of all first-order statements that are true in some second-order theory, giving us things like ...
- 4,857
7
votes
1
answer
310
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What are the simplest sentences which might distinguish Zilber’s field from the complex numbers?
Zilber’s field $\mathbb{B}$ is a field of the same size as the complex numbers $\mathbb{C}$, which satisfies the same first-order sentences about $+$ and $\cdot$. If $\mathbb{B}$ also satisfies the ...
user44143
4
votes
1
answer
158
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Source on equality-free second-order logic (nontrivially construed)
Throughout I'm only interested in the standard semantics for second-order logic, and all structures/languages are relational for simplicity.
If defined naively, second-order logic without equality is ...
- 20.7k
8
votes
1
answer
368
views
On the strength of higher-logic analogues of $\mathsf{ZFC}$ + Montague's Reflection Principle
Throughout, I work in $\mathsf{MK}$ in order to be able to conveniently quantify over logics; if one prefers, we can restrict attention to (say) $\Sigma_{17}$-definable logics and work in $\mathsf{ZFC}...
- 20.7k
5
votes
1
answer
224
views
Does second-order logic satisfy Craig interpolation for second-order languages?
(For simplicity, all languages are relational.)
In analogy with first-order languages, say that a second-order language is a set of relation symbols of two kinds: first-order relation symbols and ...
- 20.7k
11
votes
1
answer
636
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Are there quantifiers that require multiple "steps" to define?
(Below I conflate quantifiers and quantifier symbols in a couple places for readability; I can change that if that actually makes things less readable.)
For the purposes of this question, an $n$-ary ...
- 20.7k
9
votes
0
answers
256
views
Can "$\exists\mathcal{X}(R\cong C(\mathcal{X}))$" be expressed in "large" infinitary second-order logic?
Originally asked and bountied at MSE without success:
Say that a ring $R$ is spatial iff there is some topological space $\mathcal{X}$ such that $R\cong C(\mathcal{X})$, where $C(\mathcal{X})$ is the ...
- 20.7k
12
votes
1
answer
388
views
Do second-order theories always have irredundant axiomatizations?
It's a standard exercise to show that every countable first-order theory has an irredundant axiomatization. For uncountable first-order theories, the result is much more difficult and was proved by ...
- 20.7k
4
votes
0
answers
151
views
How big a "scaffold" does second-order logic need to detect its own equivalence notion?
(Previously asked and bountied at MSE:)
Let $\Sigma$ be the language consisting of a single binary relation symbol. Second-order logic can "detect" second-order-elementary-equivalence of $\...
- 20.7k
4
votes
0
answers
171
views
Can SOL characterize its own equivalence notion, without "scaffolding," for graphs?
Consider the following property $(*)_\mathcal{L}$ of a logic $\mathcal{L}$:
$(*)_\mathcal{L}:\quad$ There is no $\mathcal{L}$-sentence $\varphi$ such that for all graphs $\mathcal{A},\mathcal{B}$ we ...
- 20.7k
5
votes
1
answer
260
views
Can the forcing-absolute fragment of SOL have a strong Lowenheim-Skolem property?
Previously asked and bountied at MSE:
Let $\mathsf{SOL_{abs}}$ be the "forcing-absolute" fragment of second-order logic - that is, the set of second-order formulas $\varphi$ such that for ...
- 20.7k
5
votes
1
answer
256
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Does there always exist a categorical extension of $ZFC_2$ with no set models?
$ZFC_2$, i.e. second-order Zermelo-Fraenkel set theory with Choice, has only one proper class model upto isomorphism, namely $V$. But it may or may not also have set models. If $V$ has no ...
- 4,527
5
votes
1
answer
589
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Can there be no "surprisingly averageable" second-order sentences?
Say that a second-order sentence $\varphi$ is averageable iff there exists some infinite cardinal $\kappa$ and some nonprincipal ultrafilter $\mathcal{U}$ on $\kappa$ such that for every $\kappa$-...
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