Questions tagged [higher-order-logics]

For logics which admit quantification over sets/relations/functions/etc. rather than merely over individuals. The most common example is second-order logic.

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5
votes
1answer
173 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 ...
5
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1answer
190 views

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
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1answer
297 views

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 $\kappa$-...
6
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1answer
246 views

Can $\mathsf{Ord}$ be weakly compact from a second-order perspective?

Working in $\mathsf{ZFC}$ + "There is a weakly compact cardinal" and letting $\kappa$ be the least weakly compact cardinal, say that a logic $\mathcal{L}$ is loraxian iff every $\mathcal{L}$...
7
votes
1answer
293 views

Compatibility of Łośian phenomena in second-order logic

(Throughout, all ultrafilters are nonprincipal.) Given a property $P$ - really, a sentence in some appropriate logic - say that a ultrafilter $\mathcal{U}$ on a cardinal $\kappa$ averages $P$ iff for ...
2
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1answer
218 views

Abstraction logic

As part of my research on building an interactive theorem proving system, I have discovered a new logic that I call Abstraction logic. I have written up the details here: https://doi.org/10.47757/...
6
votes
1answer
289 views

Failure of "directedness" for second-order logic?

Say that a logic $\mathcal{L}$ is directed iff whenever $\mathfrak{A}\equiv_\mathcal{L}\mathfrak{B}$ there is some $\mathfrak{C}$ with $\mathcal{L}$-elementary substructures $\mathfrak{A}'\...
2
votes
1answer
202 views

Second-order strong minimality and amorphousness, take 2

Recently I asked a question about whether a second-order analogue of strong minimality could correspond to amorphous satisfiability (= having a model whose underlying set cannot be partitioned into ...
7
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2answers
587 views

Can second-order logic identify "amorphous satisfiability"?

Recall that a set is amorphous iff it is infinite but has no partition into two infinite subsets. I'm interested in the possible structure (in the sense of model theory) which an amorphous set can ...
7
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2answers
234 views

Are there structures in a finite signature that are recursively categorically axiomatizable in SOL but not finitely categorically axiomatizable?

Recall that a structure $\mathcal{M} = \langle M, I^\sigma_M \rangle$ in a signature $\sigma$ is categorically axiomatized by a second-order theory $T$ when, for any $\sigma$-structure $\mathcal{N} = \...
4
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1answer
238 views

Candidate "AEC-yielding" fragments of bad logics

Previously asked and bountied on MSE without success: Given a logic $\mathcal{L}$ and a signature $\Sigma$, let the $\Sigma$-system of $\mathcal{L}$ be the pair $Sys_\Sigma(\mathcal{L})=(Struc(\Sigma),...
4
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1answer
284 views

Compactness number for a fragment of second-order logic

Previously asked and bountied without response at MSE. This question is a companion to this one, about a tame(?) fragment of second-order logic with the standard semantics, $\mathsf{SOL}$, motivated ...
8
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0answers
217 views

How many steps does it take to "Tarski-Vaughtify" second-order logic?

Given a regular logic $\mathcal{L}$, let $\preccurlyeq_\mathcal{L}$ be the usual elementary submodelhood relation for $\mathcal{L}$. There is also a separate submodelhood relation coming from the ...
6
votes
2answers
357 views

Analogues of worldly cardinals for (an unusual version of) second-order $\mathsf{ZFC}$

Let $\mathfrak{ZFC}(\mathsf{SOL})$ be the theory in second-order logic (with the standard semantics) gotten from $\mathsf{ZFC}$ by modifying the Separation and Replacement schemes to apply to ...
14
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2answers
616 views

How special is first-order $\mathsf{PA}$?

This is a modified version of a question which was asked and bountied at MSE without success. Below, "$\mathsf{PA}$" refers to first-order Peano arithmetic. There are various "...
5
votes
1answer
180 views

Can a nonstandard model of $\mathsf{PA}$ be "$\Delta^1_1$-well-ordered?"

This was asked and bountied at MSE with no response: My question is the following: Is there a nonstandard model $\mathcal{M}\models\mathsf{PA}$ such that $\mathcal{M}$ has no $\Delta^1_1$-with-...
6
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0answers
156 views

Reference requestion: theorem guaranteeing self-embeddings of expansions of $\mathit{Ord}$

This is an attempt to locate a theorem I vaguely remember but cannot find (which arose in the context of Consistency of non-trivial elementary embedding $j : \mathit{Ord} \to \mathit{Ord}$ and ...
15
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1answer
566 views

Can $Ord$ have nontrivial second-order elementary self-embeddings?

I forgot to mention originally: this was motivated by this old MSE question. It's not hard to show in $\mathsf{ZFC}$ that there are nontrivial elementary embeddings from $(Ord; \in)$ to itself - or ...
13
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0answers
387 views

Does Foundation increase the strength of second-order logic?

Thinking about the recent threads on structural consequences of the Axiom of Foundation (AF) over ZF-AF, I've been trying to find some conservativity result which explains why AF doesn't seem to have ...
7
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1answer
572 views

Preservation results in abstract logics

In retrospect the original version of this question was impossibly bloated. Here's a better version: There are many results about when first-order sentences are preserved by algebraic operations on ...
18
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2answers
986 views

What sort of cardinal number is the Löwenheim–Skolem number for second-order logic?

In their paper “On Löwenheim–Skolem–Tarski numbers for extensions of first order logic”, Magidor and Väänänen make the following statement: “For second order logic, $\mathrm{LS}(L^{2})$ [the Löwenheim–...
12
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2answers
870 views

The (non-)absoluteness of second-order elementary equivalence

Elementary equivalence is set-theoretically absolute between any two transitive models of set theory; this is also true for the infinitary logics - e.g., $\mathcal{L}_{\omega_1\omega}$ - at least, ...
15
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3answers
818 views

Vopěnka's Principle for non-first-order logics

(For simplicity, the background theory for this post is NBG, a set theory directly treating proper classes which is a conservative extension of ZFC.) Vopěnka's Principle ($VP$) states that, given any ...
9
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2answers
902 views

Are there any complete, first-order and unstable theories which have non-categorical second-order formulations?

Since it's not stable, $PA$ fails at being categorical in a power in the worst possible way, having $2^{\lambda}$ models in any uncountable $\lambda$. But $PA$ regains its categoricity in the move to ...
10
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3answers
1k views

Categoricity in second order logic

Hi, It's shown by an easy cardinality argument that there are complete second-order theories that are not categorical (have more than one model up to isomorphism). Anyone knows of a concrete example ...
12
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4answers
1k views

ultrapowers and higher order logic

One of the reasons I think ultrapowers are interesting is the following corollary of Łoś's theorem: Let $V$ be a relational structure and $^*V$ an ultrapower of $V$. Then a first order statement ...
18
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2answers
2k views

Algebrization of second-order logic

Is there an algebrization of second-order logic, analogous to Boolean algebras for propositional logic and cylindric and polyadic algebras for first-order logic?