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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Topos semantics of constructive higher order logic

I would like to find a reference that describes the semantics of constructive higher order logic with function types in toposes. In particular, it seems that if we are to take function types as ...
Trebor's user avatar
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6 votes
1 answer
300 views

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 ...
user avatar
4 votes
1 answer
142 views

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 ...
Noah Schweber's user avatar
8 votes
1 answer
349 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}...
Noah Schweber's user avatar
5 votes
1 answer
210 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 ...
Noah Schweber's user avatar
11 votes
1 answer
606 views

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 ...
Noah Schweber's user avatar
9 votes
0 answers
247 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 ...
Noah Schweber's user avatar
1 vote
0 answers
195 views

Can Frege set extensions of second order unary predicates serve as a foundation for mathematics?

To second order logic, add a primitive partial one place function symbol "$\epsilon$" from unary predicate symbols [upper cases] to object symbols [lower cases]. Define: $ x=y \equiv_{df} \...
Zuhair Al-Johar's user avatar
3 votes
0 answers
120 views

At which large cardinal property this second order ordinal arithmetic stops?

Language: Second order logic, with as usual predicates written in upper case, and objects in lower case. Let $<$ be a primitive constant binary relation symbol. Equality between objects is ...
Zuhair Al-Johar's user avatar
12 votes
1 answer
372 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 ...
Noah Schweber's user avatar
4 votes
0 answers
150 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 $\...
Noah Schweber's user avatar
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 ...
Noah Schweber's user avatar
5 votes
1 answer
245 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 ...
Noah Schweber's user avatar
5 votes
1 answer
244 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 ...
Keshav Srinivasan's user avatar
5 votes
1 answer
545 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 $\kappa$-...
Noah Schweber's user avatar
6 votes
1 answer
314 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 $\mathcal{L}$...
Noah Schweber's user avatar
8 votes
1 answer
386 views

Does "agreement on cardinalities" imply second-order elementary substructurehood?

Say that a logic $\mathcal{L}$ satisfies the weak test property iff for all $\mathfrak{A}\subseteq\mathfrak{B}$ we have $(1)\implies(2)$ below: For each $\mathcal{L}$-formula $\varphi$ with ...
Noah Schweber's user avatar
7 votes
1 answer
351 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 ...
Noah Schweber's user avatar
2 votes
1 answer
344 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/...
Steven Obua's user avatar
6 votes
1 answer
357 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}'\...
Noah Schweber's user avatar
2 votes
1 answer
215 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 ...
Noah Schweber's user avatar
7 votes
2 answers
654 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 ...
Noah Schweber's user avatar
6 votes
2 answers
268 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} = \...
Beau Madison Mount's user avatar
4 votes
1 answer
269 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),...
Noah Schweber's user avatar
4 votes
1 answer
378 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 ...
Noah Schweber's user avatar
9 votes
0 answers
446 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 ...
Noah Schweber's user avatar
8 votes
2 answers
478 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 ...
Noah Schweber's user avatar
15 votes
2 answers
971 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 "...
Noah Schweber's user avatar
5 votes
1 answer
249 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-...
Noah Schweber's user avatar
6 votes
0 answers
182 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 ...
Noah Schweber's user avatar
16 votes
1 answer
757 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 ...
Noah Schweber's user avatar
6 votes
0 answers
243 views

Number of models vs. complexity for SOL theories

This was previously asked at MSE without success. Suppose $T$ is a complete first-order theory with continuum-many countable models up to isomorphism. We define two sets of Turing degrees associated ...
Noah Schweber's user avatar
13 votes
0 answers
445 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 ...
Elliot Glazer's user avatar
7 votes
1 answer
602 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 ...
Noah Schweber's user avatar
18 votes
2 answers
1k 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–...
Thomas Benjamin's user avatar
13 votes
2 answers
1k 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, ...
Noah Schweber's user avatar
16 votes
3 answers
977 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 ...
Noah Schweber's user avatar
9 votes
2 answers
988 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 ...
Kate Hodesdon's user avatar
10 votes
3 answers
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 ...
Carlos Sáez's user avatar
12 votes
4 answers
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 ...
M T's user avatar
  • 2,681
19 votes
2 answers
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?
nikmil's user avatar
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