# 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 ...

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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 ...

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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 ...

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### 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}...

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### 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 ...

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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 ...

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### 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 ...

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### 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} \...

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### 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 ...

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### 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 ...

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### 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 $\...

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### 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 ...

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### 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 ...

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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 ...

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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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### 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 $\mathcal{L}$...

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### 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 ...

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### 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 ...

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### 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/...

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### 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}'\...

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### 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 ...

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### 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 ...

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### 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} = \...

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### 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),...

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### 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 ...

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### 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 ...

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### 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 ...

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### 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 "...

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### 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-...

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### 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 ...

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### 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 ...

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### 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 ...

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### 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 ...

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### 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 ...

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### 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–...

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### 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, ...

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### 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 ...

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### 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 ...

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### 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 ...

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### 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 ...

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### 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?