# 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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### 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**

votes

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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 ...

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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 $\kappa$-...

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votes

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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 $\mathcal{L}$...

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votes

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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 ...

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votes

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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/...

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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}'\...

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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 ...

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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 ...

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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} = \...

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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),...

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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 ...

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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 ...

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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 ...

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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 "...

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**1**answer

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-...

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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 ...

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**1**answer

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**

votes

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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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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 ...

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**2**answers

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–...

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**2**answers

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**

votes

**3**answers

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 ...

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votes

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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 ...

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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 ...

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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 ...

**18**

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?