17
votes
Accepted
Can $Ord$ have nontrivial second-order elementary self-embeddings?
Answering this question would either require refuting choiceless large cardinals or getting close to refuting Woodin's HOD Conjecture.
First, if choiceless cardinals are consistent, one cannot rule ...
- 8,099
15
votes
Second-order ordinal definability
Since you allow arbitrary sets of ordinals in your second-order definitions, all sets will be in $\text{OD}^2$. The reason is that, for any set $x$, we can code $x$ into a set of ordinals as follows. ...
- 73.1k
13
votes
How special is first-order $\mathsf{PA}$?
There are several notable papers, starting with a key paper of Angus Macintyre (Ramsey quantifiers in arithmetic, Model theory of algebra and arithmetic, Lecture Notes in Math., 834, Springer, 1980), ...
- 17.7k
12
votes
Accepted
The "first-order theory of the second-order theory of $\mathrm{ZFC}$"
I understand your theory to be the set of all sentences in the first-order language of set theory that are true in every model of $\newcommand{\ZFC}{\text{ZFC}}\ZFC_2$. This is a perfectly sensible ...
- 236k
10
votes
Accepted
How special is first-order $\mathsf{PA}$?
This argument has a couple of iffy points, but I believe it does work.
In this paper, Shelah introduced a logic $\mathcal{L}(Q_{\mathrm{Brch}})$ which is fully compact, has the property that any ...
- 12.4k
9
votes
Are there structures in a finite signature that are recursively categorically axiomatizable in SOL but not finitely categorically axiomatizable?
Sure - in fact, we can find a countable (and pretty concrete) example.
Let $\mathcal{N}=(\mathbb{N};+,\times)$ be the standard model of (first-order) arithmetic and let $R\subseteq\mathbb{N}^2$ be the ...
- 21.1k
9
votes
Accepted
Analogues of worldly cardinals for (an unusual version of) second-order $\mathsf{ZFC}$
Complementing @JasonChen's answer: Assume ZFC+$I_1$ and let $j:V_{\lambda+1}\to V_{\lambda+1}$ be elementary, so $\lambda$ is the sup of the critical sequence of $j$. Then $V_{\lambda}$ models $\...
- 9,902
9
votes
Accepted
Do second-order theories always have irredundant axiomatizations?
Here is a proof of Reznikoff’s theorem I found among my notes, not quite following Reznikoff’s proof. I stared at it for a while, and it seems to apply to second-order logic just the same; in fact, ...
- 47.1k
8
votes
Accepted
Can the forcing-absolute fragment of SOL have a strong Lowenheim-Skolem property?
Yes, and a proper class of Woodin cardinals suffices. For $n<\omega$ and $X$ a set of ordinals, $M_n(X)$ denotes the minimal iterable proper class model $M$ of ZFC with $n$ Woodin cardinals above $\...
- 9,902
8
votes
Analogues of worldly cardinals for (an unusual version of) second-order $\mathsf{ZFC}$
As discussed in the comment, Theorem 8.5 in the paper Inner Models from Extended Logics: Part 1 by Kennedy, Magidor, and Väänänen gives a consistent answer that, assuming $V=L$, models of $\mathfrak{...
8
votes
Accepted
Failure of "directedness" for second-order logic?
The answer to the question is yes.
Let $\alpha_0<\alpha_1$ be the least ordinals (in the reverse lex order, say) such that $V_{\alpha_0}$ and $V_{\alpha_1}$ have the same second order theory $T$. ...
- 8,099
8
votes
Accepted
Compactness number for a fragment of second-order logic
I think that if you have a logic $\mathcal{L}$ which has downward Lowenheim-Skolem (for theories of arbitrary cardinality, i.e. if $T$ has cardinality $\lambda$ and $N$ is a model $A$ of size $\theta\...
- 9,902
7
votes
Accepted
Topos semantics of constructive higher order logic
Categorical logic texts such as Lambek & Scott's "Introduction to higher-order categorical logic" and Johstone's "Elephant" usually focus on the categorical side of things and ...
- 48.8k
6
votes
Accepted
Can second-order logic identify "amorphous satisfiability"?
Here's a monadic example. The second-order theory of the vector space $\mathbb F_2^{\oplus\omega}$ as a vector space over $\mathbb F_2$ is second-order strongly minimal because given a finite subspace ...
- 1,916
6
votes
What are the simplest sentences which might distinguish Zilber’s field from the complex numbers?
Kruckman's comment is a good answer. A counterexample to Schanuel's conjecture would be one such sentence. For example, a non-zero polynomial $p(x,y)$ with integer coefficients such that $p(\pi,e) = 0$...
- 373
6
votes
Does there always exist a categorical extension of $ZFC_2$ with no set models?
Since "consistent" is a weird notion in the context of second-order set-theories and moreover we can't even directly talk about a second-order theory being true of $V$ within $V$, I think it'...
- 21.1k
5
votes
Accepted
Does "agreement on cardinalities" imply second-order elementary substructurehood?
No, second order logic does not have the weak test property: let $\mathfrak{B}=(\mathbb{R},{<})$ (that is, the real numbers with the only predicate being the usual "less than" order) and ...
- 9,902
5
votes
Accepted
Can a nonstandard model of $\mathsf{PA}$ be "$\Delta^1_1$-well-ordered?"
I believe you don't need this, but assume that there is a strongly inaccessible cardinal $\kappa$. Fix a first-order completion $T$ of $\mathsf{PA}$ and let $\mathcal{M}$ be a saturated model of $T$ ...
- 12.4k
5
votes
Can second-order logic identify "amorphous satisfiability"?
The answer to both questions seems to be negative. In the language having only equality, let $T$ be any completion of the theory of a set admitting an endless linear order. This property is ...
- 236k
4
votes
Candidate "AEC-yielding" fragments of bad logics
It's been so long since I've seen a good AEC question on here! In fact, so long I forgot the account info that gives me enough reputation to post this as a comment.
I think this will get you rather ...
- 508
4
votes
Abstraction logic
These should get you started on syntax with binding:
Pfenning, Eliott: Higher-Order Abstract Syntax
Gabbay: Foundations of nominal techniques: logic and semantics of variables in abstract syntax
...
- 48.8k
4
votes
Accepted
Are there quantifiers that require multiple "steps" to define?
The quantifier $\exists^\infty$ is not definable over $\mathcal{L}_0$.
To prove it, we show that a sentence $S$ tautologised by $\exists^\infty$ is also a tautology for the "always false" ...
- 116
4
votes
Can there be no "surprisingly averageable" second-order sentences?
This doesn't answer your question, but gives some information that
might be useful.
Main Claim: ZFC + "There is a proper class of inaccessible limits of measurable cardinals" + "There ...
- 9,902
4
votes
Accepted
Can $\mathsf{Ord}$ be weakly compact from a second-order perspective?
If $\kappa$ is weakly compact and there is a wellorder of $V_{\kappa+1}$ definable over $V_{\kappa+1}$ without parameters, then second order logic is Loraxian for $V_\kappa$: the least branch through ...
- 8,099
4
votes
Accepted
Definability in pure-second order logic
Let me try to help by explicating the argument Noah mentioned. I think this is part of logic folklore—it amounts at bottom to the facts that every permutation of a pure set is an isomorphism, and ...
- 236k
3
votes
Accepted
On the strength of higher-logic analogues of $\mathsf{ZFC}$ + Montague's Reflection Principle
The first chromatic cardinal is the first Mahlo cardinal. (Per the connection with the reflection in the question, I assume that in the definition, $α$ and the first argument of $c_i$ need not be ...
- 7,535
3
votes
Accepted
Does second-order logic satisfy Craig interpolation for second-order languages?
This is just an expansion of Emil Jerabek's comments above; I've made it CW to avoid reputation gain, and will delete this if he posts an answer of his own.
Craig interpolation can be rephrased as a &...
3
votes
Compatibility of Łośian phenomena in second-order logic
I'm not sure I have a definitive answer, but three nice observations that are too long for comments:
If you relax from ultrafilters to extenders (seen as directed systems of ultrafilters), then there'...
- 508
3
votes
Accepted
Vopěnka's Principle for non-first-order logics
This is a really late answer, but the answer to your question is "No." As a dual to the Theorem 6 that Thomas Benjamin mentions above (which I believe is a result of Stavi), Janos Makowsky ...
- 508
3
votes
Algebraization of second-order logic
Very belatedly, I think it's worth noting that there is some work connecting higher-order logic and cylindric/etc. algebras, at least including the following:
Sagi, A completeness theorem for higher ...
- 21.1k
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