# Questions tagged [lo.logic]

first-order and higher-order logic, model theory, set theory, proof theory, computability theory, formal languages, definability, interplay of syntax and semantics, constructive logic, intuitionism, philosophical logic, modal logic, completeness, Gödel incompleteness, decidability, undecidability, theories of truth, truth revision, consistency.

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### Is there a foundational approach that takes “structure” as primitive?

As per the title, I'd be curious to know if there have been attempts at constructing a foundation of mathematics taking, somehow, purely the notion of "structure" as primitive, maybe via a system of ...
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### Bounding and domination numbers for relation $\leq$ modulo $\omega$-nullsets

We say that $A\subseteq \omega$ is a nullset if $$\lim\sup_{n\to \infty} \frac{|A\cap n|}{n+1} = 0.$$ Let $\omega^\omega$ denote the set of functions $f:\omega\to\omega$. We define a pre-ordering ...
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### Can we have cyclic generalized positive comprehension?

In positive set theory, the axiom scheme of generalized positive comprehension in $GPK^+_\infty$ [of Olivier Esser] is stated in a manner as to forbid the symbol of the asserted set to occur in the ...
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### Could groups be used instead of sets as a foundation of mathematics?

Sets are the only fundamental objects in the theory $\sf ZFC$. But we can use $\sf ZFC$ as a foundation for all of mathematics by encoding the various other objects we care about in terms of sets. The ...
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### Is this anti-foundation axiom consistent?

Starting with ZFC: Replace axiom of Extensionality by weak Extensionality (nonempty sets having the same members, are identical) Remove axiom of Foundation. Add the anti-foundation axiom which ...
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### Undetermined Banach-Mazur games in ZF?

This question was previously asked and bountied on MSE, with no response. This MO question is related, but is also unanswered and the comments do not appear to address this question. Given a ...
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### Set of definable real numbers?

Is there a set theory at least as strong as $KP\omega$ which has as a theorem that there is a set $\mathbb{D}$ of precisely the definable real numbers?
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### How bad a proper forcing of size $\aleph_1$ can be?

This question concerns proper forcings of size $\aleph_1$. In the context of $\rm ZFC+\neg CH$, I couldn't find any counter example to the following property. Suppose $\mathbb P$ is a proper forcing ...
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### Is there a ring for which the reducibility of a polynomial is undecidable?

Let $R$ be a ring such that all of its elements have a finite number of divisors, ie $\forall r\in R\, |\{x\in R: x|r\}|<\infty$. Then we can decide whether a polynomial in $R[t]$ is reducible ...
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### Forcing square introduces diamond

Let $\mathbb S_\kappa$ be the standard forcing for $\square_\kappa$ by initial segments. This is $(\kappa+1)$-strategically closed. Observation: Let $T \subseteq \kappa^+$ be stationary. If $T$ ...
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### Improving a Lindstrom-y fact about $\mathcal{L}_{\omega_1,\omega}$?

See e.g. the last section of Ebbinghaus/Flum/Thomas for the relevant background on abstract model theory. Below, all languages are finite for simplicity. "$HC$" is the set of hereditarily countable ...
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### Elementary self-embeddings conservative over ZFC

Question: Is the following theory conservative over ZFC? And if not, what is its strength? Language: $∈$, $j$ (unary function symbol) Axioms: 1. ZFC (without separation and replacement for formulas ...
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### “Robinson arithmetic” for (some) levels of $L$?
I'll write "$\mathcal{L}_\alpha$" for the fragment $\mathcal{L}_{\infty,\omega}\cap L_\alpha$. Say that a countable admissible $\alpha$ is Robinsonian if there is some sentence \$\varphi\in\mathcal{L}...