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.
5,777 questions
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Why Zermelo postulated the existence of a set with no finite limit to the ranks of its elements?
The original Zermelo set theory explicitly allowed for urelements.
What was the reason that led Zermelo to formulate the Axiom of Infinity in terms of the existence of a set of the kind that has an ...
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What is the strength of projective determinacy?
Consider the following theories:
$T_1$: $\mathsf{ZFC+PD}$ where $\mathsf{PD}$ is stated as a schema.
$T_2$: $\mathsf{ZFC+PD}$ where $\mathsf{PD}$ is a single sentence in the language of set theory.
$...
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Is the product of "non-coding" forcings also "non-coding"?
Say that a forcing notion $\mathbb{P}$ is slow iff there is some $f:\mathbb{R}\rightarrow\mathbb{R}$ (in $V$) such that for every $\mathbb{P}$-name for a real, $\nu$, we have $\Vdash_\mathbb{P}\exists ...
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General truth functions on a topos
I was reading Topoi from Goldblat and noted that to calculate the disjunction of the internal logic of a category Set, we have to construct a characteristic function of the set:
$$A = \{(1,1), (1,0), (...
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What can be computed without collapsing $\omega_1$
Below work in $\mathsf{ZFC+CH}$ for simplicity.
Say that a (set) forcing notion $\mathbb{P}$ captures a map $f:\mathbb{R}\rightarrow\mathbb{R}$ iff there is some $\mathbb{P}$-name for a real $\nu$ ...
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Is it possible that the Goldbach's conjecture or the twin prime conjecture be undecidable? [duplicate]
There are many famous unsolved problems in number theory that can be formulated by basic concepts. Two examples are
Goldbach's conjecture:
Every even natural number greater than 2 is the sum of two ...
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Characterize nonzero integers via a polynomial in two variables
In a paper published in 1985, Shih-Ping Tung observed that an integer $m$ is nonzero if and only if $m=(2x+1)(3y+1)$ for some $x,y\in\mathbb Z$. In fact, we can write a nonzero integer $m$ as
$\pm3^a(...
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Do we have ${\frak b} \leq {\frak s}$ in ZFC?
Let ${}^\omega\omega$ denote the set of functions $f:\omega\to \omega$. For $f, g \in {}^\omega\omega$ we define
$f\leq^* g$ if there is $N\in\omega$ such that $f(n)\leq g(n)$ for all $n\in \omega$ ...
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Higher analogues of Gandy basis theorem
For $n\in\omega$ and $x$ a real let $C_n^x$ be the canonical $\Pi^1_n(x)$-complete set. E.g. $C_1^x=\mathcal{O}^x$, etc. I recall seeing long ago the fact that, assuming large cardinals (precisely: ...
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Are the powers of a set with the Baire Property in a Polish group a set with the Baire Property?
Let $G$ be a Polish group and let $A\subseteq G$ be a subset with the Baire Property. Does it follow that for any $n\in \mathbb{N}$, the power $A^{n}$ also has the Baire Property?
Of course, if $A$ is ...
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Formalizing the Completeness Theorem given languages of infinite cardinality
I am reading Kunen's books on set theory and logic. In his approach, the metatheory is finitistic (which can be approximated in PRA).
This implies that in the finitistic metatheory, one can do formal ...
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The strength of representing open sets
Is the following (second-order) formula schema provable in ATR$_0$?
Let $\varphi$ be an arithmetical formula satisfying
For all $x, y\in \mathbb{R}$,
we have that $x=_\mathbb{R}y$ implies $\varphi(x)...
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Type defined by quantifying another type
The following material is quoted from A Crèche Course in Model Theory by Domenico Zambella, Section 15.3.
$\mathcal{U}$ is how we denote the Monster model.
For every $a\in\mathcal{U}^{x}$ and $b\in\...
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Is 0# still unique in ZFC without powerset?
Working in $ZFC$, the statement "$0^\sharp$ exists" is often liberally taken to be one of many known equivalent statements.
However, working in $Z_2$ or $ZFC^-$ (with collection, well-...
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Why and how do (classical) reverse mathematics and intuitionistic reverse mathematics relate?
Broadly speaking, the idea of “reverse mathematics” is to find equivalents to various standard mathematical statements over a weak base theory, in order to gauge the strength of theories (sets of ...
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