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.
5,792 questions
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Are there different levels of homomorphisms/isomorphisms between formal theories?
I will start with an analogy. We know that $R^2$ and $R$ are isomorphic as sets, but not isomorphic as vector spaces or topological spaces. So we have different notions of isomorphisms, each notion ...
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1
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122
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Order-preserving injection $\iota:[0,1]\to X$ for large $X\subseteq [0,1]$ [duplicate]
Inspired by this older question: If $X\subseteq [0,1]$ such that $|X|=2^{\aleph_0}$, is there necessarily an order-preserving injection $\iota:[0,1]\to X$?
- 54.4k
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Optimal Subsystem of $\mathsf{PA}_{2}$ for Proving Existence of Set of Gödel Codes of True Arithmetic
Let $\mathcal{N}$ denote the standard model of first-order Peano arithmetic, $\mathsf{PA}$, and let true arithmetic, $\mathsf{Th}(\mathcal{N})$, be the set of sentences in $\mathsf{L}_{\mathsf{PA}}$ ...
- 33
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Sets which inductively define themselves (with just "$+$")
This is related to the second part of this old question of mine.
For $A\subseteq\mathbb{N}$ let $\mathfrak{N}_A=(\mathbb{N};0,1,+,A)$ be the expansion of Presburger arithmetic with (a predicate naming)...
- 19.6k
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110
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Collapsing cardinals to extend the ranges of parameters of large cardinal properties
There are quite a lot large cardinal properties that have an ordinal parameter with bounded value. I'm curious that can we extend their range by collapsing cardinals.
For example, consider the $\alpha$...
- 1,802
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116
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Is Cardinal Choice equivalent to AC?
Define: $X=\operatorname {Frege} (A) \iff \\ X= \{ y \mid \exists z: y= \{ a \subseteq A \mid a\text { is bijective to } z \}\}$
So $\operatorname {Frege}(X)$ is the set of all nonempty equivalence ...
- 13.2k
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330
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Logic-Compactness in complete lattice
Assume $\Gamma$ is an infinite set composed of formulas (of finite length), and $A$ is a formula (of finite length). For example, $\Gamma=\{x,y\sqcup z,x_1,x_2,x_3,\cdots\}$, $A=(x\sqcap y)\sqcup(x\...
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What is the minimal strength a logic should have to be able to express, say, first order logic?
I am trying to figure out what would qualify as something like the mother of all logics. The motivation of this question comes from studying some model theory. There I came across this interesting ...
- 1,693
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Axioms of Soft Logic
Someone here bumped into the papers by Moshe Klein & Oded Maimon on Soft Logic? I try to understand whether their axioms actually exclude the zero-product property.
Here are the axioms from one of ...
- 115
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The strength of arithmetical Baire category theorem
The (second-order) Baire category theorem for open sets given by codes is provable in the base theory RCA$_0$ of reverse mathematics (see Simpson's SOSOA).
Is the following version of the Baire ...
- 4,792
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Do we have a hierarchy/generalisations of effectively axiomatised theories, just like we have a hierarchy of Turing machines?
The notion of an effectively axiomatised theory is based on computing abilities of a Turing machine. For instance, the wffs and proofs are required to be decidable, the theorems are required to be ...
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Ramsey ultrafilters after adding splitting reals
This is a reference request. Are there any results of the following kind?
Assume $\mathrm{CH}$ and let $\mathcal{U}$ be a Ramsey ultrafilter. Let $c$ be a Cohen real. Then in $V[c]$, can $\mathcal{U}\...
- 151
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334
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Cutsets and antichains in ${\cal P}(\omega)$
If $(P,\leq)$ is a poset, an antichain is a set $A\subseteq P$ such that for all $a\neq b\in A$ we have $a\not\leq b$ and $b\not\leq a$. A chain is a subset $C\subseteq P$ such that for all $c,d\in C$ ...
- 54.4k
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470
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Is this very short proof of Zorn’s Lemma correct? [closed]
I recently came across a recent paper presenting a proof of Zorn’s Lemma that seems very short and elementary. I found it interesting because the proof does not use transfinite induction or any set ...
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How to interpret ZU in a finitely ranked infinite set version of it?
The language of this set theory, in addition to $\in$, contains a unary predicate $A$
for urelements. $\mathrm{Set}(x)$ abbreviates $\lnot A(x)$. The axioms (and axiom schemes) of
Z, modified to allow ...
- 13.2k
14
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643
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Reference request: determinacy and Lebesgue-measurability locally
I've heard it said many times times that $\boldsymbol{\Pi}^{1}_{n}$-determinacy implies $\boldsymbol{\Sigma}^{1}_{n+1}$-Lebesgue measurability (hence for instance $n$ many Woodin cardinals with a ...
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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 ...
- 13.2k
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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.
$...
- 1,529
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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 ...
- 19.6k
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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$ ...
- 19.6k
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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 ...
4
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131
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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(...
- 18.1k
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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$ ...
- 54.4k
6
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1
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367
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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: ...
- 19.6k
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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 ...
- 235
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639
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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)...
- 4,792
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276
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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 ...
- 38.8k
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Decision problem for finite (unordered) trees
One of the strongest results on the decidability of theories is Rabin's Tree Theorem. One way to state it is the following: tThe problem of deciding whether a sentence on the monadic second order (MSO)...
- 545
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353
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On the class forcing used in Jensen's proof of Con(CH+SH)
It seems that Jensen's proof of the consistency of CH + SH used class forcing, but the revelant properties are not clearly verified. I haven't learnt about class forcing, so I wonder whether it is ...
- 83
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Adding special trees
For a regular cardinal $\kappa$, a $\kappa$ tree $T$ is called special when there is a regressive function $f : T \to T$ (regressive in the tree order) so that the inverse image of every point is the ...
- 21k
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If $X'$ computes $\mathcal{O}^{Y}$ must $X$ compute $Y$?
If $X'$ computes $\mathcal{O}^{Y}$ must $X$ compute $Y$? If not is there a function $\Gamma$ which guarantees that if $X'$ computes $\Gamma(Y)$ then $X$ computes $Y$?
It is easy enough to see that ...
- 4,039
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333
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Is van Dalen’s “open problem” about $\bf{CT}$ and indecomposability actually open?
In the paper "How connected is the intuitionistic continuum", D. van Dalen proves that in intuitionistic mathematics, the set $\mathbb{R} \setminus \mathbb{Q}$ is indecomposable, which means ...
- 1,073
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Nonstationary support iterations
In Friedman and Magidor - The Number of Normal Measures, the authors use a nonstationary support iteration of posets, rather than the more customary countable or Easton support iterations: conditions $...
- 2,205
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Relation of two notions of n-ineffable cardinal
There are two notions of n-ineffable. One is standard, defined in Baumgartner's paper, used by Harvey Friedman: κ is n-ineffable iff for every 2-coloring of $[\kappa]^{n+1}$, there is a stationary ...
- 1,802
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Determinacy transfer $\mathbf\Delta^1_{2n}\text{-}\mathsf{Det}\to\mathbf\Pi^1_{2n}\text{-}\mathsf{Det}$ over a subsystem of second-order arithmetic
It is known that for $n\ge 1$, $\mathbf{\Delta}^1_{2n}$-Determinacy implies $\mathbf{\Pi}^1_{2n}$-determinacy, but as far as I know, this is a theorem in $\mathsf{ZFC}$. It brings the following first ...
- 3,426
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216
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Inner characterization of constructible steps?
Define a set theoretic closed formula $\Phi$ internally saying that ``a given transitive set is equal to G"odel's $L_\alpha$ for some ordinal $\alpha$'',
in the sense that if $X$ is transitive ...
- 2,441
15
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872
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(AC) and existence of basis of $\{0,1\}^X$ for any set $X$
For any set $X$, let $\{0,1\}^X$ be the collection of all functions $f:X\to\{0,1\}$. We make it into a vector space over the field $\mathbb{F}_2$ by endowing it with pointwise addition modulo $2$ and ...
- 54.4k
3
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208
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Strong conservativity and extra structure in free objects
The background for my question is somewhat related to this; there a very interesting paper is provided, but the setting and examples are somewhat different. I can add any necessary background or ...
2
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1
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417
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Unprovable statements and generic properties
I should start with the following disclaimer that I know virtually no logic, sorry forgive me if my questions are ill-posed. I appreciate that all of this is probably completely obvious to a logician, ...
- 3,118
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2
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1k
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Defining Lebesgue non-measurable sets with countable information
Is there a formula $\phi$ in the language of set theory such that
$$
\text{ZFC proves } \exists x \in \mathbb{R}:\text{ the set }A_x:=\{y\in\mathbb{R}:\phi(x,y)\} \text{ is not Lebesgue measurable?}
$...
- 237
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421
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What are some nice consequences of Martin's conjecture?
I was reading last evening a paper which talked a bit about behaviors on cones and I know for a fact that Martin's conjecture is one of the most active areas in computability and that it was settled ...
- 1,483
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2
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1k
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How big can determined sets be?
I vividly remember seeing an affirmative answer to this question presented in seminar, but I can't track down a citation nor can I prove it myself, so now I'm doubting it's actually true:
Working in $...
- 19.6k
13
votes
1
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524
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When determinacy pulls back
Let $f:\mathbb{R}\rightarrow\omega_1$ be the "usual" surjection (send $r$ to the ordinal coded by $r$ according to some fixed reasonable coding system if such an ordinal exists, and to $0$ ...
- 19.6k
12
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0
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198
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Reverse mathematics of $\mathbb{\Sigma^1_2}$-measurability
Analytic sets are projections of Borel sets, and are known to be Lebesgue measurable (in fact universally measurable). The question of whether measurability of analytic sets can be shown in some ...
- 661
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187
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Multi-step-definability of quantifiers, take 2
Let $\mathsf{QFL}$ be the quantifier-free fragment of first-order logic. We can recursively build a sequence of extensions of $\mathsf{QFL}$ by adding at each successor stage all quantifiers which are ...
- 19.6k
-1
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1
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Comparing arbitrary choice functions with choice functions defined by well-orderings
In order to formulate my questions I need first to give a number of definitions. Also to help the reader I prove two preliminary relevant results.
Given an infinite set $A$, let ${\cal P}^*(A)$ be ...
