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Results tagged with lo.logic
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user 8628
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.
60
votes
7
answers
9k
views
In what respect are univalent foundations "better" than set theory?
It was an ambitious project of Vladimir Voevodsky's to provide new foundations for mathematics with univalent foundations (UF) to eventually replace set theory (ST).
Part of what makes ST so appealing …
23
votes
2
answers
3k
views
Does the "three-set-lemma" imply the Axiom of Choice?
Consider the following curious statement:
$(S)$ $\;$ Let $X$ be a non-empty set and let $f:X \to X$ be fixpoint-free (that is $f(x) \neq x$ for all $x\in X$). Then there are subsets $X_1, X_2, X_3 …
- 48.9k
20
votes
1
answer
534
views
Almost orthogonal maps $f:\omega \to \{-1,1\}$
Let $\omega$ denote the set of non-negative integers. For sets $A,B$, let $B^A$ denote the set of maps $f:A\to B$. For $f,g\in\{-1,1\}^\omega$ we say that $f,g$ are almost orthogonal if there is $C_0\ …
- 48.9k
13
votes
1
answer
932
views
Cantor-Bernstein with "weakly injective" functions
Let us call a map $f: X \to Y$ between non-empty sets a "weak injection" if $f^{-1}(\{y\})\subseteq X$ is finite for every $y \in Y$.
Recall that the (Schroeder-)Cantor-Bernstein-Theorem (sometimes ab …
- 48.9k
12
votes
1
answer
678
views
Graphs $G$ with $G \cong \text{Aut}(G)$
Let $G=(V,E)$ be a simple, undirected graph. By $\newcommand{\Aut}{\text{Aut}}\Aut(G)$ we denote the collection of graph isomorphisms $\varphi:G\to G$. We let $$E(\Aut(G)) =\big\{\{\varphi, \psi\}:\va …
- 48.9k
12
votes
1
answer
291
views
Can the cardinal $2^{\aleph_0}$ be order-embedded in ${\cal P}(\omega)/(\text{fin})$?
For $A,B\in{\cal P}(\omega)$ we say $A\subseteq^* B$ if $A\setminus B$ is finite (that is, $A$ is "almost contained" in $B$). We write $A\simeq_{\text{fin}} B$ if $A\subseteq^* B$ and $B\subseteq^* A$ …
- 48.9k
11
votes
1
answer
446
views
Fixed points of injective self-maps
Is it consistent in $\mathsf{ZF}$ that there is a set $X$ with more than $1$ point such that every injective map $f:X\to X$ has a fixed point?
- 48.9k
11
votes
2
answers
1k
views
Meta-undecidability
Could there be an undecidable statement $S$ in ${\sf ZFC}$ of which one will never be able to prove its undecidability for principal reasons (ie we will never know that $S$ is undecidable)?
If this …
- 48.9k
11
votes
1
answer
746
views
Generalized limits on $\ell^\infty(\mathbb{N})$
Let $\ell^\infty(\mathbb{N})$ denote the set of bounded real sequences $(a_n)_{n\in\mathbb{N}}$. The $\lim$ operator is a partial linear operator from $\ell^\infty(\mathbb{N})$ to $\mathbb{R}$. With t …
- 48.9k
10
votes
1
answer
2k
views
Does the axiom of choice follow from the statement "Every simple undirected graph is either ...
Using the Well-Ordering Principle, which is equivalent to the Axiom of Choice, it can be proved that
(S): for every simple, undirected graph $G$, finite or infinite, either $G$ or its comple …
- 48.9k
10
votes
2
answers
579
views
Maximal Abelian subgroups of $S_\omega$
Let $S_\omega$ be the group of permutations (bijections) $\varphi:\omega\to\omega$, together with composition as binary operation.
Zorn's Lemma implies that every commutative subgroup of $S_\omega$ is …
- 48.9k
10
votes
2
answers
973
views
Size of maximal intersecting families
Let $X$ be a non-empty set, and let ${\cal S}\subseteq {\cal P}(X)$ be family of non-empty subsets of $X$. We say that ${\cal S}$ is intersecting if any two members of ${\cal S}$ have non-empty inters …
- 48.9k
10
votes
1
answer
262
views
Does every linear cover contain a minimal cover?
This is a follow-up question to an older question.
Let $X\neq \emptyset$ be a set. We say that ${\cal C}\subseteq {\cal P}(X)$ is a cover if $\bigcup {\cal C} = X$, and we call ${\cal C}$ linear if $| …
- 48.9k
10
votes
2
answers
234
views
Additive, multiplicative, and Dedekind infiniteness in ${\sf (ZF)}$
We call a set $X$
Dedekind-infinite if there is an injective map $f:X\to X$ that is not surjective,
addititvely infinite if $X \neq\emptyset$ and there is an injective map $f:\big((X\times\{1\})\cup( …
- 48.9k
10
votes
0
answers
368
views
Model for "$\kappa$ limit cardinal iff $2^\kappa$ limit cardinal"
Is there a model of ${\sf (ZFC)}$ such that in the model we have that $\kappa$ is a limit cardinal if and only if $2^\kappa$ is a limit cardinal?
- 48.9k