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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.
2
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201
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Does $|(X\times\{0\}) \cup (X\times\{1\})| \leq |X|$ for $X$ infinite imply ${\sf AC}$?
Consider the statement
For any infinite set $X$ there is an injection $\varphi$ from $(X\times\{0\}) \cup (X\times\{1\})$ into $X$.
Does this imply the ${\sf AC}$?
- 48.9k
2
votes
1
answer
198
views
Some very weak statements on choice
This is a follow-up question to Does $|(X\times\{0\}) \cup (X\times\{1\})| \leq |X|$ for $X$ infinite imply ${\sf AC}$?
Consider the statements
$(\text{S}1)$ For any infinite set $X$ there i …
- 48.9k
4
votes
1
answer
266
views
Does "Every infinite set is splittable" imply $\mathsf{AC}$? [duplicate]
We say an infinite set $X$ is splittable if there are $X_1, X_2\subseteq X$ with $X_1\cap X_2 = \emptyset$, $X_1\cup X_2 = X$ and there are bijections $\varphi:X_1\to X_2$ and $\psi:X_1\to X$.
Does t …
- 48.9k
4
votes
1
answer
274
views
Is it consistent that the gaps between cardinals $\kappa$ and $2^\kappa$ "get larger and lar...
Is the following statement consistent in $\mathsf{ZFC}$?
For every ordinal $\beta$ there is an ordinal $\lambda_0$ such that for all ordinals $\lambda\geq\lambda_0$ we have $2^{\aleph_{\lambd …
- 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
3
votes
2
answers
348
views
Is it consistent that $\frak{d} < 2^{\aleph_0}$?
Let $\omega^\omega$ denote the set of all functions $f:\omega\to\omega$. We write $f <^* g$ if there is $N\in\omega$ such that $f(n) < g(n)$ for all $n>N$. A set $D\subseteq \omega^\omega$ is said to …
- 48.9k
5
votes
Maximality statements that cannot be proved using $\mathsf{ZL}$
A statement I like very much is:
Any compact topology is contained in a maximal compact topology.
A direct application of Zorn's Lemma doesn't work, as the following argument shows. Cons …
- 48.9k
3
votes
2
answers
232
views
${\frak b}$ and ${\frak d}$ defined with $\leq$ instead of $\leq^*$
Let $\omega^\omega$ denote the collection of all functions $f:\omega\to\omega$. For $f,g\in\omega$ we define
$f\leq g$ if $f(n)\leq g(n)$ for all $n\in\omega$;
$f\leq^* g$ if there is $N\in\omega$ s …
- 48.9k
4
votes
0
answers
124
views
Consistency of an intersection property
If $\kappa$ is an infinite cardinal then we denote by $[\kappa]^\kappa$ the collection of subsets of $\kappa$ that have cardinality $\kappa$. We say that $\kappa$ is intersectionally strange if there …
- 48.9k
2
votes
2
answers
267
views
Meta-incomputability
Is there a set $B$ about which it provably cannot be decided whether it is computable in $\mathsf{ZFC}$?
- 48.9k
3
votes
1
answer
110
views
Function $f:\kappa\to\alpha$ with small fibers where $\alpha\in\kappa$
Is it consistent in $\mathsf{ZF}$ that there is an infinite cardinal $\kappa$, cardinals $\alpha, \beta\in\kappa$ and a function $f:\kappa\to \alpha$ such that for each $x\in\alpha$ there is an inject …
- 48.9k
0
votes
1
answer
426
views
Smallest $\beta$ such that it is provable that $2^{\aleph_\beta} > 2^{\aleph_0}$
What is the smallest cardinal $\beta$ such that it is provable in ${\sf (ZFC)}$ that $2^{\aleph_\beta} > 2^{\aleph_0}$?
- 48.9k
-1
votes
1
answer
247
views
Does $|\kappa^{<\kappa}|=|\lambda^{<\lambda}|$ imply $\kappa=\lambda$?
For sets $A,B$ we write $A\approx B$ if there is a bijection between $S$ sand $B$.
If $\kappa$ is a cardinal, let $\kappa^{<\kappa}$ denote the collection of subsets of $\kappa$ having cardinality $<\ …
- 48.9k
1
vote
0
answers
54
views
Weak Power Hypothesis in ${\sf ZFC}$ [duplicate]
For sets $x, y$, let $x\approx y$ mean that there is a bijection $\varphi:x\to y$.
The Weak Power Hypothesis (WPH) states that
if $x,y$ are sets and ${\cal P}(x)\approx {\cal P}(y)$ then $x\approx y$ …
- 48.9k
6
votes
1
answer
486
views
Decidability of Frankl's union-closed sets conjecture
Is it conceivable that Frankl's union closed sets conjecture is undecidable in $\mathsf{ZFC}$, or is this quite implausible, perhaps due to the "finitistic" nature of the statement, or for some other …
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