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Results for "mathfrak p" tagged with set-theory
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forcing, large cardinals, descriptive set theory, infinite combinatorics, cardinal characteristics, forcing axioms, ultrapowers, measures, reflection, pcf theory, models of set theory, axioms of set theory, independence, axiom of choice, continuum hypothesis, determinacy, Borel equivalence relations, Boolean-valued models, embeddings, orders, relations, transfinite recursion, set theory as a foundation of mathematics, the philosophy of set theory.
28
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Can one divide by the cardinal of an amorphous set?
The ultimate question is whether this is actually provable in ZF:
If $\mathfrak{m}$ is an infinite cardinal, must there be two cardinals $\mathfrak{p} \neq \mathfrak{q}$ such that $\mathfrak{p}\cdot\mathfrak … {m} = \mathfrak{q}\cdot\mathfrak{m}$? …
- 44.4k
27
votes
4
answers
3k
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Nilradicals without Zorn's lemma
Every proof I found (e.g. in the classical "Commutative Algebra" by Atiyah and Macdonald) uses Zorn's lemma to prove that $x \notin Nil(A) \Rightarrow x \notin \cap_{\mathfrak{p}\in Spec(A)} \mathfrak{ … p}$ (the other way is immediate). …
- 1,042
14
votes
2
answers
666
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Are there interesting examples of theorems proved using ‘height’ extensions?
$\mathfrak{p}=\mathfrak{t}$, remarkable cardinals, Todorčević and Farah's book "Some applications of the method of forcing"). …
- 613
13
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1
answer
354
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Is the Martin's axiom number $\mathfrak m$ regular
This is because
$$\mathfrak m \leq \mathfrak p \leq \mathfrak c$$
where $\mathfrak p$ is the pseudo-intersection number. … Hence $\mathfrak m = \mathfrak c$ implies $\mathfrak m = \mathfrak p$ and $\mathfrak p$ can be shown to be regular. …
- 245
12
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3
answers
755
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The continuum hypothesis for packing shapes without overlapping
For $S\subseteq\mathbb{R}^n$, let $\mathfrak{p}(S)$ be the supremum of the cardinalities of disjoint packings of copies of $S$ inside $\mathbb{R}^n$. … In case the continuum hypothesis holds, either $\mathfrak{p}(S)$ is countable or $\mathfrak{p}(S)=2^{\aleph_0}$; however, in general this need not be true. …
- 21.1k
12
votes
1
answer
1k
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Hausdorff gaps and $\mathfrak{p}=\mathfrak{t}$
Recently Malliaris and Shelah proved that $\mathfrak{p}=\mathfrak{t}$
(see: http://math.uchicago.edu/~mem/Malliaris-Shelah-CST.pdf). … of the form $\omega^\omega/G$
in the case $\mathfrak{p}<\mathfrak{t}$. …
user44578
12
votes
1
answer
533
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Building the real from Dedekind finite sets
It is also known that the reals may contain a Dedekind finite set, so they are Dedekind finite union of disjoint sets (if $\mathfrak p<\mathfrak c$ is Dedekind finite then $\frak p\times c=c$ so we may … partition $\Bbb R$ into $\frak p$ many parts)
Is it possible to get a "better" construction of the reals using Dedekind finite sets? …
- 1,676
11
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1
answer
400
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The Parovichenko cardinal, is it equal to $\max\{\aleph_2,\mathfrak p\}$?
These two results yield the inequality $\mathfrak P\ge\max\{\aleph_2,\mathfrak p\}$.
So, under CH we have $\mathfrak P=\aleph_2>\mathfrak c=\mathfrak p=\aleph_1$. … Is it consistent that $\mathfrak P>\max\{\aleph_2,\mathfrak p\}$? …
- 41.8k
11
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1
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627
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A new cardinal characteristic (related to partitions)?
A family $\mathfrak P$ of partitions of $\omega$ is called directed if for any two partitions $\mathcal A,\mathcal B\in\mathfrak P$ there exists a partition $\mathcal C\in\mathfrak P$ such that each set … An infinite subset $D\subset\omega$ is called $\mathfrak P$-discrete if for any partition $\mathcal P\in\mathfrak P$ there exists a finite set $F\subset D$ such that for any $P\in\mathcal P$ the intersection …
- 41.8k
10
votes
0
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495
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Is there a model of set theory in which $\mathfrak p< \mathfrak b < \mathfrak q$?
Is there a model of set theory in which $\mathfrak p< \mathfrak b < \mathfrak q$? … $\min\{\mathfrak b, \mathfrak q \}\in \{\mathfrak p,\mathfrak q \}$ ? …
- 1,101
9
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2
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466
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Small uncountable cardinals related to $\sigma$-continuity
It is easy to show that $\bar\sigma<\mathfrak q_0$ implies that $\bar\sigma=\sigma$.
Problem 1. Is $\bar \sigma=\sigma$ in ZFC?
Problem 2. Is $\mathfrak p\le \bar\sigma$?
Problem 3. … It is known that $\mathfrak q_0\le\mathfrak q\le \log(\mathfrak c^+)\le\mathfrak c$ where $\log(\mathfrak c^+)=\min\{\kappa:2^\kappa>\mathfrak c\}$. …
- 41.8k
8
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0
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241
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Topological applications of $\mathfrak{p}=\mathfrak{t}$
{p}$ has some topological property, then $\mathfrak{t}$ also has it). … So, my question is: are there any interesting consequence of $\mathfrak{p}=\mathfrak{t}$ in Topology? …
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2
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Characterizations of infinite compact Abelian groups and probability spaces based on the for...
Let $\kappa$ be an infinite cardinal and let $\mathfrak{g}_\kappa$ consists of compact abelian groups of size $\kappa.$ What is the cardinality of $\{[G]_\equiv : G \in \mathfrak{g}_\kappa \}? … Let $\kappa$ be an infinite cardinal and let $\mathfrak{p}_\kappa$ consists of probability spaces of size $\kappa.$ What is the cardinality of $\{[G]_\equiv : G \in \mathfrak{p}_\kappa \}? …
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7
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1
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283
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Can $\Delta^{1}_{2}$ separate degrees of constructibility?
I wonder whether this can be strengthenend in the following way:
Given appropriate largeness assumptions (existence of generic filters, large cardinals...), at least one of $A$ and $\mathfrak{P}(\omega …
- 437
7
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0
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553
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Recent application of model theory in set theory by Shelah-Malliaris
Recently one of the oldest open problems in set theory about the cardinal invariants of the continuum (i.e the question of whether $\mathfrak{p}=\mathfrak{t}$) was solved by Shelah and Malliaris (see " …
user39869