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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.
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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? …
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1
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Thinning directed sets ${\frak P}$ of partitions of $\omega$ with no ${\frak P}$-discrete su...
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 … Let $\mathfrak P$ is a family of partitions of $\omega$. …
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5
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Is it true that $\mathit{MA}(\omega_1)$ iff $\omega_1<\mathfrak{p}$?
Recall that
$\mathfrak{p}=\min\{|F|: F$ is a subfamily of $[\omega]^{\omega}$ with the sfip which has no infinite pseudo-intersection $\}$. … The cardinal $\mathfrak{q}_0$ defined as the smallest cardinality of a subset of $\mathbf{R}$ which is not a $Q$-space.
Q1. Is it true that $\mathit{MA}(\omega_1)$ iff $\omega_1<\mathfrak{p}$ ?
Q2. …
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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 $\mathfrak{p}\cdot\mathfrak … {m} = \mathfrak{q}\cdot\mathfrak{m}$? …
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Is $\mathfrak q_0$ equal to the smallest cardinality of a second-countable $T_1$-space which...
It is known that $\mathfrak p\le\mathfrak q_0\le\mathfrak b$. … The standard proof of the inequality $\mathfrak p\le\mathfrak q_0$ gives a bit more:
Any second-countable $T_1$-space of cardinality $<\mathfrak p$ is a $Q$-space. …
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Cardinal characteristics of the ideal of $\sigma$-continuity of the Pawlikowski function
On the other hand, it can be shown that $\mathrm{non}(\mathcal I_P)\ge\mathfrak p$.
It is well-known that the strict inequality $\mathfrak p<\mathrm{non}(\mathcal M)$ is consistent. … Which of the inequalities $\mathfrak p<\mathrm{non}(\mathcal I_P)$ and $\mathrm{non}(\mathcal I_P)<\mathrm{non}(\mathcal M)$ is consistent?
Problem 2. …
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14
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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"). …
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Is $2^\mathfrak{m}<2^\mathfrak{n}\Rightarrow\mathfrak{m}<\mathfrak{n}$ equivalent to the axi...
Lindenbaum and Tarski assert in ``Communication sur les recherches de la théorie des ensembles'' without proof that $\mathfrak{p}^\mathfrak{m}<\mathfrak{p}^\mathfrak{n}\wedge\mathfrak{p}\neq0\Rightarrow … This is not difficult to prove, but I do not know whether the statement for $\mathfrak{p}=2$ is also equivalent to the axiom of choice. …
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When are classes with prescribed reducts "pseudo"-elementary?
Another necessary condition is that $\{0\},\{1\} \in \mathfrak L(H) \subseteq \{\{0\},\{1\}\} \cup \mathfrak P(\mathbb N_{\geq 2})$ . … set $\mathfrak P(\mathbb N)$, while for an element $L \in \mathfrak P(\mathbb N)$, we have $\mathfrak L(L)$ if and only if $L \in \mathfrak L(H)$. …
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Calculate the $\downarrow$, $\downarrow\uparrow$ and $\uparrow\downarrow$ cofinalities of th...
It is clear that $$\max\{{\uparrow\downarrow}(\mathfrak P),{\downarrow\uparrow}(\mathfrak P)\}\le \min\{{\downarrow}(\mathfrak P),{\uparrow}(\mathfrak P)\}.$$
I would like to know the values of the … Calculate the $\downarrow$-cofinality ${\downarrow}(\mathfrak P)$ of the poset $\mathfrak P$. In particular, is ${\downarrow}(\mathfrak P)=\mathfrak c$? Or ${\downarrow}(\mathfrak P)=\mathfrak d$? …
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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 …
CommunityBot
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The existence of $T$-ultrafilters in ZFC
A partition $\mathcal P$ is called finitary if $\sup_{P\in\mathcal P}|P|$ is finite.
Let $\mathfrak P$ is a family of partitions of $\omega$. … An infinite subset $T\subset\omega$ is called $\mathfrak P$-thin if for any partition $\mathcal P\in\mathfrak P$ there exists a finite set $F\subset T$ such that for any $P\in\mathcal P$ the intersection …
CommunityBot
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Is $\mathfrak p=\omega_1$ equivalent to the existence of a Hausdorff gap without infinite ps...
The definition of the small uncountable cardinal $\mathfrak p$ implies that under $\mathfrak p>\omega_1$ each Hausdorff gap has an infinite pseudointersection. … What does happen under $\mathfrak p=\omega_1$?
Problem. Is $\mathfrak p=\omega_1$ equivalent to the existence of a Hausdorff gap without infinite pseudointersection? …
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Compactifications with remainder $[0,\omega_1]$ and convergent sequences
The statement $(\star)$ does not hold under $\omega_1<\mathfrak p$. That is why I am asking only about the consistency of $(\star)$.
Remark 2. …
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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 $\mathfrak p< \mathfrak b < \mathfrak q$? … $\min\{\mathfrak b, \mathfrak q \}\in \{\mathfrak p,\mathfrak q \}$ ? …
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