All Questions
Tagged with forcing cardinal-characteristics
12 questions
4
votes
0
answers
237
views
Where can I find information about this concept of 'dual ideals'?
I have come across the following notion of (what I am calling) dual ideals, and I am looking for any work in which this notion has been considered, and particularly anything about transferring ...
- 2,206
5
votes
0
answers
109
views
Comparing Mathias forcing notions relative to various filters
Let $\mathcal F$ be a (non-principle, non trivial, ...) filter on $\omega$. The Mathias Forcing relative to $\mathcal F$ is the forcing notion $\mathbb M(\mathcal F)$ consisting of pairs $(s, X)$ with ...
- 1,882
5
votes
1
answer
215
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Cofinal well-founded subset in mod finite order
The mod finite order on ${}^\omega \omega$ is defined as $f \leq^\ast g$ if and only if $f(n) \leq g(n)$ except for finitely many $n \in \omega$. My question is: can we always extract a cofinal well-...
- 505
11
votes
1
answer
704
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Is $\mathfrak j_{2:1}=\mathfrak{j}_{2:2}$ in ZFC?
A function $f:\omega\to\omega$ is called
$\bullet$ 2-to-1 if $|f^{-1}(y)|\le 2$ for any $y\in\omega$;
$\bullet$ almost injective if the set $\{y\in \omega:|f^{-1}(y)|>1\}$ is finite.
Let us ...
- 41.9k
5
votes
1
answer
356
views
Calculate the $\downarrow$, $\downarrow\uparrow$ and $\uparrow\downarrow$ cofinalities of the poset of nontrivial finitary partitions of $\omega$
Let $(P,\le)$ be a poset. For a point $x\in P$ let
$${\downarrow}x=\{p\in P:p\le x\}\quad\text{and}\quad{\uparrow}x=\{p\in P:x\le p\}$$be the lower and upper sets of the point $x$, and for a subset $...
- 41.9k
7
votes
1
answer
195
views
Bounding and domination numbers for relation $\leq$ modulo $\omega$-nullsets
We say that $A\subseteq \omega$ is a nullset if $$\lim\sup_{n\to \infty} \frac{|A\cap n|}{n+1} = 0.$$
Let $\omega^\omega$ denote the set of functions $f:\omega\to\omega$. We define a pre-ordering ...
- 48.9k
6
votes
1
answer
266
views
A question on simple $P_{\aleph_2}$-points
This question is motivated by discussion surrounding this MO question.
An ultrafilter $U$ on $\omega$ is a simple $P_{\aleph_2}$-point if it is generated by a sequence $\langle X_\alpha:\alpha<\...
- 7,125
11
votes
1
answer
629
views
A new cardinal characteristic (related to partitions)?
In this post I will discuss some cardinal characteristic of the continuum, related to partitions of $\omega$ and would like to know if it is equal to some known cardinal characteristic.
By a partition ...
- 41.9k
3
votes
2
answers
257
views
How to increase unbounding and dominating numbers for $(\kappa^\lambda,\leq^*)$
Let $\kappa$ be regular and $\lambda\geq\kappa$. For $f, g\in\kappa^\lambda$ say that $f\le^* g$ if the set $\{\gamma<\lambda:f(\gamma)>g(\gamma)\}$ has
size less than $\kappa$. Set
$\mathfrak{...
- 31
9
votes
1
answer
358
views
Effect of adding one Hechler real versus adding two on the meager ideal
In "The Kunen-Miller Chart (Lebesgue Measure, The Baire Property, Laver Reals and Preservation Theorems for Forcing)" by Haim Judah and Saharon Shelah
JSL Vol. 55, No. 3 (Sep., 1990), pp. 909-927 ([...
- 1,882
12
votes
1
answer
330
views
A simple cardinal characteristic associated with $\omega^\omega$
We can define a very simple cardinal characteristic in the following way. Recall the relation $\leq^*$ on $\omega^\omega$ defined by $x\leq^* y$ iff $x(i)\leq y(i)$ for all but finitely many $i$. For $...
- 791
13
votes
2
answers
997
views
Consistency results separating three cardinal characteristics simultaneously
(For information on cardinal characteristics of the continuum aka cardinal invariants see Joel David Hamkins' MO answer here; Andreas Blass's handbook article is an excellent reference.)
Problem 2.3 ...
- 2,701