Questions tagged [cardinal-arithmetic]
The cardinal-arithmetic tag has no usage guidance.
73 questions
-2
votes
0
answers
102
views
Can this theory interpret Peano arithmetic?
Logic: Bi-sorted first order logic with equality, first sort written in lower case range over natural numbers, the second sort written in upper case range over sets of naturals, "$=$" has no ...
- 11.3k
10
votes
2
answers
564
views
Cardinal arithmetic under determinacy
Work in a reasonable theory of determinacy such as $\mathsf{ZF+DC+AD}$. Which of the following identities are true for arbitrary infinite sets?
$|A^2|=|A^3|$ (motivated by an MSE question that asks ...
- 667
3
votes
2
answers
157
views
Modification of Lemma 0 in Hajnal's paper "Embedding finite graphs into graphs colored with infinitely many colors"
I am looking for a proof of the following lemma.
Let $E_0$ be the set of edges of an undirected graph with no loops with vertex set a cardinal $\kappa$. Let $E_1$ be the family of two-element subsets ...
- 1,644
27
votes
1
answer
932
views
A cardinal inequality for finiteness
Nearly ten years ago, I explained in a blog post that, assuming only ZF, a cardinal number $\mathfrak{n}$ is finite if and only if it satisfies this monstrous inequality:
$$2^{2^{2^{2^{\mathfrak{n}}}}}...
- 44.4k
2
votes
2
answers
233
views
Name for a certain type of cardinal
I'm not a set-theorist, but I hope this question is appropriate. This is just a question about names:
Fix a cardinal $\lambda$. I'd like to know if there is a name for regular cardinals $\kappa$ such ...
- 15.7k
15
votes
1
answer
615
views
Changing the cofinality of a regular cardinal without collapsing any cardinals?
I have a short but hopefully interesting question on cardinal arithmetic and collapsing cardinals:
Is it possible to change the cofinality of a regular cardinal without collapsing any cardinals?
Is ...
- 307
8
votes
1
answer
260
views
Can we force $\aleph_\omega^\omega<2^{\aleph_\omega}$?
Since $\operatorname{cf}(\aleph_\omega)=\omega$, $\aleph_\omega<\aleph_\omega^\omega$. However, can we force $\aleph_\omega^\omega<2^{\aleph_\omega}$? I am especially interested in models for ...
- 2,171
5
votes
1
answer
771
views
Cardinals without Axiom of choice?
When comparing the size of sets using 1-1 functions ($ x\approx y$), AC is used to ensure the existence of a unique ordinal $|x|$ which is a $\approx$-representative of each class [$x$].
This in turn ...
- 1,031
-2
votes
1
answer
112
views
How to prove that increasing the number of constant symbols of a first-order logic by the number of formulas keeps the number of formulas the same [closed]
Let $S$ be a set of theory symbols for a first-order logic, and let $C$ be a set of constant symbols in $S$ such that $|C| = |L(S)|$, where $L(S)$ is the set of all formulas generated by $S$ in the ...
- 1
7
votes
0
answers
183
views
Shelah’s Representation Theorem: existence of scales
Let $\lambda$ be a singular cardinal of countable cofinality. Shelah’s Representation Theorem states that there is an increasing sequence of regular cardinals $\langle\delta_n\rangle_{n<\omega}$, ...
- 533
9
votes
3
answers
426
views
Exponentiation of Dedekind cardinals
Question: Let $\mathfrak n$ be an infinite cardinal. In ZF (set theory without the axiom of choice) can either of the implications
$$\mathfrak n=\mathfrak n+1\implies2^\mathfrak n=2^{\mathfrak n+1}\...
- 13.4k
15
votes
1
answer
480
views
Topology and pcf theory
$\DeclareMathOperator\pcf{pcf}$For simplicity say $\aleph_\omega$ is a strong limit. Let $A=\pcf\{\aleph_n:n\in\omega\}$. Then it follows from basic properties of pcf operation that $X\subseteq A\...
- 667
8
votes
1
answer
323
views
A surjection from square onto power: Is limit Hartogs/Lindenbaum number necessary?
I am considering the construction in [Peng—Shen—Wu] in which the authors show the consistency of a set $X$ such that there is a surjection from $X^2$ onto the power set of $X$ (henceforth $\mathscr{P}(...
- 2,171
3
votes
0
answers
222
views
Basic cardinal arithmetic without choice
Do we know everything about addition and multiplication of cardinalities in choiceless set theory?
For example, let $M$ be a model of $\textsf{ZF}+\textsf{AD}+V=L(\mathbb{R})$, consider the sets $\...
- 865
4
votes
0
answers
133
views
Characterizations for SSH and SCH above an uncountable cardinal
SSH asserts that pp$(\lambda) =\lambda^+$ for every singular cardinal $\lambda$. There are two nice characterizations for SCH and SCH in terms of covering numbers (see for example "Large ...
- 533
4
votes
1
answer
678
views
Does Tarski's squaring theorem imply Axiom of Choice in NFU?
I'm trying to see which results from mainstream set theory (ZF) about Axiom of Choice can be proved in New Foundations with Urelements (U is added simply because ...
- 351
5
votes
0
answers
205
views
Does this proof by Shelah use any "hidden assumptions"?
Recall that the approachability ideal for $\kappa$, denoted $I[\kappa]$, consists of all sets $A\subseteq\kappa$ such that there is a sequence $\overline{a}=(a_{\alpha})_{\alpha\in\kappa}$ of bounded ...
- 1,799
16
votes
1
answer
460
views
Can $\kappa^\lambda$ be large if $2^\lambda$ is small and $\lambda<\mathrm{cof}(\kappa)$?
We work in ZFC throughout. The following question was posed to me by a friend:
Can there exist cardinals $\kappa,\lambda$ such that $\lambda<\mathrm{cof}(\kappa)$ and $2^\lambda<\kappa<\...
- 28.2k
5
votes
0
answers
148
views
Existence of a large family of sets with big differences
Let $\lambda\leq\kappa$ be two cardinals and $\Gamma$ be a set with $|\Gamma|=\kappa$.
Question. Does there exist a family $\mathscr{A}\subseteq \{A\subseteq \Gamma: |A|=\lambda\}$ with $|\mathscr{A}|...
- 1,511
2
votes
1
answer
235
views
Continuum function maximum
Easton's theorem can give a very weak nontrivial constraint on continuum function, but it does not hold for singular cardinals. So:
What are the non-trivial constraints on continuum function in ...
- 695
3
votes
1
answer
493
views
Cardinality of infinite towers of Alephs - can tower be more than countable?
Lets define function T as
$$T(0) = \aleph_0$$
$$T(1) = \aleph_{\aleph_0}$$
$$T(2) = \aleph_{\aleph_{\aleph_0}}$$
etc
No finite tower of alephs can reach the first inaccessible cardinal
My questions ...
- 185
2
votes
0
answers
280
views
Can be this "handwaving" idea about "counting" reals somehow put on solid ground?
We know that the Cantor's cardinality of a countable set is $\aleph_0$ and the cardinality of continuum is $2^{\aleph_0}=\aleph_0^{\aleph_0}$. Unfortunately, this measure is based on the idea of ...
- 10.1k
11
votes
1
answer
503
views
Does $\text{AC}_{\text{WO}}$ prove $\Theta \neq \aleph_{\omega+1}?$
The choice principle $\text{AC}_{\text{WO}}$ proves a large amount of cardinal arithmetic. It's well-known to imply DC, that successor cardinals are regular, and that for all $X$, there is $\lambda$ ...
- 4,316
11
votes
1
answer
316
views
Can this result in cardinal arithmetic be established without using pcf theory?
Suppose $\kappa\leq\mu$ are infinite cardinals. Let us agree to call a family $\mathcal{P}\subseteq[\mu]^{<\mu}$ a countably generating family for $[\mu]^\kappa$ if every member of $[\mu]^\kappa$ ...
- 7,125
4
votes
0
answers
176
views
Reference request for an elementary identity in cardinal arithmetic
Let $\kappa$ be an infinite cardinal. Then we have
$$\beth_{\kappa+1} = \prod_{\alpha<\kappa} \beth_{\alpha} = \beth_{\kappa}^{\kappa}$$
The only non-trivial inequality is the first less-than-or-...
- 4,265
5
votes
1
answer
375
views
Minimum cardinality of a cofinal collection of countable subsets of a set
Setup
Let $X$ be a set of cardinality $\kappa\geq \aleph_0$.
Edit:
Based on Todd Eisworth's suggestion:
What is the minimum cardinality of a collection $\hat{X}$ of countable subsets of $X$ such that ...
- 5,407
5
votes
1
answer
473
views
How can we know the well-foundedness of $\epsilon_0$?
I think the question can be quite philosophical, but I see that $WF(\epsilon_0)$ is widely accepted as one of the attributes of the natural numbers.
Gentzen proved $Con(PA)$ with $PRA+WF(\epsilon_0)$....
- 171
3
votes
0
answers
242
views
Cardinal numbers and the Bolzano-Weierstrass theorem
Let $\kappa$ be a cardinal number, define $\textsf{M}(\kappa)$ and $\textsf{BW}(\kappa)$ as follows:
$\textsf{M}(\kappa)$ : For every sequence $(f_{n}:\kappa\to \mathbb{R})_{n\in\mathbb{N}}$ of real-...
- 561
4
votes
1
answer
280
views
Model in $\mathsf{ZFC}$ such that ${\cal P}(\ldots)$ has "jumping property"
Is there a model of $\mathsf{ZFC}$ such that for every cardinal $\beta > \aleph_0$ there is a cardinal $\alpha < \beta$ such that $|{\cal P}(\alpha)|
>\beta$?
- 48.8k
4
votes
1
answer
213
views
Cardinal exponentiations inequality
Let $\kappa < \beth_2$ and $\lambda<\beth_1$ be cardinals. What can we say about $\kappa^{\lambda}$ without assuming CH? Is it true that $\kappa^{\lambda} < \beth_2$ or $\kappa^{<\beth_1} &...
- 73
1
vote
1
answer
213
views
Self-embeddings of uncountable total orders
A nice theorem of Dushnik and Miller (from 1940) states that if $(\Omega,\leq)$ is a countable total order, then either
there is an element $\omega \in \Omega$ such that $(\Omega \setminus \{\omega\}...
- 4,547
17
votes
2
answers
557
views
Raising the index of accessibility
In the standard reference books Locally presentable and accessible categories (Adamek-Rosicky, Theorem 2.11) and Accessible categories (Makkai-Pare, $\S$2.3), it is shown that for regular cardinals $\...
- 66.7k
7
votes
1
answer
459
views
Is it consistent that $|[\kappa]^{<\kappa}| > \kappa$?
Let $\kappa>\aleph_0$ be a cardinal, and let $[\kappa]^{<\kappa}$ denote the collection of subsets of $\kappa$ having cardinality strictly less than $\kappa$. Is it consistent that $$|[\kappa]^{&...
- 48.8k
10
votes
0
answers
377
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?
- 48.8k
9
votes
0
answers
247
views
Categorifying Hyperoperations
Is there some categorical version of tetration or higher hyperoperations?
This is motivated by the fact that coproducts categorify addition of finite cardinals, and products/exponentials categorify ...
- 10.1k
4
votes
0
answers
182
views
Generic two-cardinal behavior of first-order sentences
This is a hopefully improved version of a question I asked before and then deleted because it was based on some fundamentally incorrect assumptions.
Some first-order theories are able to control the ...
- 12.4k
3
votes
1
answer
123
views
Approximation on separable topological space with size $\mathfrak{c}$
Let $X$ be a separable topological space of size $\mathfrak{c}$. By a simple function $\phi:X\to X$, we mean a finite range valued measurable function.
Q. Is it possible to find a sequence of ...
- 4,058
2
votes
1
answer
150
views
On minimal generating sets of certain submodules
All our rings are commutative with unity.
For an $R$-module $M$ and a submodule $N$ of $M$ and ideal $I$ of $R$, let $(N:I):=\{m\in M : Im \subseteq N\}$. Let $\mu (M)$ denote the least cardinality ...
user111524
17
votes
0
answers
558
views
Gitik's work on Shelah's weak hypothesis
It seems that Moti Gitik has recently refuted some variants of Shelah's weak hypothesis. For this see the title and abstract of his talk at the Set Theory, Model Theory and Applications conference.
I ...
- 32.1k
5
votes
1
answer
335
views
Simultaneous failure of weak diamond
Let $\lambda$ be an infinite cardinal. Recall that Weak diamond $\Phi_S$ on $S\subseteq\lambda^+$ is the following principle:
For every function $F:2^{<\lambda^+}\rightarrow 2$, there exists $g\in ...
- 2,381
2
votes
1
answer
185
views
Cardinality of generating sets of faithful modules over integral domain
Let $R$ be an integral domain. Let $\alpha$ be an infinite cardinal . Let $M$ be a faithful $R$-module such that $\mu(M)< \alpha$ . Let $N$ be a submodule of $M$ and $m\in M$ and $r\in R$ be such ...
user111524
1
vote
1
answer
158
views
On cardinality of generating subsets of some submodules
Let $R$ be a commutative ring with unity. Let $\alpha$ be an infinite cardinal . Let $M$ be an $R$-module such that $\mu(M)< \alpha$ . Let $N$ be a submodule of $M$ and $m\in M$ and $r\in R$ be ...
user111524
15
votes
3
answers
618
views
Dominating families in bigger cardinals
A dominating family on $\omega^\omega$ is a set $\mathcal D \subset \omega^\omega$ such that for every $f \in \omega^\omega$ there exists $g \in \mathcal D$ such that $f<^* g$ (that is, $f(n)<g(...
user46747
9
votes
1
answer
404
views
Injection from $\aleph_2$ into the power set of $\mathbb{R}$
Assume $\mathsf{ZF} + \mathsf{DC}$. Must there exist an injection from $\aleph_2$ to $\mathcal{P}(\mathbb{R})$? If not, what is the consistency strength of the nonexistence of such an injection?
I ...
- 5,471
7
votes
1
answer
373
views
Does Laver Forcing add an infinitely often equal real?
Given a model $V$ of set theory and an inner model $W \subseteq V$, a real $x \in V \cap \omega^\omega$ is infinitely often equal over $W$ if for each real $y \in \omega^\omega$ there are infinitely $...
- 1,882
11
votes
1
answer
528
views
What is the cofinality of $([\kappa]^\omega, \subseteq)$?
For an uncountable cardinal $\kappa$, we are interested in the least size of a cofinal subset of the partial order $([\kappa]^\omega, \subseteq)$. It is obvious that this cofinality is at least $\...
- 11.5k
9
votes
2
answers
540
views
Reals which must, can't or might be added by forcing
Let $W \subseteq V$ be an inner model of ZFC. There are a variety of theorems that characterize when a real $x \in V$ is the generic of a forcing notion $\mathbb P \in W$, for example, the ...
- 1,882
3
votes
1
answer
783
views
Cardinality of ${\mathbb{C}_p}$ [closed]
I know, that field ${\mathbb{Q}_p}$ (field of p-adic numbers) has the same cardinality as $\mathbb{C}$. Taking algebraic closure doesn't change the cardinality of infinite field, so cardinality $\...
- 41
7
votes
1
answer
448
views
Fixed points of cardinal logarithm
For any cardinal $\kappa$ set $$\log(\kappa) = \min\{\mu\in \kappa\cup\{\kappa\}: 2^\mu \geq \kappa\}.$$
Clearly $\log(\omega) = \omega$ and in $\textsf{GCH}$ we have $\log(\aleph_\omega) = \aleph_\...
- 48.8k
7
votes
2
answers
248
views
Getting PFA + GCH above $\omega$
The Proper Forcing Axiom kills CH in a particularly specific way: it implies that $2^{\aleph_0}=\aleph_2$. However, its impact on the continuum function above $\aleph_0$ is much less clear. It is ...
- 21.1k