# Questions tagged [cardinal-arithmetic]

The cardinal-arithmetic tag has no usage guidance.

72
questions

9
votes

2
answers

472
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 ...

3
votes

2
answers

155
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 ...

26
votes

1
answer

904
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}}}}}...

2
votes

2
answers

218
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 ...

14
votes

1
answer

597
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 ...

8
votes

1
answer

246
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 ...

5
votes

1
answer

693
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 ...

-2
votes

1
answer

107
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 ...

7
votes

0
answers

170
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 $\langle\delta_n\rangle_{n<\omega}$, ...

9
votes

3
answers

417
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}\...

15
votes

1
answer

443
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\...

7
votes

1
answer

282
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}(...

3
votes

0
answers

202
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 $\...

4
votes

0
answers

129
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 ...

4
votes

1
answer

661
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 ...

5
votes

0
answers

201
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 ...

15
votes

1
answer

435
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<\...

5
votes

0
answers

146
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}|...

2
votes

1
answer

222
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 ...

3
votes

1
answer

479
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 ...

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 ...

11
votes

1
answer

485
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 $\lambda$ ...

11
votes

1
answer

306
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 $[\mu]^\kappa$ ...

4
votes

0
answers

173
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-...

5
votes

1
answer

364
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
votes

1
answer

456
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 $PRA+WF(\epsilon_0)$....

2
votes

0
answers

235
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-...

4
votes

1
answer

278
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 $|{\cal P}(\alpha)|
>\beta$?

4
votes

1
answer

206
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} &...

1
vote

1
answer

207
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\}...

17
votes

2
answers

541
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 $\...

7
votes

1
answer

446
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]^{&...

8
votes

0
answers

351
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?

9
votes

0
answers

242
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 ...

4
votes

0
answers

177
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 ...

3
votes

1
answer

122
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 ...

2
votes

1
answer

147
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 ...

17
votes

0
answers

547
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 ...

5
votes

1
answer

330
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
votes

1
answer

184
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 ...

1
vote

1
answer

156
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 ...

15
votes

3
answers

613
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(...

9
votes

1
answer

392
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 ...

7
votes

1
answer

352
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 $...

11
votes

1
answer

508
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 $\...

9
votes

2
answers

530
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 ...

3
votes

1
answer

741
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 $\...

7
votes

1
answer

442
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_\...

7
votes

2
answers

236
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 ...

1
vote

1
answer

216
views

### Wondering if the following set-theoretic assertion is known to be consistent w/ ZFC

I'm wondering about the following (I've written a couple set theory papers but don't consider myself a set theorist, so please keep this in mind when answering):
Is it consistent w/ ZFC that there ...