Questions tagged [cardinal-arithmetic]

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4
votes
1answer
228 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
0answers
206 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
1answer
253 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
1answer
142 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} &...
0
votes
0answers
90 views

Can the cardinality of the set of all intervening cardinals between sets and their power sets be always singular?

This is a question that I've posted to Mathematics Stack Exchange, that was un-answered. To re-iterate it here: Is the following known to be consistent relative to some large cardinal assumption? $...
1
vote
1answer
127 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\}...
16
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2answers
419 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 $\...
8
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0answers
280 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?
6
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0answers
150 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
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0answers
166 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
1answer
107 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
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1answer
138 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 ...
16
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0answers
465 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
1answer
255 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
1answer
152 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
1answer
136 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
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3answers
560 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(...
10
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1answer
344 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
1answer
226 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 $...
9
votes
1answer
310 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 $\...
8
votes
2answers
434 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
vote
1answer
368 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
1answer
407 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
2answers
186 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
1answer
209 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 ...
8
votes
4answers
764 views

Direct axiomatization of ordinal and cardinal numbers

Again, this question is related (**) to a previous one: in standard books on basic set theory, after stating the axioms of ZFC, ordinal numbers are introduced early on. Afterwards cardinals appear: ...
12
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1answer
576 views

When can Power Sets be Limit Cardinals?

My original question (posted in https://math.stackexchange.com/questions/1584430/can-all-power-sets-be-limit-cardinals) was: Is it possible to create a model of ZFC, so that the cardinality of each ...
5
votes
1answer
155 views

Ordinals which embed in surreal subfields

If $k$ is an ordered field, the least ordinal $s(k)$ which doesn't embed in $(k,<)$ is regular. This is because every interval of an ordered field embeds in every infinite interval so given a ...
14
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1answer
1k views

Does “cardinal arithmetic is well-defined” imply axiom of choice?

Let me quickly explain what I mean with my question. Let $(\kappa_i)_{i\in I}$ be a collection of cardinal numbers, indexed by elements of some set $I$. We can try to define $\sum\limits_{i\in I}\...
30
votes
4answers
2k views

Is it possible to define higher cardinal arithmetics

In number theory there are several operators like ‎addition, ‎multiplication and ‎exponentiation defined from ‎$‎‎‎\omega‎‎\times‎‎\omega‎$ ‎to ‎‎$‎‎‎\omega‎$. Each ‎of ‎them ‎is defined as an ‎...
7
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2answers
769 views

Exponentiation and Dedekind-finite cardinals

It is known that the sum and the product of two Dedekind-finite cardinals are also Dedekind-finite cardinals. What about cardinal exponentiation ? Question: Let A and B be two Dedekind-finite ...
16
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0answers
663 views

Ideas behind Gitik's solution of PCF conjecture

Recently Moti Gitik has refuted Shelah's PCF conjecture (see Short extenders forcings II ) by proving the following theorem: Theorem. Assuming the consistency of infinitely many strong cardinals, one ...
6
votes
2answers
381 views

For which cardinal numbers $\kappa$ is it consistent with ZFC that $\kappa^{\mathrm{cf}(\kappa)} < \kappa^\kappa$?

ZFC proves that $\kappa^{\mathrm{cf}(\kappa)} \leq \kappa^\kappa$ for all infinite cardinal numbers $\kappa$. Further, it is consistent with ZFC that we always have equality (e.g. assume GCH). ...
6
votes
2answers
368 views

When are all greater cardinals sharply greater?

Makkai and Paré introduced the following binary relation on regular cardinals: given $\kappa$ and $\lambda$, $\kappa \vartriangleleft \lambda$ (read, $\kappa$ is sharply less than $\lambda$) when $\...
5
votes
2answers
380 views

Prevalent singular cardinals hypothesis

The following notion is introduced by Assaf Rinot: Definition. A singular cardinal $\kappa$ is a prevalent singular cardinal iff there exists a family $\mathbb{A}\subset P(\kappa)$ with $|\mathbb{A}...
6
votes
3answers
504 views

Consistency strength of the failure of Shelah's Strong Hypothesis (SSH)

Some known facts about SSH (Shelah's Strong Hypothesis): i) "$0^\sharp$ does not exist" implies SSH. ii) SSH implies SCH (Singular Cardinal Hypothesis). iii) The failure of SCH is equiconsistent ...
7
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0answers
240 views

Other variants of the Shelah's Weak Hypothesis

The paper Menachem Kojman. Splitting families of sets in ZFC. arXiv:1209.1307 presents these variants of the Shelah's Weak Hypothesis: $$ (\textrm{SWH}_n) \textrm{ There are no infinite } \nu ...
22
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1answer
2k views

Possible troubles in Shelah's book “Cardinal Arithmetic”

I found some possible troubles in Observation 5.3(7) in the Chapter II of the Shelah's book "Cardinal Arithmetic" (page 86). For convenience, I quote the result and the proof in the book here (...
9
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1answer
487 views

Some variants of the Shelah's Weak Hypothesis

Are equivalent (in ZFC) the following two statements, for any infinite cardinal $\mu$? (i) For every infinite cardinal $\kappa$, $|\{ \lambda \in \kappa : \lambda \textrm{ is a singular cardinal and} ...
10
votes
1answer
631 views

“cov vs pp” problem

This is the problem $(\beta)$ of the section 14.7 in the "Analytical Guide" of the Shelah's book "Cardinal Arithmetic": $(\beta)$ Is $\operatorname{cov} ( \lambda , \lambda, \aleph_{1} , 2) =^{+} \...
3
votes
3answers
187 views

Subquotients in ZF

In ZF we have the two relations $A \leq B$ and $A \leq^\ast B$ which relate the size of sets: the first says there is an injection from $A$ to $B$, the second that there is a surjection from $B$ to $A$...
5
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4answers
400 views

do behavior of gimel or GCH determine all infinte products of cardinals?

Let $Card$ be the class of infinite cardinals and $p\colon Card^2\to Card$ be given by $(\kappa,\lambda)\mapsto\kappa^\lambda$. Assuming GCH it is known that $p(\kappa,\lambda)$ is either $\kappa$ (if ...
1
vote
1answer
493 views

Cardinal Arithmetic, foundations and constructive math

This is not my area but a question occurred to me that I can not find the answer to. There is a very strong axiom of constructibility which ironically gives us highly non-constructive math (GCH is one ...
2
votes
1answer
460 views

About the notions of Grothendieck Universe and Tarski Universe

I assume ZFC. Let $U$ a set with the following (1), (2), (3): 1) $\omega\in U$ 2) $x\in U\ \Rightarrow x\subset U$ 3) $x\in U\ \Rightarrow \mathcal{P}(x)\in U$ (where $\mathcal{P}(x):=${$y| y\...
2
votes
1answer
387 views

A question about cardinal arithmetic

Let [J]: Jech "Set theory" (Millenium edition) Let $\kappa$ a limit ordinal. From [J], T.3.11, p. 33 we have that $\kappa<\kappa^{cf(\kappa)}$. I improved that proof, and obtain : $\kappa <...
9
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1answer
581 views

Does this “jumping-ahead” ordinal function exist?

While working on a project in operator algebras with a collaborator (and fellow MO user), we are able to successfully complete a transfinite induction assuming that the following has an affirmative ...
8
votes
1answer
525 views

Is there always an uncountable $\kappa$ such that $\kappa^{\aleph_0}=2^\kappa$?

The cardinal equation $\kappa^{\aleph_0}=2^\kappa$ is satisfied by $\kappa=\aleph_0$. It is also satisfied by any $\kappa$ for which $MA(\kappa)$ holds. Under $GCH$, the equation is satisfied by $\...