Questions tagged [cardinal-arithmetic]

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118 views

Cantor normal form [closed]

Every ordinal number $α$ can be uniquely written as ${\displaystyle \omega ^{\beta _{1}}c_{1}+\omega ^{\beta _{2}}c_{2}+\cdots +\omega ^{\beta _{k}}c_{k}}$, where $k$ is a natural number, ${\...
3
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1answer
288 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 ...
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0answers
227 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
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1answer
365 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
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1answer
242 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
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0answers
150 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
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1answer
283 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 ...
6
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1answer
294 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
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218 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
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1answer
260 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
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1answer
150 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
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1answer
133 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
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2answers
443 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 $\...
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297 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?
7
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0answers
171 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
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1answer
108 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
139 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
484 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
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1answer
274 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
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1answer
160 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
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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
566 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
359 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
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1answer
248 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 $...
10
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1answer
348 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
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2answers
454 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
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1answer
415 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
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1answer
410 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
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2answers
191 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
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1answer
210 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 ...
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4answers
814 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
639 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
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1answer
160 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 ...
15
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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}\...
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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
791 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
697 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
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2answers
385 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
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2answers
385 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
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2answers
387 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}...
7
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3answers
532 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
244 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
501 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
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1answer
646 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
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3answers
191 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
407 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
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1answer
508 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
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1answer
468 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\...