# Questions tagged [cardinal-arithmetic]

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### Summing over continuum and uncountable numerocities [closed]

Here I want to address of the question if it is possible to make a sum over an uncontable set and discuss integration rules involving uncountably infinite constants. I will provide introduction in ...
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### 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?
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ... 531 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 ...
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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 $g\in ... 2 votes 1 answer 177 views ### Cardinality of generating sets of faithful modules over integral domain Let$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$be such ... 1 vote 1 answer 153 views ### On cardinality of generating subsets of some submodules Let$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$be ... 15 votes 3 answers 596 views ### Dominating families in bigger cardinals A dominating family on$\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(... 380 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 ...
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### 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 ...
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### 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 ...
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### 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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### 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: ...
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### 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 ...
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### 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 ...
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 $\sum\limits_{i\in I}\... 32 votes 4 answers 2k views ### Is it possible to define higher cardinal arithmetics In number theory there are several operators like ‎addition, ‎multiplication and ‎exponentiation defined from ‎$‎‎‎\omega‎‎\times‎‎\omega‎$‎to ‎‎$‎‎‎\omega‎$. Each ‎of ‎them ‎is defined as an ‎... 7 votes 2 answers 897 views ### Exponentiation and Dedekind-finite cardinals It is known that the sum and the product of two Dedekind-finite cardinals are also Dedekind-finite cardinals. What about cardinal exponentiation ? Question: Let A and B be two Dedekind-finite ... 17 votes 0 answers 746 views ### Ideas behind Gitik's solution of PCF conjecture Recently Moti Gitik has refuted Shelah's PCF conjecture (see Short extenders forcings II ) by proving the following theorem: Theorem. Assuming the consistency of infinitely many strong cardinals, one ... 6 votes 2 answers 419 views ### For which cardinal numbers$\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$. Further, it is consistent with ZFC that we always have equality (e.g. assume GCH). ... 6 votes 2 answers 426 views ### When are all greater cardinals sharply greater? Makkai and Paré introduced the following binary relation on regular cardinals: given$\kappa$and$\lambda$,$\kappa \vartriangleleft \lambda$(read,$\kappa$is sharply less than$\lambda$) when$\...
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}...