Articles of cardinals

Closing a subcategory under finite colimits by transfinite induction

Let $\mathcal{C}$ be a locally small category with all finite colimits, and let $\mathcal{A}$ be a small full subcategory. I wish to prove the following: Proposition. There exists a full subcategory $\mathcal{B}$ of $\mathcal{C}$ satisfying these conditions: $\mathcal{A} \subseteq \mathcal{B} \subseteq \mathcal{C}$ Finite colimits exist in $\mathcal{B}$ and are the same as in $\mathcal{C}$. Every […]

Cartesian product of large sets

For a non-empty set $A$ let $A’$ denote the Cartesian product of $A$ with itself taken denumerably many times. Now given a set $S$ whose cardinality is strictly greater than the cardinality of continuum (i.e., $\Bbb R$), in which cases do we have ${\rm card}(S’)={\rm card}(S)$ and in which cases do we have ${\rm card}(S’)\ne{\rm […]

I read that ordinal numbers relate to length, while cardinal numbers relate to size. How do 'length' and 'size' differ?

I read that ordinal numbers relate to length, while cardinal numbers relate to size. How do ‘length’ and ‘size’ differ? Note : I am an absolute novice, and I’m having a little trouble visualizing ordinal numbers.

Bijection between $\mathbb R^\mathbb N$ and $\mathbb R$

This question already has an answer here: Bijection from $\mathbb R$ to $\mathbb {R^N}$ 3 answers

Can every set be expressed as the union of a chain of sets of lesser cardinality?

If a set $S$ has countably many elements $\{x_n\}$, it can be expressed as a union of a chain of finite sets $$ \{x_1\} \subset \{x_1,x_2\} \subset $$ But what about a set of arbitrary cardinality $S$? Can we express it as a union : $$S=\bigcup_{i\in I}S_i$$ where $I$ is a totally ordered set of […]

Is $\aleph_0$ the minimum infinite cardinal number in $ZF$?

$\aleph_0$ is the least infinite cardinal number in ZFC. However, without AC, not every set is well-ordered. So is it consistent that a set is infinite but not $\ge \aleph_0$? In other words, is it possible that exist an infinite set $A$ with hartog number $h(A)=\aleph_0$?

How to show $\kappa^{cf(\kappa)}>\kappa$?

For $\kappa \geq \omega$, which is a cardinal, then how to show $\kappa^{cf(\kappa)}>\kappa$? My idea: When $\kappa=\aleph_\omega$, then $cf(\kappa)=\omega$, is $\kappa^\omega>\kappa$? It seems $\kappa^{cf(\kappa)}>\kappa$ is wrong. Could you help me?

Jech's proof of Silver's Theorem on SCH

Jech’s textbook proves Silver’s Theorem–that if the Singular Cardinals Hypothesis holds for all singular cardinals of countable cofinality, then it holds for all singular cardinals–by breaking up the question into several lemmas and considering the case $\kappa=\aleph_{\omega_1}$ to simplify notation. I have a question about a technical detail in the proof of one of these […]

Let $X$ and $Y$ be countable sets. Then $X\cup Y$ is countable

Since $X$ and $Y$ are countable, we have two bijections: (1) $f: \mathbb{N} \rightarrow X$ ; (2) $g: \mathbb{N} \rightarrow Y$. So to prove that $X\cup Y$ is countable, I figure I need to define some function, h: $\mathbb{N} \rightarrow X\cup Y$ Thus, I was wondering if I could claim something similar to the following: […]

Why do $2^{\lt\kappa}=\kappa$ and $\kappa^{\lt\kappa}=\kappa$ when the Generalized Continuum Hypothesis holds?

I’m going to assume GCH here. If that holds, then why do we have the equalities $2^{\lt\kappa}=\kappa$ for every $\kappa$, and $\kappa^{\lt\kappa}=\kappa$ for all regular $\kappa$?