4
votes
1answer
37 views

Extension of ZFC models preserves cardinals

Let $M \subseteq N$ be countable transitive ZFC set models. Assume that this extension preserves cardinals, i.e. if $\alpha$ is an ordinal number (this notion is absolute) such that $(\alpha \text{ is ...
5
votes
1answer
62 views

Partial order on cardinalities without the axiom of choice

Cardinality can still be defined without choice, e.g. as equivalence class of equipotent sets, see Defining cardinality in the absence of choice. Injections define partial order on cardinalities by ...
5
votes
0answers
54 views

Raising a partial function to the power of an ordinal

Consider a set $X$, and let $f : X \rightarrow X$ denote a partial function. Then for natural $n$, we can define $f^n$ as iterated composition, e.g. $f^2 = f \circ f$. Now suppose that $X$ is also ...
4
votes
2answers
119 views

The regularity of successor cardinal

I was looking at two different proofs of the fact that successor cardinals are regular. It struck me as odd that both proofs used AC. Looking at the concepts involved in defining cofinality I feel as ...
3
votes
3answers
119 views

Cardinality of all sequences of non-negative integers with finite number of non-zero terms. (NBHM 2012)

Consider the set $S$ of all sequences of non-negative integers with finite number of non-zero terms. Is the set $S$ countable or not? What is the cardinality of the set $S$ if it is not countable? ...
2
votes
1answer
82 views

Are the closed ordinals (apart from $0$ and $1$) precisely the regular cardinals?

Given a partially ordered set $P$, a collection of partially ordered sets, call it $\mathcal{Q}$, and an arbitrary function $f : P \rightarrow \mathcal{Q},$ we can form a new partially ordered set ...
5
votes
1answer
169 views

In ZF, does there exist an ordinal of provably uncountable cofinality?

Question is in the title. In ZFC, one can prove that $\aleph_{\alpha+1}$ is regular, so there is a large source of cardinals with uncountable cofinality, but in ZF, it is consistent that ${\rm ...
5
votes
2answers
101 views

Reference request, self study

I'm looking for references (books/lecture notes) for : Cardinality without choice, Scott's trick; Cardinal arithmetic without choice. Any suggestions ? Thanks in advance.
7
votes
3answers
110 views

What questions become answerable/computable given an uncountable character set?

Having reached the concluding portion of my first course in real analysis, one subject that I feel was not adequately addressed was the issue of cardinalities. This is a subject I was interested in ...
6
votes
2answers
184 views

Confusion about cofinality

I'm confused about the notion of the cofinality of a cardinal. Since I think the source of the confusion is the Von Neumann cardinal assignment, my first question is: Question 0. Is there an article ...
4
votes
0answers
38 views

Cardinal characteristics — generalisation?

I'm just reading about what is called cardinal characteristics of the continuum. For example there are the bounding number $\mathfrak b$ and the dominating number $\mathfrak d$ etc. which are defined ...
5
votes
1answer
118 views

Differences in worlds with and without $\aleph_0<|S|<2^{\aleph_0}$

Paul Cohen told us that whether or not there is $S$ with \begin{equation} \aleph_0<|S|<2^{\aleph_0} \end{equation} cannot be decided within ZFC, and hence it is reasonable to work in two ...
2
votes
1answer
102 views

Cardinality of all possible partitions of a infinite set

I suppose this problem should be a commonplace, but I only find this one, in which notations are kind of idiosyncratic, along with a glaring defect. My question is where I can find a more formal ...
6
votes
1answer
240 views

is there a cardinality between the rational and the irrationals?

Was asked this question, and I have no idea about how to start proving it. Could someone give me some good reference material to start with.
15
votes
1answer
1k views

Overview of basic results on cardinal arithmetic

Are there some good overviews of basic formulas about addition, multiplication and exponentiation of cardinals (preferably available online)?