Linked Questions

56
votes
1answer
11k 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)?
31
votes
7answers
12k views

Incredible frequency of careless mistakes [closed]

Ever since high school, I've had a serious problem with math classes. Be it discrete math, algebra, calculus or linear algebra, I seldom have trouble understanding the texts or lectures, but when it ...
33
votes
3answers
3k views

Is cardinality a well defined function?

I was wondering if the cardinality of a set is a well defined function, more specifically, does it have a well defined domain and range? One would say you could assign a number to every finite set, ...
23
votes
4answers
6k views

Cardinality of all cardinalities

Let $C = \{0, 1, 2, \ldots, \aleph_0, \aleph_1, \aleph_2, \ldots\}$. What is $\left|C\right|$? Or is it even well-defined?
11
votes
4answers
3k views

Is the class of cardinals totally ordered?

In a Wikipedia article http://en.wikipedia.org/wiki/Aleph_number#Aleph-one I encountered the following sentence: "If the axiom of choice (AC) is used, it can be proved that the class of cardinal ...
25
votes
2answers
2k views

Where is axiom of regularity actually used?

Where is axiom of regularity actually used? Why is it important? Are there some proofs, which are substantially simpler thanks to this axiom? This question was to some extent provoked by Dan ...
13
votes
2answers
745 views

What is the difference between cardinals and alephs?

It is stated that by Zermelo’s theorem, every cardinal is an aleph. But what is the difference between cardinals and alephs? I thought that alephs were just a way to denote cardinals (just for ...
6
votes
3answers
1k views

Defining cardinals without choice

According to Wikipedia if we assume AC we define a cardinals number to be an ordinal that is not in bijection with any smaller ordinal. Without AC, one takes the cardinality of a set $X$ to be the ...
4
votes
2answers
1k views

A question about cardinal arithmetics without the Axiom of Choice

Is multiplication of infinite cardinals defined in ZF without Choice?
4
votes
2answers
2k views

Cardinality of union of ${{\aleph }_{0}}$ disjoint sets of cardinality $\mathfrak{c}$

I have a home work question which is: " what is the cardinality of the union of ${{\aleph }_{0}}$ disjoint sets of cardinality $\mathfrak{c}$?" I believe somehow we can get to: cardinality = $({{\...
5
votes
4answers
645 views

Does every ordinal have cardinality no greater than $\aleph_\mathbb{0}$?

My notes say that the ordinals $\omega + 1, \omega + 2, ... , 2 \omega, ... , 3 \omega, ... \omega^2, ... $ are all countable, and hence have cardinality equal to $\omega = \aleph_\mathbb{0}$. So I ...
1
vote
3answers
224 views

Ordinal with given cardinality (without AC)

Is it possible to show that every cardinality has an ordinal with this cardinality (without the axiom of choice)? If so, how?
8
votes
2answers
322 views

Axiom of Choice needed to “categorify” the cardinals?

I was playing around in $\mathsf{Set},$ trying to reduce it modulo isomorphisms to make a category $\mathsf{Card},$ letting the objects of $\mathsf{Card}$ be the isomorphism classes of $\mathsf{Set}$ ...
5
votes
2answers
2k views

A proper formal definition of the Ordinal numbers?

Maybe I'm just being a bit stupid with where I'm searching here, but I can't actually find a proper definition of these things wherever I look, books or online. The definition I find most is that it's ...
3
votes
2answers
467 views

Cardinality of the complex numbers in ZF

As you all know, cardinality of $\mathbb{R} = 2^{\aleph_0}$ can be proved in ZF, since cardinality of $\mathbb{N} \times \mathbb{N} = \aleph_0$ can be proved in ZF. I know that the statement 'For any ...

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