25 questions linked to/from Defining cardinality in the absence of choice
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### 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)?
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### 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 ...
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### 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, ...
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### 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?
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...