Confusion about definition of cardinal number.

Question

This question is from "Theory of Sets" by E.Kamke. This is about the definition of cardinal number . Here is the definiton of a cardinal number that was given in the book.

I have a few confusions about this definition .

1. First of all , I think in "By a cardinal number of a power $\mathfrak m$" , the of is a typo .It probably was supposed to be "or".

2. It says "... we mean an arbitrary representative $\mathfrak M$ of a class of mutually equivalent sets" , does this mean $\mathfrak m$ is the set $\mathfrak M$.

3. In wikipedia I have found another definition of cardinal number.

The oldest definition of the cardinality of a set $X$ (implicit in Cantor and explicit in Frege and Principia Mathematica) is as the class $[X]$ of all sets that are equinumerous with $X$.

But I am not sure if this definition is equivalent to the definition provided in "Theory of Sets".I guess one similarity is that both definitions use the notion of "Equivalence classes".

These are all of the confusions I am having , can someone clarify them for me ?

Answer 1

I believe

1. a typo

2. Maybe but I think it is more likely (but I could be dead wrong) that he is saying we may use an arbitrary symbol, such as $\mathscr m$ to represent the idea of the class of all sets "equivalent" (more on my use of scare quotes when I address point 3) to $\mathscr M$. This is equivalent to using $\aleph_0$ to represent the concept of the cardinality of $\mathbb N$ (which is of course the cardinality of $\mathbb Q$ or the cardinality of the evens, or the odds, or prime numbers or $\mathbb Z$ or... which we can indicate by $|\mathbb N| = |\mathbb Q| = |\mathbb Z| = |\mathbb E| = ...etc.... = \aleph_0$.

.... Or... it could be a typo.

3. These definitions seem to be the same if we figure Kamke's defintion of "equivalent" and wikipedias definition of "equinumerous" to mean "a bijection exists between the two sets.

The both are saying the if $M$ is a set then $|M|$ would be the collection of all sets with bijections to $M$ (let's ignore the ZF problems of defining "the set of all sets so that ..." for the moment) and we use that collection to represent the cardinality of the set. Kamke, in the days before the finalization of ZF (maybe) probably (again maybe) played loose with the word "set" whereas now we would be more careful and talk of proper class.

Answer 2

To answer your questions:

1. Yes I agree, that looks like a typo.

2. Yes, that appears to be the case.

3. These are different. In the definition from Wikipedia, the cardinality of a set $X$ is defined to be a proper class, namely the class of all sets which are equinumerous with $X$. Two sets $X$ and $Y$ are called equinumerous, if there exists a bijection $f: X \rightarrow Y$. On the other hand, in the definition quoted from your book, the cardinality is defined to be a set, and in fact it is a member of the proper class as defined in the Wikipedia quote.