In Fundamentals of Mathematics, Volume 1 Foundations of Mathematics: The Real Number System and Algebra, after defining set equivalence as the ability to put the elements of the related sets in one-to-one correspondence, the following statement appears:

The cardinal number $\tilde{x}$ of a set $x$ is then regarded as representing "that which is common" to all sets that are equivalent to $x$. Thus, we might say that the cardinal number of $x$ is simply the set of all sets that are equivalent to $x$, although such a definition is problematical on account of its relationship to the universal set.

The term problematical can have a slightly different connotation than the term problematic. The former implying requires expert handling. In other words, this may not be grounds for completely rejecting the definition. Unfortunately I do not have access to the German Language original to know what "problematical" was translated from.

Regardless of that nuance, the authors are certainly indicating that their proposed definition leads to difficulty in "relationship to the universal set". Is this difficulty simply Russell's antinomy?

  • $\begingroup$ It should be noted that there is a way of getting around this issue via Scott's Trick. $\endgroup$ – Hayden Feb 17 at 7:50

Basically, yes. The problem is that the collection of all equivalent sets is a proper class (except in the case of the empty set). Thus to work with the notion of cardinality in set theory, it is convenient to define some representative of the class that is a set.

The most common solution is to use the axiom of choice and define the cardinality as the smallest ordinal in the class. In the absence of choice, one can instead appeal to foundation and define it as the subset consisting of the sets of lowest rank.


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