# Why is the definition of cardinal number as the set of all sets equivalent to a given set “problematical”?

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?

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