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**HallsofIvy** Why do you think it is circular? If I have the set {a, b}, surely, I can determine which sets have a one-to-one correspondence with that set. Nothing circular in that. $\displaystyle \overline{\{a,b\}}$ is the collection of all such sets (which would, of course, include {a,b} itself). I see nothing circular in that. Now, notice I did not use the term "same number of elements" which, in your quote, is purely descriptive and not a part of the mathematical definition. Perhaps that was what was bothering you - that we seem to need to determine what sets have the "same number of elements" in order to determine a "cardinal number". We **don't** we merely need to determine one-to-one correspondences which does NOT have to depend on "same number". Indeed, we can wait until **after** we have defined "cardinal number" before we even mention "same number".