User blog comment:Edwin Shade/How Do I Evaluate BEAF Arrays In Two Dimensions ?/@comment-30754445-20170827183955/@comment-30754445-20170830201354

A very good question.

You are right. IF constructing such a set were possible, then we would have a logical paradox. Indeed, what you've just said is proof that such a set does not exist.

You see, one of the mind-boggling discoveries of modern set theory is that not every collection of things can be called a "set". Some infinite collections are too large to be called a set.

For example, "the set of all sets" is a paradoxial entity. It has to include itself as an element (since "the set of all sets" is - in itself - a set), but one can prove that a set can never have itself as a member.

Similarly, the collection of all ordinals is not a set. It's too big to be a set (and your own argument explains exactly why). Your analysis shows that even if we limit ourselves to "the collection of all unique infinite ordinals", it would still be to large to form a set.

And now comes the really interesting part:

In googology, we aren't really interested in all ordinals. Only the smaller ones interest us: the 'countable ordinals''. '''When talking about the ordinal strength of some fast-growing function (even an uncomputable function), we are always refering to a countable ordinal.

Now, it turns out that the collection of all countable ordinals is  a set. So we can rephrase your question:

"If a unique countable ordinal infinity is defined as the first element that does not belong in a particular countable set, and you were to construct a set of all countable ordinal infinities, then wouldn't a countable ordinal infinity outside the set not be possible ?"

And the answer is yes! A countable ordinal infinity outside this set would not be possible. We've just proven that the limit of all countable ordinals is - in itself - not countable. In fact, it is the smallest noncountable ordinal, known as ω₁.