User blog comment:Edwin Shade/How Do I Evaluate BEAF Arrays In Two Dimensions ?/@comment-30754445-20170827183955/@comment-30754445-20170830183956

@Deedlit/Edwin Shade

"In order for us to answer your questions, you need to define what 'unique ordinal infinities' and 'fundamentally different' mean. I will say that, if one considers and  to be fundamentally different ordinal infinities, then probably one would consider, the Bachmann-Howard ordinal, the TFB ordinal, etc to all be different as well, and there would be infinitely many fundamentally different ordinal infinities"

It's even worse than that.

Not only there are an infinite number of them, but there's no way to "catch" them all with a single definition.

Why? Because if it were possible, this very same definition could be used to define a larger "unique" ordinal: The smallest ordinal not in the set! (it is a basic gurantee of set theory that such an ordinal always exists)

Indeed, this is exactly the trick by which all these "unique ordinal infinities" are defined (one at time): is the smallest ordinal not in {1,2,3,...}. is the smallest ordinal which can't be reached with 's and so on.