User blog comment:Edwin Shade/The Grand List Of Transfinite Ordinals/@comment-11227630-20171222030119/@comment-1605058-20171230123355

Yes, they can be considered finite numbers, though this is a little misleading. What \(0^\sharp,0^\dagger\) really are is certain sets of true sentences (the definition is rather technical so let me spare you the details). By using Godel encodings, we can consider them to be subsets of natural numbers, and by further identifying those with 0-1 sequences and then with binary expansions, those can be viewed as real numbers in the interval [0,1]. I would argue though that it's better to stop at the "subset of natural numbers" interpretation.

This is quite similar to Turing machine oracles - the "best" interpretation of the halting oracle is to just consider it the set of the halting Turing machines, but for many purposes it is better to view it as a set of natural numbers or a binary string, so it might as well be viewed as a real number.

For the record, the reason why \(0^\sharp,0^\dagger\) often appear in the large cardinal lists is that, although not large cardinals themselves, their existence's consistency strength is comparable to very strong large cardinal axioms.