Well, if we don’t care about utility I could define infinitely many transcendental numbers with no utility other than I just made them up. The number that is the concatenation of the digits of all prime numbers in sequence, for instance: 0.23571113171923… I christen this Dave’s Number. (It probably already has a name, but I’m stealing it.) Let’s add it to the list. Now we can define Dave’s Second Number as the first prime added to Dave’s Number: 2.235711131723… Dave’s Third Number is the second prime added to Dave’s Number: 3.235711131723… Since we’re cataloguing numbers with no utility, let’s add them all to the list.