Is $1234567891011121314151617181920212223......$ an integer? This question came from that one and from that talk where it's noted that "integers have a finite count of digits", so that the "number" in the title is not at all a number (not integer nor rational or real) . This statement seems to me justified because I suspect that if we admit an infinite count of digits we build a set of "numbers" that is not countable (but I've not a proof). I'm wrong?

Reading comments and answers I'm a bit confused. So I add a more specific question:
Is the string in the title a number in some model of Peano Axioms?

Your number is not an integer. Suppose that you can have an integer with an infinite number of digits. Then by Cantor's diagonal argument you can show that the set of all integers in uncountable, which is a contradiction if you accept the Peano axioms as suggested by Ahaan S. Rungta. Thus your number cannot be an integer. 
