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?

First of all, before we can callthis an integer, we can only call it a blob of ink or a bunch of pixels. How are some buches of pixels integers and others are not? Well, the are not integers, they represent integers. For example MMXV, and $2015$, and 0x7DF are distinct representations of the same integer, and all given with enough "context" to allow us to conclude (more or less easily) what representation scheme was used and that integers "behind" these notations are in fact the same.

I won't go into details of the other two representation, but concentrate on the decimal one. How do we know which number it represents? While the answer seems to be "two thousands and fifteen" obviously, it is not really that obvious. The convention(!) behind the notation is that a finite sequence of digit symbols $d_1d_2\ldots d_k$ represents the number $\sum_{i=0}^{k-1}10^{i}d_{k-i}$ (with a bit of ignoring the distinction between a digit symbol and its numerical value). Your candidate does not match this convention because it is not a finite sequence. We cannot even read the value of $k$ from it. Admittedly, there are other ways to formulate the convention of deimal representation, but they all fail in some way or other with an infinite digit string (it may be appropriate to consider left-infinite sequences of digits and view the natural numbers as a subset of the "numbers" obtained this way (by prependingthem with infinitely many zeroes), but beware - the realm of Archimedes-Plutoniumism lurks behind that; and this does not apply to your right-infinite string).

To put it differently: Asking what integer an ill-formed (because infinite) sequence of decimal digits represents is not much better than asking what integer the ill-formed "roman numeral" MQLXXRT represents.

This is an infinite string of digits. But this string, when we try to interpret it in decimals, does not encode a number, i.e., an element of ${\mathbb N}$ or ${\mathbb R}$. Natural numbers, when written in decimals, appear as finite strings, and have a last digit. Noninteger real numbers, when written in decimals, require a decimal point, after which an infinity of digits may appear. Therefore, if you decide to insert a decimal point in front or between any two digits in your sequence, you obtain a well defined positive real number.