proof

consider the list of all prime numbers, ie. 2, 3, 5, 7, 11... now remove all but the first one digit prime, the first two digit prime, etc, leaving 2, 11, 101, etc. arrange these primes in a table, like the one below.

Code:

0002
0011
0101
1009
...

let us create a new number, N. let the first digit of N equal the first digit of the first number, plus one, the second digit equal the second digit of the second number, plus one, etc. with only the numbers on the table above, N = ...0223.

we have thus constructed a (probably) prime number that is different from every prime number on our list, proving that there are uncountably many primes, and, as a corollary, that there exists some prime number with infinitely many digits.