Cardnality of integers > Cardnality of integers

Hello,

I have a question from my classes that i do not understand.

The professor has shown that the cardnality of reals is > cardnality
of integers by using the "diagnolization" argument to show that if
someone had listed all of the reals, they could not be put into a one
to one correspondance with the integers because one can always
construct another real from the list using the diagnolization method.

My question is:
Suppose one were to construct a list of integers using this method:

1 = 2^0 * 3^0 * 5^0 * 7^0...
2 = 2^1 * 3^0 * 5^0 * 7^0...
3 = 2^0 * 3^1 * 5^0 * 7^0...
....

Once someone has completed the list of all of the integers, it seems i
can use the same diagnoalization argument to show that there is an
integer not in the list.

Any thoughts are appreciated.
Thank You
.

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