If you can add 1 then it wasn't infinite to start with.
And I agree - it's not a number.
1 is a number.
Can you add a non-number to a number?
Does truth become 15% more truthful if you multiply it by 1.15 ?
Does God live an extra few years by not smoking? No. He exists for a past and future eternity.
Like I said. It's an oxymoron to give an abstraction like infinity a finite quality like the ability to ADD it to something.
Many vacuous discussions stem from just taking everything literally.
Here, you should make the distinction between on the one hand what we usually mean with the notion of addition, notion which is obviously based on our everyday life experience that has absolutely nothing to do with the concept of infinity, and on the other formal systems which may or may not represent the common-sense notions we may have in mind.
So, there's no problem for a mathematician to invent a notion of infinity and a notion of addition that make sense when used together.
And since none of us can experience infinity, I don't see how we could possibly exclude any such operation. Rather, it's just a matter of definition.
According to this, I don't see any difficulty in the notion of adding 1 to infinity. I don't even see any problem in considering you could actually add one element to an infinity of elements. For example, consider the set of all even numbers. I think it makes sense to add for example the number 3 to this set, making the result a different set, also with an infinite number of elements, but with one element more than the first set, and a clearly identifiable element. You'd have to look carefully at all the consequence of that to give this example a clear sense but I'm sure it could be done.
EB