Thought of this today.

two years ago someone told me that you cannot know if there exist a biggest integer or not.

then i said that there must be a bigger integer, because we can add one to it.

But now i thought $\displaystyle \infty+1 = \infty$, so when we get to $\displaystyle \infty$ big numbers adding one does not increase it.

keep in mind i have a very elementary knowedge of infinity, so maby this is supid i dont know.

In this proof i will assume that:

1. No integer can be both odd and even at the same time.

2. For any integer n, $\displaystyle n+1\not< n.$

1 and 2 are not proven here, but proof exist.

assertion: there exist $\displaystyle \infty$ many integers.

proof: suppose n is the biggest integer such that $\displaystyle n = n+1.$ But if n is even, then from def $\displaystyle n = 2k,$ for an integer k.

But then $\displaystyle n+1 = 2k+1,$ for some k, which is odd by def. Then we know that $\displaystyle n\neq n+1,$ because of assumption 1. And we know that $\displaystyle n+1 \not< n$ by assumption 2.

Then the only option that remains is $\displaystyle n+1 > n,$ But then n is not the biggest integer.

But if n is odd then from def $\displaystyle n = 2k+1,$ for some k, But then $\displaystyle n+1 = 2k+2 = 2(k+1),$ which is even an even integer from def. Thus by assumptions 1,2 the only option is $\displaystyle n+1 > n.$

therefore if n is odd or even, there exist a bigger integer n+1. Q.E.D.

please point out any flaws in my arguments you may find.

thanks