I'm currently reading a book on non-standard analysis. The proof states that there is an infinite integer greater than all reals. How is this possible? I thought that the reals contained the integers?
The proof relies on the fact that since you can write "For every real x, there is an integer y that is greater" as a sentence in the reals, it is true in the hyperreals.
Edit:
The book I'm reading is "Infinitesimal Calculus" by Kleinberg and Henle. The proof is literally
"Since $\forall x_1\exists x_2(x_1<x_2\land I(x))$ where $I(x)$ is 'x is an integer' is a sentence in the reals, it is a sentence in the hyperreals. We know that $1/hyperreal$ is infinite and therefore there exists an integer greater than it."
Several issues were raised in your question as well as the follow-up discussion but I will respond to the first pair of questions you ask: The proof states that there is an infinite integer greater than all reals. How is this possible? I thought that the reals contained the integers? The answer is simply that one needs to replace "integer" by "hyperinteger" in the sentence. I did not look inside the book by Kleinberg and Henle but I assume that if they write "infinite integer" what they mean is an "infinite hyperinteger".
Recall that the hyperreals are a proper ordered field extension of the reals. Any such extension has to contain infinitesimals. Such extensions can be easily constructed in ZF without the need for the axiom of choice. The inverse of an infinitesimal is then an infinite number greater than every real number.
I see from Luxemburg's review at mathscinet that the Kleinberg and Henle adopt Robinson's framework rather than Nelson's (though according to the dates they could have used Nelson's framework). At any rate, in Nelson's framework one would clarify the sentence by writing "there is an integer greater than all the standard reals".