On Saturday 22 October 2011 02:13:07 David Wilson wrote:
How do you show that there is an integer greater than any given real number?
The real question is, given the axioms for the real numbers (Dedekind-complete ordered field), how do you define the "real integers" within them.
Two possibilities spring to mind.
Definition A: The "real integers" are the minimal subset of R that contains 1 and is closed over negation and addition. ... Definition A seems more in the spirit of the number theory definition of Z. However, I do not immediately see how it answers my original question.
Define U = {x in R: x < some integer}. Dedekind completeness says it has a least upper bound unless it's empty (no!) or all of R; call it y. If for some n y<n then y+1<n+1 so y+1 in U, contradicting "upper bound". If for all n y>=n then the same is true of y-1, contradicting "least". So U was all of R, and we're done. -- g