Re: Every open interval contains a rational?

Tony wrote:

I learned that some mathematicians make a difference between definition
via axioms and definition via construction.
In this sense Cantor's is a construction of real numbers from rational.
Construction is preferred by some because a set of axioms may contain
one which is not true.
Thus the construction proves the existence while the axioms don't
guarantee the existence.


What makes me say that the statement "every open interval contains a
rational" can and should be proved without any reference to a specific
way of defining the reals has *nothing* to do with this. It's just that
a proof based upon a specific way of defining the reals will not be a
proof for someane who has seen a different way of defining them. For
instance, if you prove something about the reals with a proof based upon
Cantor's constriction, then that proof will be acceptable to someone who
defines them by Dedekind cuts only after that person has made the extra
effort of proving that there's a certain specific connection between
real numbers and Cauchy sequences of rational numbers.

Best regards,

Jose Carlos Santos
.

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