Re: Axioms for the real numbers



un student wrote:


How come real numbers are uncountable? Which part of axioms "brings in"
the fact that reals are uncountable? I just can't understand where it
comes from.... and yes, I have studied different definitions of reals
but still can't get it.

The completeness axiom, which states that every sequence bounded above has a l.u.b. (least upper bound), below a g.l.b. (greatest lower bound).
What this axioms says, in relation to your question, the (countable) set
of rational numbers is not "dense" enough to reflect the continuity of
the real numbers. To reflect that we need a more "dense" set of numbers,
such as the power set of rational numbers - which is uncountable. The
completeness axiom is just a formalization of this need. Imho.

---Nam

TIA


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