Re: Rudin and Dedekind cuts
But surely there is a fundamental difference. To
define rationals we can start with the Peano axioms
for integers and then define rationals as ordered
pairs of integers. The idea of a 'number' is very
clear, there is no ambiguity.
Is it really? And exactly what is the DEFINITION of "number"?
With Dedekind cuts, there IS ambiguity.
What ambiguity do you see?
"Number" versus "interval"? What is the definition of each of those?
I appreciate that I can define reals using Cauchy
sequences, but I am trying to understand the Dedekind
cuts.
.
Relevant Pages
- Re: Rudin and Dedekind cuts
... integers and then define rationals as ordered pairs of integers. ... The idea of a 'number' is very clear, there is no ambiguity. ... we define rationals as certain _equivalence classes_ of ordered ... The definition of the reals using Cauchy sequences has the very same ...
(sci.math) - Re: Rudin and Dedekind cuts
... To define rationals we can start with the Peano axioms for integers and then define rationals as ordered pairs of integers. ... The idea of a 'number' is very clear, there is no ambiguity. ... I appreciate that I can define reals using Cauchy sequences, but I am trying to understand the Dedekind cuts. ...
(sci.math) - Re: Cantor Confusion
... though hard to unveil intentional mistake by Dedekind? ... extended to rationals in order to allow division and include fractions, ... the irrationals cannot be located numerically. ... the reals vanish completely within the sauce of irreals. ...
(sci.math) - Re: Cantor Confusion
... Eckard Blumschein wrote: ... one answer in case of rationals and a different one in case of reals. ... If, for example, one is trying to create Dedekind cuts, about the only ... what Dedekind saw are the ones lacking insight. ...
(sci.math) - Re: Axioms for positive elements of an ordered field?
... > rationals that greatly simplifies the definition of the product ... > of two reals. ... I find that of the four methods described (Cauchy, Dedekind, Weierstrass ... In para 2 the real number is actually introduced as the lower bound of ...
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