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Topic: to contruct reals as infinite decimals, do you need axiom of choice
Replies: 5   Last Post: Aug 18, 2013 5:46 PM

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Virgil

Posts: 9,012
Registered: 1/6/11
Re: to contruct reals as infinite decimals, do you need axiom of choice
Posted: Aug 18, 2013 5:16 PM
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In article <65876c86-4347-42cd-9558-b5a1c5289c72@googlegroups.com>,
Zeit Geist <tucsondrew@me.com> wrote:

> On Sunday, August 18, 2013 12:36:14 PM UTC-7, G Patel wrote:
> > On Sunday, August 18, 2013 10:09:26 AM UTC-4, dull...@sprynet.com wrote:
> >

> > > On Sun, 18 Aug 2013 00:33:40 -0700 (PDT), G Patel
>
> > I'm just trying to construct them without reading a professional
> > construction. I failed.
> >
> >
> >
> > So we just let S = {be set of all equivalence classes of infinite decimals}
> > , where two infinite decimals are equivalent in such and such way.
> >

>
> What way? It's important.
>
> First of all, you need to construct the Natural Numbers.
> Then, the Integers.
> Next, the Rational Numbers.
> Finally you construct the Real Numbers.


Or one can do
1. Naturals
2. Positive Rationals
(or non-negative Rationals if 0 is taken as a natural)
3. all Rationals
5. finally the Reals
>
> Have done the first steps yet?
> Do you have Definition for each of those Sets?
>
> And no, the Axiom of Choice is not need for any of the steps.
>

> >
> > So no AC needed because in the { } I do not use infinite choice.

>
> Hang on.
> Although { } = 0, { } is NOT a real number.
> It is a Natural Number, a Finite Ordinal.
> Now, 0.000... is a representation of a Real Number.
>
> Let me know you thoughts.
> We can walk through the process.
>
> ZG

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