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Topic: Uncountable Diagonal Problem
Replies: 52   Last Post: Jan 6, 2013 2:43 PM

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 camgirls@hush.com Posts: 12 Registered: 4/8/11
Re: Uncountable Diagonal Problem
Posted: Jan 1, 2013 7:07 PM
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On Jan 2, 9:56 am, Virgil <vir...@ligriv.com> wrote:
> In article
> <8733dfe2-163a-4e34-b402-2f018fcac...@i2g2000pbi.googlegroups.com>,
>  Graham Cooper <grahamcoop...@gmail.com> wrote:
>
>
>
>
>

> > On Dec 31 2012, 9:27 am, Virgil <vir...@ligriv.com> wrote:
> > > In article
> > > <4036660e-9527-479d-9c47-a1adf9d34...@px4g2000pbc.googlegroups.com>,
> > >  "Ross A. Finlayson" <ross.finlay...@gmail.com> wrote:

>
> > > > On Dec 30, 1:33 pm, Virgil <vir...@ligriv.com> wrote:
> > > > > In article
> > > > > <2fc759b9-3c22-4f0b-83e0-bf9814a3f...@y5g2000pbi.googlegroups.com>,
> > > > >  "Ross A. Finlayson" <ross.finlay...@gmail.com> wrote:

>
> > > > > > Formulate Cantor's nested intervals with "mega-sequences" (or
> > > > > > transfinite sequence or ordinal-indexed sequence) instead of sequences
> > > > > > of endpoints.  Well-order the reals and apply, that the sequences
> > > > > > converge yet have not emptiness between them else there would be two
> > > > > > contiguous points, in the linear continuum.

>
> > > > > Not possible with the standard reals without violating such properties
> > > > > of the reals as the LUB and GLB properties:
> > > > > Every non-empty set of reals bounded above has a real number LUB.
> > > > > Every non-empty set of reals bounded below has a real number GLB.
> > > > > --

>
> > > > Those are definitions, not derived.  Maybe they're "wrong", of the
> > > > true nature of the continuum.

>
> > >  if false for your "continuum" then that continuum is not the standard
> > > real number field.

>
> > > > A well ordering of the reals doesn't have uncountably many points in
> > > > their natural order.

>
> > > But, if one could find an explicit well-ordering of the reals, it would
> > > have to contain all those  uncountably many reals in SOME order.

>
> > LETS TRY!
>
> > LIST
> > R1   0.11111111...
> > R2   0.22222222...
> > R3   0.01010101...
> > R4   0.99999999...
> > ...

>
> > DIAGONAL =  0.1209....
>
> > WHAT ARE ALL THE MISSING REALS VIRGIL?
>
> > HINT:  you should be able to calculate 9*9*9*9 of them?
>
> Way more than that!
>
> As long as the digit replacement rule does not replace any digit with
> either a 0 or a 9, one can have as many as 8*7 = 56 different rules for
> any digit position, giving 4*56^8 nonmembers of your list.
>

Let's try a complete example then.

not worrying about 0.3333.. = 1.00000.. [base 4]

GIVEN THIS LIST

A LIST OF REALS IN [BASE 4]

R1 0.0000...
R2 0.3333...
R3 0.3210...
...

0.100... is MISSING FROM THE LIST
0.200... is MISSING FROM THE LIST
0.300... is MISSING FROM THE LIST
0.110... is MISSING FROM THE LIST
0.210... is MISSING FROM THE LIST
0.310... is MISSING FROM THE LIST
0.120... is MISSING FROM THE LIST
0.220... is MISSING FROM THE LIST
0.320... is MISSING FROM THE LIST
0.102... is MISSING FROM THE LIST
0.202... is MISSING FROM THE LIST
0.302... is MISSING FROM THE LIST
0.112... is MISSING FROM THE LIST
0.212... is MISSING FROM THE LIST
0.312... is MISSING FROM THE LIST
0.122... is MISSING FROM THE LIST
0.222... is MISSING FROM THE LIST
0.322... is MISSING FROM THE LIST
0.103... is MISSING FROM THE LIST
0.203... is MISSING FROM THE LIST
0.303... is MISSING FROM THE LIST
0.113... is MISSING FROM THE LIST
0.213... is MISSING FROM THE LIST
0.313... is MISSING FROM THE LIST
0.123... is MISSING FROM THE LIST
0.223... is MISSING FROM THE LIST
0.323... is MISSING FROM THE LIST

**********************

HINT: DIGIT 1 IS DIFFERENT TO LIST[1,1]
HINT: DIGIT 2 IS DIFFERENT TO LIST[2,2]
HINT: DIGIT 3 IS DIFFERENT TO LIST[3,3]

Herc

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