EXTEND REALS WITH AN UNKNOWN DIGIT
DIGIT = {0 1 ?}

LIST OF REALS

0. 0 0 0
0. 1 ? 1
0. 0 0 0

DIAG = 0. 0?0
ADIAG = 0 1?1 ......... ROW 2

------------------------------------------

THIS IS PROVABLE USING POWERSETS

SET = {3,5}

2^{SET} =
{ {}
{3}
{5}
{3 5}
}

3^SET =
{ {}
{3}
{5}
{3 5}
{3?}
{3? 5}
{5?}
{3 5?}
{3? 5?}
}

CANTORS "MISSING" SET

m = {x | x!e 3^S_x} = {}

On 2/13/23 7:59 PM, Graham Cooper wrote:
> EXTEND REALS WITH AN UNKNOWN DIGIT
> DIGIT = {0 1 ?}
>
> LIST OF REALS
>
> 0. 0 0 0
> 0. 1 ? 1
> 0. 0 0 0
>
> DIAG = 0. 0?0
> ADIAG = 0 1?1 ......... ROW 2
>
> ------------------------------------------
>
> THIS IS PROVABLE USING POWERSETS
>
> SET = {3,5}
>
> 2^{SET} =
> {
> {}
> {3}
> {5}
> {3 5}
> }
>
> 3^SET =
> {
> {}
> {3}
> {5}
> {3 5}
> {3?}
> {3? 5}
> {5?}
> {3 5?}
> {3? 5?}
> }
>
>
> CANTORS "MISSING" SET
>
> m = {x | x!e 3^S_x} = {}

If I understand what you are trying to do, I think you just created a
nearly worthless number system, as it loses almost all of the important
properties for real numbers, like ordering.

> > 0. 0 0 0
> > 0. 1 ? 1
> > 0. 0 0 0
> >
> > DIAG = 0. 0?0
> > ADIAG = 0 1?1 ......... ROW 2

> > CANTORS "MISSING" SET
> >
> > m = {x | x!e 3^S_x} = {}
> If I understand what you are trying to do, I think you just created a
> nearly worthless number system, as it loses almost all of the important
> properties for real numbers, like ordering.

R C ER
EXTENDED REALS is a LARGER SET than R
so all the properties are conserved

IF a LARGER SET than R is trivially countable then so is R

On 2/13/23 8:24 PM, Graham Cooper wrote:
>>> 0. 0 0 0
>>> 0. 1 ? 1
>>> 0. 0 0 0
>>>
>>> DIAG = 0. 0?0
>>> ADIAG = 0 1?1 ......... ROW 2
>
>>> CANTORS "MISSING" SET
>>>
>>> m = {x | x!e 3^S_x} = {}
>> If I understand what you are trying to do, I think you just created a
>> nearly worthless number system, as it loses almost all of the important
>> properties for real numbers, like ordering.
>
> R C ER
> EXTENDED REALS is a LARGER SET than R
> so all the properties are conserved
>
> IF a LARGER SET than R is trivially countable then so is R
>

Except that it DOESN'T have all the properites, as your "extended reals"
are not orderable.

Which is larger? 0.1?2 or 0.131 ?

Also, your "Powerset" definition seems to violate the definition of a
power set, and I suspect it gets you back into Russels paradox.

Also, I don't think you have actaully established that your extended
reals actually ARE a super set of the reals aod also countable.

On Tuesday, February 14, 2023 at 12:49:46 PM UTC+11, Richard Damon wrote:
> On 2/13/23 8:24 PM, Graham Cooper wrote:
> >>> 0. 0 0 0
> >>> 0. 1 ? 1
> >>> 0. 0 0 0
> >>>
> >>> DIAG = 0. 0?0
> >>> ADIAG = 0 1?1 ......... ROW 2
> >
> >>> CANTORS "MISSING" SET
> >>>
> >>> m = {x | x!e 3^S_x} = {}
> >> If I understand what you are trying to do, I think you just created a
> >> nearly worthless number system, as it loses almost all of the important
> >> properties for real numbers, like ordering.
> >
> > R C ER
> > EXTENDED REALS is a LARGER SET than R
> > so all the properties are conserved
> >
> > IF a LARGER SET than R is trivially countable then so is R
> >
> Except that it DOESN'T have all the properites, as your "extended reals"
> are not orderable.
>
>
> Which is larger? 0.1?2 or 0.131 ?

Just treat '?' as a 3rd digit for orderable

>
> Also, your "Powerset" definition seems to violate the definition of a
> power set, and I suspect it gets you back into Russels paradox.

Russells paradox wont work with '?' as you cannot negate it or change it into anything other than '?'

>
>
> Also, I don't think you have actaully established that your extended
> reals actually ARE a super set of the reals aod also countable.

its trivailly a superset as RDIG C ERDIG

0.1 e R
->
0.1 e ER

0.01 e R
->
0.01 e ER

On 2/13/23 9:04 PM, Graham Cooper wrote:
> On Tuesday, February 14, 2023 at 12:49:46 PM UTC+11, Richard Damon wrote:
>> On 2/13/23 8:24 PM, Graham Cooper wrote:
>>>>> 0. 0 0 0
>>>>> 0. 1 ? 1
>>>>> 0. 0 0 0
>>>>>
>>>>> DIAG = 0. 0?0
>>>>> ADIAG = 0 1?1 ......... ROW 2
>>>
>>>>> CANTORS "MISSING" SET
>>>>>
>>>>> m = {x | x!e 3^S_x} = {}
>>>> If I understand what you are trying to do, I think you just created a
>>>> nearly worthless number system, as it loses almost all of the important
>>>> properties for real numbers, like ordering.
>>>
>>> R C ER
>>> EXTENDED REALS is a LARGER SET than R
>>> so all the properties are conserved
>>>
>>> IF a LARGER SET than R is trivially countable then so is R
>>>
>> Except that it DOESN'T have all the properites, as your "extended reals"
>> are not orderable.
>>
>>
>> Which is larger? 0.1?2 or 0.131 ?
>
>
> Just treat '?' as a 3rd digit for orderable

So, you are just writing numbers in base 11? that doesn't help.

>
>
>
>
>>
>> Also, your "Powerset" definition seems to violate the definition of a
>> power set, and I suspect it gets you back into Russels paradox.
>
>
> Russells paradox wont work with '?' as you cannot negate it or change it into anything other than '?'
>
>

Why? is ? == 1 or is it different.

>
>
>
>
>>
>>
>> Also, I don't think you have actaully established that your extended
>> reals actually ARE a super set of the reals aod also countable.
>
> its trivailly a superset as RDIG C ERDIG
>
> 0.1 e R
> ->
> 0.1 e ER
>
> 0.01 e R
> ->
> 0.01 e ER
>

Nope, since R is an uncountable set, you can't complete that matching.
Until you establish a 1:1 ordered mapping from R to a subset of ER, you
haven't established that ER is at least as big as R.