An unknown digit [?] can be anything from 0..9

A = 3.?
3.0 <= A <= 3.9

B = 3.?4
3.04 <= B <= 3.94

------------ ADDING 2 UDRs ------------

C = 1.? + 2.?
1.0+2.0 <= C <= 1.9+2.9
3 <= C <= 4.8

----------- MULTIPLYING 2 UDRs -----------

D = 1.? * 2.?
1.0*2.0 <= D <= 1.9*2.9
2 <= D <= 5.51

------------- LISTING ALL UDRs ------------

UDRs are countable because you can list the ANTIDIAGONAL

r1 = 0.[1]234..
r2 = 0.5[5]55..
r3 = 0.56[?]65.. <--------- ANTIDIAGONAL [5/?] [6/5]
r4 = 0.234[5]6..
....

|UDR| = |N|

------------ |R| = |N| --------------

UDRs are a SUPERSET of REALS

R C UDR
|R| = |UDR|

|R| = |N|

On 6/3/2022 3:14 PM, Graham Cooper wrote:
> An unknown digit [?] can be anything from 0..9
>
> A = 3.?
> 3.0 <= A <= 3.9
>
> B = 3.?4
> 3.04 <= B <= 3.94
>

Shut up dickwad.

On Saturday, June 4, 2022 at 12:14:10 AM UTC+2, Graham Cooper wrote:

> An unknown digit [?] can be anything from 0..9

Indeed! But we usually use "letters" (with and without indices) in math to refer to such digits.
> A = 3.?

A = 3.d with d in {0, 1, 2, 3, 4, 5, 6, 7, 8, 9}.

> 3.0 <= A <= 3.9

> B = 3.?4

B = 3.d4 with d in {0, 1, 2, 3, 4, 5, 6, 7, 8, 9}.

> 3.04 <= B <= 3.94

Right.
> ------------ ADDING 2 UDRs ------------
>
> C = 1.? + 2.?

C = 1.d + 2.e with d,e in {0, 1, 2, 3, 4, 5, 6, 7, 8, 9}.

[if you want to allow for different "unknown digits"].

> 1.0+2.0 <= C <= 1.9+2.9
> 3 <= C <= 4.8

Right.

> ----------- MULTIPLYING 2 UDRs -----------
>
> D = 1.? * 2.?

D = 1.d * 2.e with d,e in {0, 1, 2, 3, 4, 5, 6, 7, 8, 9}.

[if you want to allow for different "unknown digits"].

> 1.0*2.0 <= D <= 1.9*2.9
> 2 <= D <= 5.51

Right.

> ------------- LISTING [...] UDRs ------------
>
> r1 = 0.[1]234..
> r2 = 0.5[5]55..
> r3 = 0.56[d]65..
> r4 = 0.234[5]6..
> ...

=> diag = 0.15d5...

=> adiag = 0.56e6... where e = 6 if d = 5 and e = 5 if d in {0, 1, 2, 3, 4, 6, 7, 8, 9}

The general "transformation rule" in this case is d' = 6 if d = 5 and d' = 5 if d in {0, 1, 2, 3, 4, 6, 7, 8, 9}.

On Tuesday, June 7, 2022 at 8:58:15 PM UTC+10, Fritz Feldhase wrote:
> On Saturday, June 4, 2022 at 12:14:10 AM UTC+2, Graham Cooper wrote:
>
> > An unknown digit [?] can be anything from 0..9
> Indeed! But we usually use "letters" (with and without indices) in math to refer to such digits.
>
> > A = 3.?
>
> A = 3.d with d in {0, 1, 2, 3, 4, 5, 6, 7, 8, 9}.
> > 3.0 <= A <= 3.9
>
> > B = 3.?4
> B = 3.d4 with d in {0, 1, 2, 3, 4, 5, 6, 7, 8, 9}.
> > 3.04 <= B <= 3.94
> Right.
> > ------------ ADDING 2 UDRs ------------
> >
> > C = 1.? + 2.?
> C = 1.d + 2.e with d,e in {0, 1, 2, 3, 4, 5, 6, 7, 8, 9}.
>
> [if you want to allow for different "unknown digits"].
> > 1.0+2.0 <= C <= 1.9+2.9
> > 3 <= C <= 4.8
> Right.
> > ----------- MULTIPLYING 2 UDRs -----------
> >
> > D = 1.? * 2.?
> D = 1.d * 2.e with d,e in {0, 1, 2, 3, 4, 5, 6, 7, 8, 9}.
>
> [if you want to allow for different "unknown digits"].
> > 1.0*2.0 <= D <= 1.9*2.9
> > 2 <= D <= 5.51
> Right.
>
> > ------------- LISTING [...] UDRs ------------
> >
> > r1 = 0.[1]234..
> > r2 = 0.5[5]55..
> > r3 = 0.56[d]65..
> > r4 = 0.234[5]6..
> > ...
> => diag = 0.15d5...
>
> => adiag = 0.56e6... where e = 6 if d = 5 and e = 5 if d in {0, 1, 2, 3, 4, 6, 7, 8, 9}
>
> The general "transformation rule" in this case is d' = 6 if d = 5 and d' = 5 if d in {0, 1, 2, 3, 4, 6, 7, 8, 9}.

Unknown is more like a 3rd digit

[0] [1] [?]

Or a JOKER CARD!

Its trivial to EXTEND the REALS with A WILDCARD SYMBOL so the ANTI-DIAGONAL is LISTED

R1 0 . [x] xxxxxxxx
R2 0 . x [x] xxxxxxx
R3 0 . xx [?] xxxxxx <----------------------- LISTABLE ANTI-DIAGONAL
R3 0 . xxx [x] xxxxx

THESE ARE NOT REALS!
? IS NOT A VARIABLE - ITS A TOKEN!
? IS NOT A 3RD DIGIT - ITS A TOKEN!

UDR C REALS

The CONCEPT to have a LIST with a LISTABLE ANTI-DIAGONAL by using a WILD CARD is really very simple!

And 100 maths "gurus" cannot fathom a kids game

> [0] [1] [?]
>
> Or a JOKER CARD!
>
> Its trivial to EXTEND the REALS with A WILDCARD SYMBOL so the ANTI-DIAGONAL is LISTED
>
> R1 0 . [x] xxxxxxxx
> R2 0 . x [x] xxxxxxx
> R3 0 . xx [?] xxxxxx <----------------------- LISTABLE ANTI-DIAGONAL
> R3 0 . xxx [x] xxxxx
>
>
> THESE ARE NOT REALS!
> ? IS NOT A VARIABLE - ITS A TOKEN!
> ? IS NOT A 3RD DIGIT - ITS A TOKEN!
>
> UDR C REALS

should read

REALS C UDR

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

eg

REALS
0.323
0.232
0.233

UDR
0.323
0.77?
0.12?

UDR is BIGGER THAN REALS because it has JOKER CARDS in place of digits

On Tuesday, June 7, 2022 at 1:15:28 PM UTC+2, Graham Cooper wrote:

> Unknown is more like a 3rd digit
>
> [0] [1] [?]

Nope, you made that clear in the context of the decimal system:

> An unknown digit [?] can be anything from 0..9

So ? = 0 or ? = 1 or ... or ? = 9.

'?' is not an _additional_ digit.

Hence in the context of a binary system:

? in {0, 1}.

On Tuesday, June 7, 2022 at 10:09:05 PM UTC+10, Fritz Feldhase wrote:
> On Tuesday, June 7, 2022 at 1:15:28 PM UTC+2, Graham Cooper wrote:
>
> > Unknown is more like a 3rd digit
> >
> > [0] [1] [?]
> Nope, you made that clear in the context of the decimal system:
> > An unknown digit [?] can be anything from 0..9
> So ? = 0 or ? = 1 or ... or ? = 9.
>
> '?' is not an _additional_ digit.
>
> Hence in the context of a binary system:
>
> ? in {0, 1}.

The idea is:

R1 0. _ _ _
R2 0 . _ ? _
R3 0 . _ _ _

to use a WILDCARD for a digit that is UNKNOWN
so that the ANTI-DIAGONAL can be listed

IF your saying this is impossible I do not see your reasoning, its TRIVIAL actually

LET [?] have 2 phases - unknown value token and newly established value

The anti-diagonal is really a PROGRAM and PROGRAMS are LISTABLE

PROGRAM_n
m = 1
while true
... D = LIST_m_m
... if D = ? then AD = ?
... else AD = 1-D
... LIST_n_m = AD
... m = m + 1
loop

so the ANTIDIAGONAL is LISTABLE with this usage of "?"

NOTE: if D = ? then AD = ?
you cannot transform an unknown digit

On Tuesday, June 7, 2022 at 2:24:32 PM UTC+2, Graham Cooper wrote:

> NOTE: if D = ? then AD = ?

That's not math, but mumbo jumbo.

In math we usually use letters (with or without indices) instead of "?". This letters may refer to unknown but "fixed" values.

Let d in {0, 1}. Then d is unknown but either 0 or 1.

> you cannot transform an unknown digit

Of course we can: Let d' = 1 - d. Then d' e {0, 1} and d' =/= d.

Look dumbo, if we consider an arbitrary number r in [0, 1] then all its digits are unknown.

Now in math we do not write

r = 0. ? ? ? ? ...

but rather, say,

r = d_1 d_2 d_3 ... with d_i e {0, 1} for all i e IN

in this case.
In addition we may formulate the "transformation rule" d' = 1 - d for d in {0, 1}. We do not have to know "the value" of d_i to apply this rule to d_i (for any i e IN).

EOD

On Tuesday, June 7, 2022 at 11:10:28 PM UTC+10, Fritz Feldhase wrote:
> On Tuesday, June 7, 2022 at 2:24:32 PM UTC+2, Graham Cooper wrote:
>
> > NOTE: if D = ? then AD = ?
> That's not math, but mumbo jumbo.
>
> In math we usually use letters (with or without indices) instead of "?". This letters may refer to unknown but "fixed" values.
>
> Let d in {0, 1}. Then d is unknown but either 0 or 1.
> > you cannot transform an unknown digit
> Of course we can: Let d' = 1 - d. Then d' e {0, 1} and d' =/= d.
>
> Look dumbo, if we consider an arbitrary number r in [0, 1] then all its digits are unknown.
>
> Now in math we do not write
>
> r = 0. ? ? ? ? ...
>
> but rather, say,
>
> r = d_1 d_2 d_3 ... with d_i e {0, 1} for all i e IN
>
> in this case.
>
> In addition we may formulate the "transformation rule" d' = 1 - d for d in {0, 1}. We do not have to know "the value" of d_i to apply this rule to d_i (for any i e IN).
>
>
> EOD

Wow what a load of worthless tripe!

REAL 1 0 . 1 1 1 1
REAL 2 0 . 1 1 1 1
REAL 3 0. 0 0 ? 0
REAL 4 0 . 1 1 1 1

SEE THE DIAGONAL 11?1
AND REAL 3 00?0

Your so stupid as to call that impossible to list the AD using an UNKNOWN TOKEN

the whole newsgroup is entirely retarded!

a dozen poofter responses not 1 post ON TOPIC!

NOT 1 ACKNOLEDGEMENT I LISTED THE ANTI-DIAGONAL !

HOORAY HOORAY!

CANTORS PROOF IS NO MORE!

NO! JUST IDIOTS DRIVEL FROM ALL OF YOU