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Re: WHAT HAPPENS IF YOU CHANGE THE DIAGONAL TO THE ANTIDIAGONAL?
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Graham Cooper
Nov 9, 2012, 4:13:09 PM11/9/12
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On Nov 9, 7:18 pm, forbisga...@gmail.com wrote:
> On Thursday, November 8, 2012 10:18:43 PM UTC-8, Graham Cooper wrote:
> > On Nov 9, 4:03 pm, forbisga...@gmail.com wrote:
> > > On Thursday, November 8, 2012 8:38:12 PM UTC-8, Graham Cooper wrote:
>
> > > > You change the DIAGONAL to the ANTI-DIAGONAL and no existing real can
> > > > be left on the list!
>
> > > Tell me how to produce the antidiagonal of an infinite list.
>
> > One method is to input a stream of digits and interpret them as a
> > transpose sort of the list.
>
> Wait a second.
> I'm accepting a minor repurposing of the word as defined athttp://en.wiktionary.org/wiki/antidiagonal
> What do you mean by the antidiagonal of an infinite list of
> real numbers with infinite digits after the decimal point?
> Where do you find the upper right or bottom left corner of
> an infinitely large matrix? I get the diagonal because it
> defines the first digit after the decimal point in the first
> member of the set, the second digit after the decimal point in
> the second member of the set, the nth digit after the decimal
> point in the nth member of the set.
>
> The part I cut dealt with finite sets and an arbitrary cutoff
> of digits after the decimal point. The routine you provided
> doesn't apply to Cantor's diagonalization let alone refute it.
>
> Here's a version of Cantor's argument.
> Given any ordered set of unique reals on the interval (0,1)
> represented base ten where the order is identified via a map
> to the natural numbers in their normal order, then...
> given the identification of the elements using the index function
> R=f(x) where x is index into f and R is the value the real number
> at that index... then the real number developed by summing
> case (floor(f(x) * 10^(x+1)) mod 10 ) = 0 then 7 * 0.1^n
> else ((floor(f(x) * 10^(x+1)) mod 10) - 1) * 0.1^n
> for all x won't be in the set. The reason is it differs from
> f(x) in at least the xth digit after the decimal.
If it differs from all reals on the list, then by what Epsilon>0
is the difference?
> If the number
> isn't in the ordered set it won't be in the unordered set containing
> the same reals.
The "MISSING REALS" here :
http://tinyurl.com/antidiagonals
are different to some R at the 1st digit pos
are different to some R at the 2nd digit pos
are different to some R at the 3rd digit pos
What do you notice about the "MISSING REALS" ?
Herc
> On Thursday, November 8, 2012 10:18:43 PM UTC-8, Graham Cooper wrote:
> > On Nov 9, 4:03 pm, forbisga...@gmail.com wrote:
> > > On Thursday, November 8, 2012 8:38:12 PM UTC-8, Graham Cooper wrote:
>
> > > > You change the DIAGONAL to the ANTI-DIAGONAL and no existing real can
> > > > be left on the list!
>
> > > Tell me how to produce the antidiagonal of an infinite list.
>
> > One method is to input a stream of digits and interpret them as a
> > transpose sort of the list.
>
> Wait a second.
> I'm accepting a minor repurposing of the word as defined athttp://en.wiktionary.org/wiki/antidiagonal
> What do you mean by the antidiagonal of an infinite list of
> real numbers with infinite digits after the decimal point?
> Where do you find the upper right or bottom left corner of
> an infinitely large matrix? I get the diagonal because it
> defines the first digit after the decimal point in the first
> member of the set, the second digit after the decimal point in
> the second member of the set, the nth digit after the decimal
> point in the nth member of the set.
>
> The part I cut dealt with finite sets and an arbitrary cutoff
> of digits after the decimal point. The routine you provided
> doesn't apply to Cantor's diagonalization let alone refute it.
>
> Here's a version of Cantor's argument.
> Given any ordered set of unique reals on the interval (0,1)
> represented base ten where the order is identified via a map
> to the natural numbers in their normal order, then...
> given the identification of the elements using the index function
> R=f(x) where x is index into f and R is the value the real number
> at that index... then the real number developed by summing
> case (floor(f(x) * 10^(x+1)) mod 10 ) = 0 then 7 * 0.1^n
> else ((floor(f(x) * 10^(x+1)) mod 10) - 1) * 0.1^n
> for all x won't be in the set. The reason is it differs from
> f(x) in at least the xth digit after the decimal.
If it differs from all reals on the list, then by what Epsilon>0
is the difference?
> If the number
> isn't in the ordered set it won't be in the unordered set containing
> the same reals.
The "MISSING REALS" here :
http://tinyurl.com/antidiagonals
are different to some R at the 1st digit pos
are different to some R at the 2nd digit pos
are different to some R at the 3rd digit pos
What do you notice about the "MISSING REALS" ?
Herc
William Hughes
Nov 9, 2012, 6:56:28 PM11/9/12
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On Nov 9, 5:13 pm, Graham Cooper <grahamcoop...@gmail.com> wrote:
> On Nov 9, 7:18 pm, forbisga...@gmail.com wrote:
> > Here's a version of Cantor's argument.
> > Given any ordered set of unique reals on the interval (0,1)
> > represented base ten where the order is identified via a map
> > to the natural numbers in their normal order, then...
> > given the identification of the elements using the index function
> > R=f(x) where x is index into f and R is the value the real number
> > at that index... then the real number developed by summing
> > case (floor(f(x) * 10^(x+1)) mod 10 ) = 0 then 7 * 0.1^n
> > else ((floor(f(x) * 10^(x+1)) mod 10) - 1) * 0.1^n
> > for all x won't be in the set. The reason is it differs from
> > f(x) in at least the xth digit after the decimal.
>
> If it differs from all reals on the list, then by what Epsilon>0
> is the difference?
There is no Epsilon of course. Compare: sqrt(2) differs from all
> On Nov 9, 7:18 pm, forbisga...@gmail.com wrote:
> > Here's a version of Cantor's argument.
> > Given any ordered set of unique reals on the interval (0,1)
> > represented base ten where the order is identified via a map
> > to the natural numbers in their normal order, then...
> > given the identification of the elements using the index function
> > R=f(x) where x is index into f and R is the value the real number
> > at that index... then the real number developed by summing
> > case (floor(f(x) * 10^(x+1)) mod 10 ) = 0 then 7 * 0.1^n
> > else ((floor(f(x) * 10^(x+1)) mod 10) - 1) * 0.1^n
> > for all x won't be in the set. The reason is it differs from
> > f(x) in at least the xth digit after the decimal.
>
> If it differs from all reals on the list, then by what Epsilon>0
> is the difference?
rationals
but there is no Epsilon>0 by which it differs from all rationals
Graham Cooper
Nov 9, 2012, 11:39:28 PM11/9/12
to
Herc
--
"Nothing supernatural there!" ~ Bob C
www.chatzombie.com/BUDDHA-CLOUD.gif
William Hughes
Nov 10, 2012, 12:33:35 AM11/10/12
to
proposition, if x differs from every element of S. then there is
an
Epsilon>0 such that x differs by at least epsilon
counterexample: x=sqrt(2), S the irrationals
William Hughes
Nov 10, 2012, 12:57:24 AM11/10/12
to
Graham Cooper
Nov 10, 2012, 1:06:04 AM11/10/12
to
<1 2 1 2 1 2...>
<3 3 3 3 3 3...>
<9 8 7 9 8 7...>
for a specific reason
=================
<9 9 8 6 2 5 8 3 4...>
is not different to all of
<4 2 0 5 3 4 3 4 2..>
<2 4 3 5 6 3 4 4 3..>
<0 3 3 3 0 4 4 5 5..>
...
AND SO ON..
just because you can formulate a pinpoint intersection of digit
inequality.
MODIFY-ALL-OF-SEQUENCE <4 4 3 ... > = <9 9 8 ...>
It's total nonsense if you work with infinite )sets_ of reals.
WTF is 4 4 3...? to a infinite SET NXN?
You can't PROVE ANYTHING with the nomenclature surrounded by
'uncountable' because it's an INTRINSIC DEFINITION.
That doesn't mean it HOLDS UP TO SCRUITY FOR 100 YEARS - it just HAS
NO SCRUTINY!
Herc
Uirgil
Nov 10, 2012, 1:28:37 AM11/10/12
to
In article
<f2ab8d18-7c72-4851...@m4g2000pbd.googlegroups.com>,
On the contrary, it works quite well for countably infinite sets of
reals in lists.
<f2ab8d18-7c72-4851...@m4g2000pbd.googlegroups.com>,
reals in lists.
Graham Cooper
Nov 10, 2012, 1:34:49 AM11/10/12
to
On Nov 10, 4:28 pm, Uirgil <uir...@uirgil.ur> wrote:
> In article
> <f2ab8d18-7c72-4851-92f1-e49dda4e2...@m4g2000pbd.googlegroups.com>,
You equate LIST with COUNTABLE SET
hence you missed the point completely.
There is no diagonal of an infinite set.
Herc
> In article
> <f2ab8d18-7c72-4851-92f1-e49dda4e2...@m4g2000pbd.googlegroups.com>,
hence you missed the point completely.
There is no diagonal of an infinite set.
Herc
Uirgil
Nov 10, 2012, 1:54:21 AM11/10/12
to
In article
<27de2871-d7a9-4cef...@u4g2000pbo.googlegroups.com>,
Lists ARE countable sets, though not all sets need to be lists.
infinite binary sequences. There is no anti-diagonal for, say, the set
of all reals or the set of al binary sequnces.
>
<27de2871-d7a9-4cef...@u4g2000pbo.googlegroups.com>,
>
> There is no diagonal of an infinite set.
There is an anti-diagonal for every finite or countably infinite list of
> There is no diagonal of an infinite set.
infinite binary sequences. There is no anti-diagonal for, say, the set
of all reals or the set of al binary sequnces.
>
William Hughes
Nov 10, 2012, 8:39:16 AM11/10/12
to
On Nov 10, 2:06 am, Graham Cooper <grahamcoop...@gmail.com> wrote:
<snip>
> MODIFY-ALL-OF-SEQUENCE <4 4 3 ... > = <9 9 8 ...>
>
> It's total nonsense if you work with infinite )sets_ of reals.
sqrt(2) an infinite sequence of digits you can define.
ANTIDIAG(L) an infinite sequence of digits you cannot define.
<snip>
> MODIFY-ALL-OF-SEQUENCE <4 4 3 ... > = <9 9 8 ...>
>
> It's total nonsense if you work with infinite )sets_ of reals.
ANTIDIAG(L) an infinite sequence of digits you cannot define.
Graham Cooper
Nov 10, 2012, 7:40:13 PM11/10/12
to
.... TM# TAPE
UTM( index, digitpos ) e {0 1 2 3 4 5 6 7 8 9} TAPE AFTER HALT
<->
reals(index)_digitpos e {0 1 2 3 4 5 6 7 8 9}
reals(index)_digitpos = 0 OTHERWISE
AD(i) = flip( reals(i)_i )
Herc
Graham Cooper
Nov 10, 2012, 7:42:50 PM11/10/12
to
> > You equate LIST with COUNTABLE SET
> > hence you missed the point completely.
>
> Lists ARE countable sets, though not all sets need to be lists.
>
This PROVES you MISSED THE POINT!
> > hence you missed the point completely.
>
> Lists ARE countable sets, though not all sets need to be lists.
>
Countable SETS are NOT LISTS.
This set has no diagonal.
0.102 0.434 0.543
0.245 0.545 0.544
0.095 0.111 0.999
Herc
Uirgil
Nov 10, 2012, 11:35:52 PM11/10/12
to
In article
<a1128ee0-b224-45a9...@v6g2000pbb.googlegroups.com>,
proof that they are countable.
0.434000000
0.543000000
0.245000000
0.545000000
0.544000000
0.095000000
0.111000000
0.999000000
Now it does!
<a1128ee0-b224-45a9...@v6g2000pbb.googlegroups.com>,
Graham Cooper <graham...@gmail.com> wrote:
> > > You equate LIST with COUNTABLE SET
> > > hence you missed the point completely.
> >
> > Lists ARE countable sets, though not all sets need to be lists.
> >
>
>
> This PROVES you MISSED THE POINT!
>
> Countable SETS are NOT LISTS.
But their members necessarily are capable of being listed by the very
> > > You equate LIST with COUNTABLE SET
> > > hence you missed the point completely.
> >
> > Lists ARE countable sets, though not all sets need to be lists.
> >
>
>
> This PROVES you MISSED THE POINT!
>
> Countable SETS are NOT LISTS.
proof that they are countable.
>
> This set has no diagonal.
>
> 0.102 0.434 0.543
> 0.245 0.545 0.544
> 0.095 0.111 0.999
>
>
> Herc
0.102000000
> This set has no diagonal.
>
> 0.102 0.434 0.543
> 0.245 0.545 0.544
> 0.095 0.111 0.999
>
>
> Herc
0.434000000
0.543000000
0.245000000
0.545000000
0.544000000
0.095000000
0.111000000
0.999000000
Now it does!
Graham Cooper
Nov 11, 2012, 4:59:42 PM11/11/12
to
On Nov 11, 2:35 pm, Uirgil <uir...@uirgil.ur> wrote:
> In article
> <a1128ee0-b224-45a9-bae9-56160c2ac...@v6g2000pbb.googlegroups.com>,
> In article
> Graham Cooper <grahamcoop...@gmail.com> wrote:
>
> > > > You equate LIST with COUNTABLE SET
> > > > hence you missed the point completely.
>
> > > Lists ARE countable sets, though not all sets need to be lists.
>
> > This PROVES you MISSED THE POINT!
>
> > Countable SETS are NOT LISTS.
>
> But their members necessarily are capable of being listed by the very
> proof that they are countable.
>
>
>
> > This set has no diagonal.
>
> > 0.102 0.434 0.543
> > 0.245 0.545 0.544
> > 0.095 0.111 0.999
>
> > Herc
>
> 0.102000000
> 0.434000000
> 0.543000000
> 0.245000000
> 0.545000000
> 0.544000000
> 0.095000000
> 0.111000000
> 0.999000000
>
> Now it does!
>
like looking at the position of an electron with a microscope and
>
> > > > You equate LIST with COUNTABLE SET
> > > > hence you missed the point completely.
>
> > > Lists ARE countable sets, though not all sets need to be lists.
>
> > This PROVES you MISSED THE POINT!
>
> > Countable SETS are NOT LISTS.
>
> But their members necessarily are capable of being listed by the very
> proof that they are countable.
>
>
>
> > This set has no diagonal.
>
> > 0.102 0.434 0.543
> > 0.245 0.545 0.544
> > 0.095 0.111 0.999
>
> > Herc
>
> 0.102000000
> 0.434000000
> 0.543000000
> 0.245000000
> 0.545000000
> 0.544000000
> 0.095000000
> 0.111000000
> 0.999000000
>
> Now it does!
>
claiming the position is fixed.
Herc
Graham Cooper
Nov 12, 2012, 4:03:44 PM11/12/12
to
On Nov 12, 6:16 pm, David Bernier <david...@videotron.ca> wrote:
> On 11/12/2012 03:02 AM, Uirgil wrote:
>
> > In article
> > <acc4f4cb-7f4d-4a0d-bb27-c9682b41e...@r10g2000pbd.googlegroups.com>,
> > Graham Cooper<grahamcoop...@gmail.com> wrote:
>
> >> On Nov 12, 10:23 am, forbisga...@gmail.com wrote:
> >>> set has a fixed index. The same number will alway appear at the same
> >>> index. You appear to be claiming there is no fixed index for a well
> >>> ordered set.
>
> >> I claim 20 flaws in Cantor's proof!
>
> > A lot of nuts have claimed flaws in Cantor's proofs, but, as yet, none
> > of those nuts has proved not to be cracked.
>
> [...]
>
> I think one ontological (non-mathematical)
> problematic is that if something can't be
> listed, enumerated, then it can't be shown (like
> in "show the story", don't just "tell the story"
> in journalism).
>
> The story of the reals: if it could be shown completely
> in a movie, that would be enumerable.
>
> The non-believers are not satisfied with a logical proof ...
>
> Dave
>
It's NOTHING CLOSE to a logical proof!
2OL
ALL(f):N->R E(r):R A(n):N
f(n) =/= r
clearly is 2OL!
---------------
ZFC AXIOM 9
ALL(X) E(R) (R well-orders X)
THIS IS NOT 1OL or even 2OL!
----------------
YOU ALL LIE YOU HAVE PROVEN X>OO IN FIRST ORDER LOGIC
YOU ALL LIE YOU HAVE A FORMAL PROOF OF X>OO
YOU DON'T MAKE ANY TESTABLE CLAIM
YOU DON'T UTILISE ANY CARDINALITY THEOREM
YOU FAIL TO ANSWER ANY QUESTIONS
*****************************************************
*****************************************************
Q4
How can there be uncountable many GODEL NUMBERS like this?
20130415
a01(0,1)
MIDPOINT(0,1)
A CHOICE FUNCTION
*****************************************************
*****************************************************
Q5
Which 1 of these does not hold?
a) N <-BIJECTS-> GODEL NUMBERS
b) GODEL NUMBERS <-BIJECT-> FUNCTIONS
c) FUNCTIONS <-BIJECT-> CHOICE FUNCTIONS
c) CHOICE FUNCTIONS <-BIJECT-> SETS
d) |SETS| > |N|
*****************************************************
*****************************************************
Q6
Does this Anti-Diagonal Method produce any unique
digit segment not listed?
AD METHOD
Choose the number 0.a_1a_2a_3...., where a_i = 1 if the i-th
number in your list had zero in its i-position, a_i = 0 otherwise.
LIST
R1= < <314><15><926><535><8979><323> ... >
R2= < <27><18281828><459045><235360> ... >
R3= < <333><333><333><333><333><333> ... >
R4= < <888888888888888888888><8><88> ... >
R5= < <0123456789><0123456789><01234 ... >
R6= < <1><414><21356><2373095><0488> ... >
....
G. Cooper (BInfTech)
> On 11/12/2012 03:02 AM, Uirgil wrote:
>
> > In article
> > <acc4f4cb-7f4d-4a0d-bb27-c9682b41e...@r10g2000pbd.googlegroups.com>,
> > Graham Cooper<grahamcoop...@gmail.com> wrote:
>
> >> On Nov 12, 10:23 am, forbisga...@gmail.com wrote:
> >>> On Sunday, November 11, 2012 1:59:42 PM UTC-8, Graham Cooper wrote:
> >>>> like looking at the position of an electron with a microscope and
> >>>> claiming the position is fixed.
>
> >>> The problem is Cantor dealt with a well ordered set. A well ordered
> >>>> like looking at the position of an electron with a microscope and
> >>>> claiming the position is fixed.
>
> >>> set has a fixed index. The same number will alway appear at the same
> >>> index. You appear to be claiming there is no fixed index for a well
> >>> ordered set.
>
> >> I claim 20 flaws in Cantor's proof!
>
> > A lot of nuts have claimed flaws in Cantor's proofs, but, as yet, none
> > of those nuts has proved not to be cracked.
>
> [...]
>
> I think one ontological (non-mathematical)
> problematic is that if something can't be
> listed, enumerated, then it can't be shown (like
> in "show the story", don't just "tell the story"
> in journalism).
>
> The story of the reals: if it could be shown completely
> in a movie, that would be enumerable.
>
> The non-believers are not satisfied with a logical proof ...
>
> Dave
>
It's NOTHING CLOSE to a logical proof!
2OL
ALL(f):N->R E(r):R A(n):N
f(n) =/= r
clearly is 2OL!
---------------
ZFC AXIOM 9
ALL(X) E(R) (R well-orders X)
THIS IS NOT 1OL or even 2OL!
----------------
YOU ALL LIE YOU HAVE PROVEN X>OO IN FIRST ORDER LOGIC
YOU ALL LIE YOU HAVE A FORMAL PROOF OF X>OO
YOU DON'T MAKE ANY TESTABLE CLAIM
YOU DON'T UTILISE ANY CARDINALITY THEOREM
YOU FAIL TO ANSWER ANY QUESTIONS
*****************************************************
*****************************************************
Q4
How can there be uncountable many GODEL NUMBERS like this?
20130415
a01(0,1)
MIDPOINT(0,1)
A CHOICE FUNCTION
*****************************************************
*****************************************************
Q5
Which 1 of these does not hold?
a) N <-BIJECTS-> GODEL NUMBERS
b) GODEL NUMBERS <-BIJECT-> FUNCTIONS
c) FUNCTIONS <-BIJECT-> CHOICE FUNCTIONS
c) CHOICE FUNCTIONS <-BIJECT-> SETS
d) |SETS| > |N|
*****************************************************
*****************************************************
Q6
Does this Anti-Diagonal Method produce any unique
digit segment not listed?
AD METHOD
Choose the number 0.a_1a_2a_3...., where a_i = 1 if the i-th
number in your list had zero in its i-position, a_i = 0 otherwise.
LIST
R1= < <314><15><926><535><8979><323> ... >
R2= < <27><18281828><459045><235360> ... >
R3= < <333><333><333><333><333><333> ... >
R4= < <888888888888888888888><8><88> ... >
R5= < <0123456789><0123456789><01234 ... >
R6= < <1><414><21356><2373095><0488> ... >
....
G. Cooper (BInfTech)
William Hughes
Nov 12, 2012, 4:55:55 PM11/12/12
to
On Nov 12, 5:03 pm, Graham Cooper <grahamcoop...@gmail.com> wrote:
*****************************************************
> Q5
> Which 1 of these does not hold?
>
> a) N <-BIJECTS-> GODEL NUMBERS
>
> b) GODEL NUMBERS <-BIJECT-> FUNCTIONS
>
> c) FUNCTIONS <-BIJECT-> CHOICE FUNCTIONS
>
> c) CHOICE FUNCTIONS <-BIJECT-> SETS
>
> d) |SETS| > |N|
> *****************************************************
> *****************************************************
b) is false. What is true is that
*****************************************************
> Q5
> Which 1 of these does not hold?
>
> a) N <-BIJECTS-> GODEL NUMBERS
>
> b) GODEL NUMBERS <-BIJECT-> FUNCTIONS
>
> c) FUNCTIONS <-BIJECT-> CHOICE FUNCTIONS
>
> c) CHOICE FUNCTIONS <-BIJECT-> SETS
>
> d) |SETS| > |N|
> *****************************************************
> *****************************************************
GODEL NUMBERS <-BIJECT-> CONSTRUCTABLE FUNCTIONS
However
CONSTRUCTABLE FUNCTIONS <-NOT BIJECT-> CHOICE FUNCTIONS
Graham Cooper
Nov 13, 2012, 4:10:23 AM11/13/12
to
Are you using an uncountable infinite alphabet of Logic Symbols?
Or are you imagining infinite length of logic symbols choice functions
for every set?
Herc
William Hughes
Nov 13, 2012, 8:52:23 AM11/13/12
to
On Nov 13, 5:10 am, Graham Cooper <grahamcoop...@gmail.com> wrote:
> On Nov 13, 7:55 am, William Hughes <wpihug...@gmail.com> wrote:
>
>
>
>
>
>
>
>
>
> > On Nov 12, 5:03 pm, Graham Cooper <grahamcoop...@gmail.com> wrote:
> > *****************************************************
>
> > > Q5
> > > Which 1 of these does not hold?
>
> > > a) N <-BIJECTS-> GODEL NUMBERS
>
> > > b) GODEL NUMBERS <-BIJECT-> FUNCTIONS
>
> > > c) FUNCTIONS <-BIJECT-> CHOICE FUNCTIONS
>
> > > c) CHOICE FUNCTIONS <-BIJECT-> SETS
>
> > > d) |SETS| > |N|
> > > *****************************************************
> > > *****************************************************
>
> > b) is false. What is true is that
>
> > GODEL NUMBERS <-BIJECT-> CONSTRUCTABLE FUNCTIONS
>
> > However
>
> > CONSTRUCTABLE FUNCTIONS <-NOT BIJECT-> CHOICE FUNCTIONS
>
> So what's an unlistable choice function example?
There is of course, no such thing (given any x, you can always
> On Nov 13, 7:55 am, William Hughes <wpihug...@gmail.com> wrote:
>
>
>
>
>
>
>
>
>
> > On Nov 12, 5:03 pm, Graham Cooper <grahamcoop...@gmail.com> wrote:
> > *****************************************************
>
> > > Q5
> > > Which 1 of these does not hold?
>
> > > a) N <-BIJECTS-> GODEL NUMBERS
>
> > > b) GODEL NUMBERS <-BIJECT-> FUNCTIONS
>
> > > c) FUNCTIONS <-BIJECT-> CHOICE FUNCTIONS
>
> > > c) CHOICE FUNCTIONS <-BIJECT-> SETS
>
> > > d) |SETS| > |N|
> > > *****************************************************
> > > *****************************************************
>
> > b) is false. What is true is that
>
> > GODEL NUMBERS <-BIJECT-> CONSTRUCTABLE FUNCTIONS
>
> > However
>
> > CONSTRUCTABLE FUNCTIONS <-NOT BIJECT-> CHOICE FUNCTIONS
>
> So what's an unlistable choice function example?
start a list with x).
However, given any x there is a list L(x) such that x is
in L(x), does not mean there is a single list L that
contains all x.
There is no list of all choice functions.
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