The Computable Reals (alpha version)

On 12 Aug, 19:25, ju...@diegidio.name wrote:

> 'oo-n' is the (n-10)-th index for the descending enumeration.

Sorry, should read:

'oo-n' is the (n+1)-th index for a descending enumeration.

-LV

julio@diegidio.name writes:

> On 12 Aug, 18:43, "David C. Ullrich" <dullr...@sprynet.com> wrote:
>
>> > As to why I believe (think) that the list is "complete": because it is
>> > the _complete_ (over N*) list of _all_ the possible infinite (over N*)
>> > binary expansions.
>>
>> No, that's not possible.
>
> In this context, funny statement to say the least.
>
>> > Indeed, what I have given IS _per definition_ Â*the list of the
>> > "computable reals" (modulo the usual adjustments).
>>
>> No, you haven't. There _is_ a standard definition of "computable
>> real", and it simply doesn't appear anywhere in your post.
>
> Another funny statement, even more funny considering that the standard
> definition happens to correspond to mine were it not for your pre-
> judgement.

One way seems reasonable -- that all the numbers on your list are
computable -- but the other way is far from obvious. I still can't
see how any number with a pattern in its digits can be there and I am
equally puzzled about how numbers like pi-3 (computable) can be on
your list. There are details in your construction that still bother
me, but I don't think they introduce another countable infinity of
numbers so I doubt these worries will go away.

--
Ben.

On 12 Aug, 19:47, Ben Bacarisse <ben.use...@bsb.me.uk> wrote:
> ju...@diegidio.name writes:
> > On 12 Aug, 18:43, "David C. Ullrich" <dullr...@sprynet.com> wrote:
>
> >> > As to why I believe (think) that the list is "complete": because it is
> >> > the _complete_ (over N*) list of _all_ the possible infinite (over N*)
> >> > binary expansions.
>
> >> No, that's not possible.
>
> > In this context, funny statement to say the least.
>
> >> > Indeed, what I have given IS _per definition_ the list of the
> >> > "computable reals" (modulo the usual adjustments).
>
> >> No, you haven't. There _is_ a standard definition of "computable
> >> real", and it simply doesn't appear anywhere in your post.
>
> > Another funny statement, even more funny considering that the standard
> > definition happens to correspond to mine were it not for your pre-
> > judgement.
>
> One way seems reasonable -- that all the numbers on your list are
> computable -- but the other way is far from obvious. I still can't
> see how any number with a pattern in its digits can be there and I am
> equally puzzled about how numbers like pi-3 (computable) can be on
> your list.

Pardon me, how does pi-3 expands in binary?

> There are details in your construction that still bother
> me, but I don't think they introduce another countable infinity of
> numbers so I doubt these worries will go away.

There may be some sort of "point of singularity" or "double-point"
resulting from the stated equality LimE_p = LimE_q, an equality that
is key to closing the infinite case. I can't realize what that entails
to the structure of our R*, that's too specialistic for me.
Singularity, double point, moebius-torus, inconsistency, ice-cream,
I'll better be listening!

A point here might be worth noting, about the progression:

-- 1st, the construction: we first have the rules for the construction
of an "bi-enumerable list" defined so and so, and with a definition
extended so that the list keeps working consistently in the infinite
case.

-- 2nd, the reals: we then disregard the finite case and consider only
the list in the infinite (boundary) case, that is, for w = oo. Under
*this* case we establish a correspondence between our (infinite) list
and such and such interval of a definiendum set R*.

Another point I'll mention is that I would think that extending to
bases greater than two adds nothing to the construction (and so to the
underlying argument) I have given, modulo etc. etc. This does *not*
entail that there may not be significant properties of our R* emerging
from considerations over bases greater than two and, actually, by
generalizing to any base. Again, beyond me.

-LV

> --
> Ben.

julio@diegidio.name writes:

> On 12 Aug, 19:47, Ben Bacarisse <ben.use...@bsb.me.uk> wrote:
>> ju...@diegidio.name writes:
>> > On 12 Aug, 18:43, "David C. Ullrich" <dullr...@sprynet.com> wrote:
>>
>> >> > As to why I believe (think) that the list is "complete": because it is
>> >> > the _complete_ (over N*) list of _all_ the possible infinite (over N*)
>> >> > binary expansions.
>>
>> >> No, that's not possible.
>>
>> > In this context, funny statement to say the least.
>>
>> >> > Indeed, what I have given IS _per definition_ the list of the
>> >> > "computable reals" (modulo the usual adjustments).
>>
>> >> No, you haven't. There _is_ a standard definition of "computable
>> >> real", and it simply doesn't appear anywhere in your post.
>>
>> > Another funny statement, even more funny considering that the standard
>> > definition happens to correspond to mine were it not for your pre-
>> > judgement.
>>
>> One way seems reasonable -- that all the numbers on your list are
>> computable -- but the other way is far from obvious. I still can't
>> see how any number with a pattern in its digits can be there and I am
>> equally puzzled about how numbers like pi-3 (computable) can be on
>> your list.
>
> Pardon me, how does pi-3 expands in binary?

It is a computable real (by accepted definition which you said,
elsewhere is effectively the same as yours). If pi-3 is excluded how
many other number are missing from your list. I still claim 1/3, 1/5,
1/6 and so on are also missing.

>> There are details in your construction that still bother
>> me, but I don't think they introduce another countable infinity of
>> numbers so I doubt these worries will go away.
>
> There may be some sort of "point of singularity" or "double-point"
> resulting from the stated equality LimE_p = LimE_q, an equality that
> is key to closing the infinite case. I can't realize what that entails
> to the structure of our R*, that's too specialistic for me.
> Singularity, double point, moebius-torus, inconsistency, ice-cream,
> I'll better be listening!
>
> A point here might be worth noting, about the progression:
>
> -- 1st, the construction: we first have the rules for the construction
> of an "bi-enumerable list" defined so and so, and with a definition
> extended so that the list keeps working consistently in the infinite
> case.
>
> -- 2nd, the reals: we then disregard the finite case and consider only
> the list in the infinite (boundary) case, that is, for w = oo. Under
> *this* case we establish a correspondence between our (infinite) list
> and such and such interval of a definiendum set R*.
>
> Another point I'll mention is that I would think that extending to
> bases greater than two adds nothing to the construction (and so to the
> underlying argument) I have given, modulo etc. etc. This does *not*
> entail that there may not be significant properties of our R* emerging
> from considerations over bases greater than two and, actually, by
> generalizing to any base. Again, beyond me.

No idea what you are talking about, but that is fine. I am happy to
play in the foothills. Where is 0.101010101010101... on your list?
Genuine question, since I get lost in part of your construction so it
may well be there along with all the other "repeat pattern" numbers.

--
Ben.

<julio@diegidio.name> wrote in message
news:9d3af44f-8119-469d-acd1-4bfed792a06e@25g2000hsx.googlegroups.com...
On 12 Aug, 18:43, "David C. Ullrich" <dullr...@sprynet.com> wrote:

> > As to why I believe (think) that the list is "complete": because it is
> > the _complete_ (over N*) list of _all_ the possible infinite (over N*)
> > binary expansions.
>
> No, that's not possible.

In this context, funny statement to say the least.

> > Indeed, what I have given IS _per definition_ the list of the
> > "computable reals" (modulo the usual adjustments).
>
> No, you haven't. There _is_ a standard definition of "computable
> real", and it simply doesn't appear anywhere in your post.

Another funny statement, even more funny considering that the standard
definition happens to correspond to mine were it not for your pre-
judgement.

I wander what you really don't know. For instance, it is far from a
pleasure, apart from how improductive, to keep on these tones. You
maybe don't take me for serious on this.

-LV

*****************************************
I think the reason that some people use a dismissive tone is because of the
tone you use.

As I understand it, you purport to show a way of enumerating the computable
reals. Your construction obviously doesn't work, as a simple diagonalisation
argument will produce computable Reals not on the list.

Do you accept that your construction contains an error? If you don't, then
you are claiming that set theory is inconsistent, which is a pretty bold
claim, particularly for a self-admitted beginner.

If you accept that your construction is wrong, then you are basically asking
for somebody to help you find the error in your construction. I can provide
some assistance with that.

Your construction can fail in two ways:

1. It doesn't construct a list at all, or
2. It constructs a list, but this does not include all the computable Reals.

The first is the place to start.

What is the first computable Real produced by your construction? What are
the second and third computable Reals it produces?

If you can't answer that, then your construction does not provide a list at
all, and hence you don't even get to first base.

So I would check that first.

If you can demonstrate that it produces a list, then we can move onto the
second possibility. I suspect we won't even get that far.

HTH


Peter Webb

ju...@diegidio.name wrote:

[near the end]
> / Very tentative, just some ideas. /

Fair enough. I haven't got very far through, but here are a few
questions.

> Julio Di Egidio (JDE)
> julio at diegidio dot name
> (c)2008 JDE, on behalf of sci.math, sci.logic

What does this ("on behalf of") mean? Are you claiming copyright in
what you wrote (which you are certainly entitled to), or assigning it
to sci.math (what for?)??

> We are going to deal with infinite decimal expansions in binary
> coding.

What is that, _exactly_? You really mean a _decimal_ expansion in
binary??

> N* := N u { 0, oo }
>
> w e N* // omega
> n e N*
>
> =============================================
> === Preliminaries ===========================
> =============================================
>
> Some preliminary inductive definitions:
>
> --- numToBin(n) / returns the binary representation of n /
>
> Ascending:
> numToBin(0) := "0"
> numToBin(1) := "1"
> numToBin(2) := "10"
> numToBin(3) := "11"
> numToBin(4) := "100"
> ...
> numToBin(n)

That seems rather simple. We know about the notion of the binary
representation of a natural... but

> Descending:

Huh? Descending from where?

> numToBin(oo) := "1...1".

Um, what does 1...1 mean? A string of 1s with two ends and no middle?

> numToBin(oo-1) := "1...10"

What does "oo-1" mean? You have not defined it anywhere...

And so on. This looks awfully like the standard "worm engineering"
we've seen before, from Phil, then from Tony Orlow. It got nowhere,
other than some thousands of posts.

Brian Chandler

On 13 Aug, 00:36, Ben Bacarisse <ben.use...@bsb.me.uk> wrote:

> Where is 0.101010101010101... on your list?

I'll be surely playing with that problem, is a next step, still I am
afraid we are going astray here.

For all I can tell, not only the construction is valid (up to contrary
evidence), but there is really just nothing to prove about its
completeness.

An inductive construction, now I am realizing, is quite an exotic
beast. I have taken the base from some code I have actually written
and run to generate the sequence in the finite cases, and play with
it. What happens at infinity is then the tricky case, and if I have
left some "holes" is either, as I have noted, at that equality we need
impose, otherwise -- I tell you -- it is just a "bug" in the
construction that can always be fixed, not an intrinsic problem. This
all, of course, unless/untill someone finds some real flaw.

In any case, back to mathematics: to proceed, I'd rather need a formal
definition for this property, "completeness", as it pertains to our
context.

Which definition of completeness am I supposed to satisfy?

-LV

On Tue, 12 Aug 2008 11:08:00 -0700 (PDT), julio@diegidio.name wrote:

>On 12 Aug, 18:43, "David C. Ullrich" <dullr...@sprynet.com> wrote:
>
>> > As to why I believe (think) that the list is "complete": because it is
>> > the _complete_ (over N*) list of _all_ the possible infinite (over N*)
>> > binary expansions.
>>
>> No, that's not possible.
>
>In this context, funny statement to say the least.
>
>> > Indeed, what I have given IS _per definition_ *the list of the
>> > "computable reals" (modulo the usual adjustments).
>>
>> No, you haven't. There _is_ a standard definition of "computable
>> real", and it simply doesn't appear anywhere in your post.
>
>Another funny statement, even more funny considering that the standard
>definition happens to correspond to mine

Prove that.

(The idea that you can prove that without ever mentioning the standard
definition is hilarious. The idea that you can prove it is simply
wrong.)

>were it not for your pre-
>judgement.
>
>I wander what you really don't know. For instance, it is far from a
>pleasure, apart from how improductive, to keep on these tones. You
>maybe don't take me for serious on this.

Guffaw. That's correct, I don't take you seriously on this.
For exactly the same reason as I would not take you seriously
if you were proving that 2 + 2 = 5.

>-LV

David C. Ullrich

"Understanding Godel isn't about following his formal proof.
That would make a mockery of everything Godel was up to."
(John Jones, "My talk about Godel to the post-grads."
in sci.logic.)

On Tue, 12 Aug 2008 12:45:38 -0700 (PDT), julio@diegidio.name wrote:

>On 12 Aug, 19:47, Ben Bacarisse <ben.use...@bsb.me.uk> wrote:
>> ju...@diegidio.name writes:
>> > On 12 Aug, 18:43, "David C. Ullrich" <dullr...@sprynet.com> wrote:
>>
>> >> > As to why I believe (think) that the list is "complete": because it is
>> >> > the _complete_ (over N*) list of _all_ the possible infinite (over N*)
>> >> > binary expansions.
>>
>> >> No, that's not possible.
>>
>> > In this context, funny statement to say the least.
>>
>> >> > Indeed, what I have given IS _per definition_ the list of the
>> >> > "computable reals" (modulo the usual adjustments).
>>
>> >> No, you haven't. There _is_ a standard definition of "computable
>> >> real", and it simply doesn't appear anywhere in your post.
>>
>> > Another funny statement, even more funny considering that the standard
>> > definition happens to correspond to mine were it not for your pre-
>> > judgement.
>>
>> One way seems reasonable -- that all the numbers on your list are
>> computable -- but the other way is far from obvious. I still can't
>> see how any number with a pattern in its digits can be there and I am
>> equally puzzled about how numbers like pi-3 (computable) can be on
>> your list.
>
>Pardon me, how does pi-3 expands in binary?

What does that matter? It's a computable number. So you're
claiming it's on your list - you need to prove that.

>> There are details in your construction that still bother
>> me, but I don't think they introduce another countable infinity of
>> numbers so I doubt these worries will go away.
>
>There may be some sort of "point of singularity" or "double-point"
>resulting from the stated equality LimE_p = LimE_q, an equality that
>is key to closing the infinite case. I can't realize what that entails
>to the structure of our R*, that's too specialistic for me.
>Singularity, double point, moebius-torus, inconsistency, ice-cream,
>I'll better be listening!
>
>A point here might be worth noting, about the progression:
>
>-- 1st, the construction: we first have the rules for the construction
>of an "bi-enumerable list" defined so and so, and with a definition
>extended so that the list keeps working consistently in the infinite
>case.
>
>-- 2nd, the reals: we then disregard the finite case and consider only
>the list in the infinite (boundary) case, that is, for w = oo. Under
>*this* case we establish a correspondence between our (infinite) list
>and such and such interval of a definiendum set R*.
>
>Another point I'll mention is that I would think that extending to
>bases greater than two adds nothing to the construction (and so to the
>underlying argument) I have given, modulo etc. etc. This does *not*
>entail that there may not be significant properties of our R* emerging
>from considerations over bases greater than two and, actually, by
>generalizing to any base. Again, beyond me.
>
>-LV
>
>> --
>> Ben.

David C. Ullrich

"Understanding Godel isn't about following his formal proof.
That would make a mockery of everything Godel was up to."
(John Jones, "My talk about Godel to the post-grads."
in sci.logic.)

On Tue, 12 Aug 2008 22:14:35 -0700 (PDT), julio@diegidio.name wrote:

>On 13 Aug, 00:36, Ben Bacarisse <ben.use...@bsb.me.uk> wrote:
>
>> Where is 0.101010101010101... on your list?
>
>I'll be surely playing with that problem, is a next step,

Wow. You insist that your list contains all computable
numbers, but when someone asks you to show it
contains a certain _rational_ number all you can say
is you'll surely be playing with that?

This is why people are not taking you seriously: You
have not come close to proving anything.

>still I am
>afraid we are going astray here.

"Astray"? The whole point is that you've supposedly
constructed a list of all the computable numbers, and
then the question of whether the list contains a certain
computable number is "going astray"?

>For all I can tell, not only the construction is valid (up to contrary
>evidence), but there is really just nothing to prove about its
>completeness.

And as far as _I_ can tell, 2 + 2 = 5. My proof of that is
valid as far as I can tell, there really is just nothing to prove
about it.

You really _expect_ to be taken seriously when

(i) you're claiming things that are known to be false

(ii) you don't ever answer questions about your supposed "proof"?

>An inductive construction, now I am realizing, is quite an exotic
>beast. I have taken the base from some code I have actually written
>and run to generate the sequence in the finite cases, and play with
>it. What happens at infinity is then the tricky case, and if I have
>left some "holes" is either, as I have noted, at that equality we need
>impose, otherwise -- I tell you -- it is just a "bug" in the
>construction that can always be fixed, not an intrinsic problem. This
>all, of course, unless/untill someone finds some real flaw.
>
>In any case, back to mathematics: to proceed, I'd rather need a formal
>definition for this property, "completeness", as it pertains to our
>context.
>
>Which definition of completeness am I supposed to satisfy?

We've all been _guessing_ that when you say the list is
"complete" you mean that it contains all the computable
numbers.

That's not what "complete" usually means, but it's the point
at issue here. _How_ do you prove that the list contains
all computable numbers?

Hint: Saying "there's nothing to prove" doesn't count as a proof.

>-LV

David C. Ullrich

"Understanding Godel isn't about following his formal proof.
That would make a mockery of everything Godel was up to."
(John Jones, "My talk about Godel to the post-grads."
in sci.logic.)