How Wide Do All The Possible Computed Digit Sequences Go? [Math]

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From: Transfer Principle on 23 Jun 2010 17:59

On Jun 22, 9:24 pm, Graham Cooper <grahamcoop...(a)gmail.com> wrote:
> On Jun 23, 2:08 pm, Sylvia Else <syl...(a)not.here.invalid> wrote:
> > So Herc's problem is that it is not inevitably true that a list that
> > contains all finite prefixes also contains all infinite sequences, and
> > indeed it is specifically false for a sequences generated by permuting
> > the computables.
> All digits in order does not mean
> a single infinitely long sequence
> like this list contains all digits in order
> of pi
> 3
> 31
> 314
> ...
> So herc_cant_3 stands
> so it should I gave 2 proofs

Herc claims that that herc_cant_3 is a theorem to which he
has given two proofs.

I would like to know in what theory Herc has given the two
proofs -- i.e., which axioms he used in the proofs.

It's not because I want to belittle Cooper's theory, but
just because I like to learn more about alternate theories,
and also to give him a warning as to what sort of theories
are likely to be belittled by others.

For example, many posters criticize theories which appear
to assume more structure than necessary. Apparently, this
herc_cant_3 proves that if X is a set (or list) such that
3, 3.1, 3.14, etc., are elements of X, then pi must also be
an element of X. Notice that in standard theory, if we
claim that X is a _closed_ set (i.e., contains all of its
limit points), then X does contain pi, so perhaps Herc is
assuming that every set is closed. But this is the type of
assumption that many posters are likely to disparage. I
don't mind assuming such additional structures (the
current TO thread also involves assuming some additional
structure), but many other posters dislike this.

From: Rupert on 23 Jun 2010 21:22

On Jun 23, 12:17 pm, Graham Cooper <grahamcoop...(a)gmail.com> wrote:
> On Jun 23, 11:31 am, Rupert <rupertmccal...(a)yahoo.com> wrote:
>
>
>
>
>
> > On Jun 22, 7:36 pm, Graham Cooper <grahamcoop...(a)gmail.com> wrote:
>
> > > On Jun 22, 4:28 pm, Graham Cooper <grahamcoop...(a)gmail.com> wrote:
>
> > > > On Jun 22, 3:21 pm, Rupert <rupertmccal...(a)yahoo.com> wrote:
>
> > > > > On Jun 22, 6:44 am, Graham Cooper <grahamcoop...(a)gmail.com> wrote:
>
> > > > > > On Jun 22, 12:08 am, Graham Cooper <grahamcoop...(a)gmail.com> wrote:
>
> > > > > > > On Jun 21, 10:40 pm, Sylvia Else <syl...(a)not.here.invalid> wrote:
>
> > > > > > > > On 21/06/2010 5:03 PM, Rupert wrote:
>
> > > > > > > > > On Jun 21, 4:28 pm, "|-|ercules"<radgray...(a)yahoo.com> wrote:
> > > > > > > > >> Every possible combination X wide...
>
> > > > > > > > >> What is X?
>
> > > > > > > > >> Now watch as 100 mathematicians fail to parse a trivial question.
>
> > > > > > > > >> Someone MUST know what idea I'm getting at!
>
> > > > > > > > >> This ternary set covers all possible digits sequences 2 digits wide!
>
> > > > > > > > >> 0.00
> > > > > > > > >> 0.01
> > > > > > > > >> 0.02
> > > > > > > > >> 0.10
> > > > > > > > >> 0.11
> > > > > > > > >> 0.12
> > > > > > > > >> 0.20
> > > > > > > > >> 0.21
> > > > > > > > >> 0.22
>
> > > > > > > > >> HOW WIDE ARE ALL_POSSIBLE_SEQUENCES COVERED IN THE SET OF COMPUTABLE REALS?
>
> > > > > > > > >> Herc
> > > > > > > > >> --
> > > > > > > > >> If you ever rob someone, even to get your own stuff back, don't use the phrase
> > > > > > > > >> "Nobody leave the room!" ~ OJ Simpson
>
> > > > > > > > > It would probably be a good idea for you to talk instead about the set
> > > > > > > > > of all computable sequences of digits base n, where n is some integer
> > > > > > > > > greater than one. Then the length of each sequence would be aleph-
> > > > > > > > > null. But not every sequence of length aleph-null would be included.
>
> > > > > > > > That answer looks correct.
>
> > > > > > > > But I guarantee that Herc won't accept it.
>
> > > > > > > > Sylvia.
>
> > > > > > > It's truly hilarious. It's like using a Santa clause metaphor
> > > > > > > to explain why Santa clause is not real,
> > > > > > > but it will do for now.
>
> > > > > > > Herc
>
> > > > > > Actually on second reading I think Rupert threw a red herring
>
> > > > > > He didn't adress the question at all. How wide are all possible
> > > > > > permutations of digits covered? This is different to all possible
> > > > > > listed sequences he just answered that numbers are inf. long!
>
> > > > > > Herc- Hide quoted text -
>
> > > > > > - Show quoted text -
>
> > > > > I'm afraid I don't understand the question.
>
> > > > If it takes 10^x reals to have every permutation x digits wide
> > > > how many digits wide would oo reals make?
>
> > > > Herc
>
> > > Where is my reference to computable reals here Rupert?
>
> > > This is a question with a quantity answer.
>
> > > If you can't answer say so.
>
> > > Herc- Hide quoted text -
>
> > > - Show quoted text -
>
> > There does not exist a cardinal number x, such that the set of all
> > sequences of decimal digits of length x has cardinality aleph-null.
>
> > If you have some cardinal number x and a set of sequences of decimal
> > digits of length x of cardinality aleph-null, then it must be the case
> > that this set does not contain all the sequences of decimal digits of
> > length x.
>
> > That is my answer to your question as best I understand it. But I am
> > not sure I really understand what you are talking about.
>
> The topic of the thread is the width of permutations

Well, that certainly wasn't clear to me before. It seemed to me tha we
were talking about sequences of decimal digits. Permutations of which
set, pray tell?

> as in every permutation of a certain width
> and it's relation to the size of the list of reals.
>
> You are refuting that this width approaches infinity
> as the list of reals approaches infinity
> based on
>
> a/. You don't know what I'm referring to
> b/. Reverse engineering that there is no defined width
> because it refutes transfiniteness theory
>
> you're avoiding the question plain and simple
>

I think my problem is that I don't understand the question. But that
doesn't mean I'm not trying.

From: Graham Cooper on 23 Jun 2010 22:05

On Jun 24, 7:59 am, Transfer Principle <lwal...(a)lausd.net> wrote:
> On Jun 22, 9:24 pm, Graham Cooper <grahamcoop...(a)gmail.com> wrote:
>
> > On Jun 23, 2:08 pm, Sylvia Else <syl...(a)not.here.invalid> wrote:
> > > So Herc's problem is that it is not inevitably true that a list that
> > > contains all finite prefixes also contains all infinite sequences, and
> > > indeed it is specifically false for a sequences generated by permuting
> > > the computables.
> > All digits in order does not mean
> > a single infinitely long sequence
> > like this list contains all digits in order
> > of pi
> > 3
> > 31
> > 314
> > ...
> > So herc_cant_3 stands
> > so it should I gave 2 proofs
>
> Herc claims that that herc_cant_3 is a theorem to which he
> has given two proofs.
>
> I would like to know in what theory Herc has given the two
> proofs -- i.e., which axioms he used in the proofs.
>
> It's not because I want to belittle Cooper's theory, but
> just because I like to learn more about alternate theories,
> and also to give him a warning as to what sort of theories
> are likely to be belittled by others.
>
> For example, many posters criticize theories which appear
> to assume more structure than necessary. Apparently, this
> herc_cant_3 proves that if X is a set (or list) such that
> 3, 3.1, 3.14, etc., are elements of X, then pi must also be
> an element of X. Notice that in standard theory, if we
> claim that X is a _closed_ set (i.e., contains all of its
> limit points), then X does contain pi, so perhaps Herc is
> assuming that every set is closed. But this is the type of
> assumption that many posters are likely to disparage. I
> don't mind assuming such additional structures (the
> current TO thread also involves assuming some additional
> structure), but many other posters dislike this.

No. I'm starting to doubt sci.math posters give more than a
cursory glance at the keywords before making their retort.

Indie not say pi was in the set. I said all the digits of pi
were in the set.

1
1 2

1 2 3
.....

The above list "has" every natural number in order
that does not mean it has an element with all N

Herc

From: Graham Cooper on 23 Jun 2010 22:22

From: Joshua Cranmer on 23 Jun 2010 22:47

On 06/23/2010 10:22 PM, Graham Cooper wrote:
> 3 proof by induction that the maximum digit width
> of all listed permutations is infinite.

I find it amusing when people claim to prove facts about infinity via
induction. Proofs by inductions can prove results for some subset of
natural numbers (or, sometimes, subsets of real numbers containing
non-natural numbers). Infinity is not a natural number, so it is, so to
speak, outside the range of induction.

--
Beware of bugs in the above code; I have only proved it correct, not
tried it. -- Donald E. Knuth