On 15 Jun., 21:11, "Mike Terry" <news.dead.person.sto...@darjeeling.plus.com> wrote:
> 1) What you write is sloppy (mathematically) and so it's not > clear what you intend to say with your individual statements.
But actually everybody can easily imagine what is meant. You prove it for your person below.
> > > Consider the list of increasing lengths of finite prefixes of pi > > > 3 > > 31 > > 314 > > 3141 > > .... > > > Everyone agrees that: > > this list contains every digit of pi (1) > > Literally this just doesn't make sense.
Obviously he means that this list contains every digit of pi with the correct index enumerating its position. In discussions like the present one it is not usual to state everything explicitly that can be understood by an experienced reader.
Perhaps you mean to say: > > this list contains every finite prefix of the infinite > digit sequence for pi (1)? > > I would agree with that... > > (What you actually said is gibberish because the list is not a list of > digits. If we try to treat it as such, then the only digit in the list is > 3).
A list of people can contain and usually does contain letters. Thiks list is not a set in orthodox set theory.
> > [My suspicion at this point is that your gibberish wording is actually the > *key* in some way to how you want to introduce some incorrect conclusion, > but time will tell on that. If I'm right, you won't like my clarification, > because it will make it harder for you to express your mistake...]
This suspicion is wrong.
Every initial segment of the decimal expansion of pi is in at least one line of your list 3. 3.1 3.14 3.141 ... What we can find in the diagonal, namely 3.141 and so on, exactly that can be found in one line. This is obvious by construction of the list.
> > Same again :) This is harder to guess what you're trying to say though...
So consider this:
Every part of the diagonal is in at least one line. That means, every part is in one single line, or there are parts that are in different lines but not in one and the same.
The latter proposition can be excluded. If there are more than one lines that contain parts of pi, then it can be proved, be induction, that two of them contain the same as one of them. This can be extended to three lines and four lines and so on for every initial segment of n lines.
Hence we prove that all of pi, that is contained in at least one of the finite lines of this list, is contained in one single line.
Conclusion: Either the complete diagonal pi does not exist, or it exists also in one and the same single line (because every line that can contain a digit sequence, is a finite line). As the latter is wrong, so is the former. There does not exist an actually infinite sequence. Actually infinite mean finished infinite. That is nonsense.
(The proof has the same status as Cantor's diagonal proof. Also his proof is valid only for all finite n. In a similar way we find above: For all finite n: All of pi that is in the list up to line n, is in a single line of the list.)
> > There is no unique list of computable reals - however, people would agree > that the computable reals are countable, and so can certainly be enumerated. > Perhaps the actual list doesn't matter and we should just choose one?
Choose this one for example:
The proof of countability of S does not require a bijection of S with N but only an injection of some superset of S into N.
This is established by my list
0 1 00 01 ...
where every line may be enumerated by an element of the countable set omega^omega^omega (and, if required, finitely many more exponents for alphabets, languages, dictionaries, thesauruses, and further properties)
An obvious enumeration of the lines is 1, 2, 3, ... where every line n can have many sub-enumerations
n n.1.a n.11.a n.111.a ...
where every section of 1's and a's contains enuogh symbols to enumerate every single line as often as required to cover all its meanings in all possible languages. If necessary we can add 2's and b's and so on and remain in the countable domain. > > > Also, now your talking about reals rather than digit sequences. These are > distinct objects, but there are obvious correspondences.
In particular users of Cantor-lists do intermingle them. Digit sequences do not converge at all. Therefore infinite digit sequences cannot be used to express something meaningful.