Re: limitation to induction on finite bounds

From: |-|erc (gotch_at_beauty.com)
Date: 07/09/04 

Date: Fri, 09 Jul 2004 00:46:38 GMT

"The Ghost In The Machine" <ewill@aurigae.athghost7038suus.net> wrote in
> >> >> >
> >> >> > 3
> >> >> > 3.1
> >> >> > 3.14
> >> >> > ..
> >> >> >
> >> >> >
> >> >> > that is pi, to a computer its pi, to cantorians its a indexed
> >> >> > negative property matrix whos only interpretation is rowwise
> >> >> > extrapolation.
> >> >>
> >> >> That could equally well be the number 3 + 14/99. Data has been lost.
> >> >
> >> > I meant
> >> > {
> >> > int(pi()),
> >> > int(pi()*10)/10 ,
> >> > int(pi()*100)/100,
> >> > ...
> >> > }
> >> >
> >> > or are you calling 3.1415... not pi either? data has been lost?
> >>
> >> pi is (the symbolic representation of) a number. The
> >> digit expansion is known to more than a billion places,
> >> but the entire number can never be printed out as a decimal expansion;
> >> there's not enough carbon in the Universe.
> >>
> >> All you're showing is approximations therefor -- and you're
> >> far from unique in that respect, since humans can't go
> >> around writing pi either; it would take too long. :-)
> >
> >
> > Taking too long is probably the worst argument one can give as a
> > mathematician.Simply because all the hair on all human heads cannot be
> > counted since it would take too long,you cannot call it an infinite
> > set.The same argument applies.If a procedure takes "too long " to be
> > evaluated does not imply infinite.So please refrain from giving such
> > gibberish on the group.
>
> It would still take too long. :-) But certainly we can
> discuss various facts about the expansion of pi, even
> if we don't know and never can know all of its digits.
> There are also some interesting unanswered questions, such
> as whether it is normal (AIUI, an expansion is normal if,
> given two different finite digit sequences of the same length,
> both are equally probable) or absolutely normal (normal in
> every base, not just base 10).

Its the methodoloy of argument Karan is picking here.
> >If a procedure takes "too long " to be
> > evaluated does not imply infinite

And she's right that the premise "takes too long" cannot imply much,
slow user?? slow process?? slow computer?? big data structrue?? small universe??

Its from your meanderings on infinity you *then* talk about infinite time,
as long as you don't imply anything from that its valid but still
a distraction for argument. So we refrain from useless conjecture hey?

>
> As it is, {pi} is certainly not an infinite set, although
> pi might be, in the sense that it is the limit of a
> sequence (pick one :-) ), which can be construed as an
> ordered set, depending on how one defines one's reals
> (more traditional thought would simply associate pi with
> the sequence, not equate it thereto). However, even the
> most optimistic definitions of pi would not have pi be
> a member in its own defining set -- especially if that
> defining set contains nothing but rational numbers.
>
> But yes, pi is irrational, transcendental, and has an
> endless, non-repeating digit expansion.
>
> The poster |-|erc is of the opinion -- shared by very
> few others -- that somehow pi is contained in the set {3,
> 3.1, 3.14, 3.141, ...}, which is one such aforementioned
> sequence, therefore making the real numbers countable.
> This appears to contradict quite a few mathematical issues
> that I'm personally aware of, such as the existence of
> non-closed sets. However, all finite sequences of pi
> are contained in the set, by definition. Understand that,
> and one might get a taste of how infinity flavors the
> discussion on occasion. :-)

no, I'm just stuck on this issue that the infinite number of digits of pi are
in the set, which you have agreed to in several ways, including above.
"all finite sequences" ~ there are an infinite number of them so *every*
digit of pi is in the set. it contains, as in they *occur somewhere* all
digits of pi. the diagonal *is* pi and the diagonal is part of the set.

all finite sequences -> all sequences up to every finite number
-> all sequences up to every number
-> all sequences up to an infinite number of numbers
-> all sequences up to infinity

once the dozen people here stop saying the infinite_set_is_only_an_approximation
we can move onto my argument.

    that somehow pi is contained in the set {3,
> 3.1, 3.14, 3.141, ...}, which is one such aforementioned
> sequence, therefore making the real numbers countable

not at all. you've got the premise of my argument and the conclusion of my
argument but not the deduction.

Cantor thinks digit sequences don't occur in the computable set of numbers.
>From that assumption that the digit sequences don't occur, he concludes there are
missing members from the set.

The sequence of digits of pi does *occur* in the set, hence the argument that
the sequence of digits is missing is void. Hence his conclusion that there must
be missing members and uncountable infinity is void.

See the difference? I didn't prove reals are countable, I disproved a particular 'proof '
that they aren't

>
> Of course such a notion contradicts the standard Cantor
> diagonalization proof as well -- but there are other
> proofs; the diagonalization proof was Cantor' second proof
> at proving that C > aleph_0.
>
> I've also defined the rather silly set T_b, which is
> simply {j / b^n, n > 0, 0 <= j < b^n, n, j in J} which is
> merely the set of all possible finite decimal expansion
> for an integer b > 1. This set can be said to "cover"
> the interval [0,1], if by "cover" one means "find points
> arbitrarily near to in the set", as it's obviously dense
> (or one can use |-|erc's SetMinus algorithm; A SetMinus r
> = inf(dist(x,r), x in A)). However, pi - 3 or pi/10 are
> not in any T_b since T_b by construction contains nothing
> but rational points. T_b is one attempt at enumerating the
> reals, as well -- an attempt that fails, of course.
>
> 1/3 is also not in {0.3, 0.33, 0.333, ... }, despite 1/3
> being rational and the set containing nothing but rationals
> (an alternate portrayal is { (10^n - 1)/(3 * 10^n), n > 0 in J}).
>
> But these distinctions are apparently quite lost on |-|erc.
> Data has been lost. :-)

Karan explained this herself, in her second post.

Herc

Relevant Pages

  • Re: limitation to induction on finite bounds
    ... > if we don't know and never can know all of its digits. ... Its from your meanderings on infinity you *then* talk about infinite time, ... all finite sequences of pi ...
    (sci.logic)
  • "Algorithmic Randomness, Quantum Physics, and Incompleteness"
    ... all finite sequences are to be found infinitely often and ... different members of the infinite set of random numbers. ... Just using random numbers with initial digits 1 thru 9, ... it generated by a shorter input algorithm than the length ...
    (sci.logic)
  • Re: Question regarding infinite length integers
    ... natural numbers into the set of ordered sequences over the alphabet ... normal choice maps the naturals to the set of all finite ... What is wrong with the following: Given an integer n, with m digits, n ... and therefore m can be infinite. ...
    (sci.logic)
  • Re: abundance of irrationals!)
    ... >>> some infinite set of symbols to represent them. ... >> representation of the naturals? ... > some with infinitely many significant digits. ... All finite sequences are bounded. ...
    (sci.math)
  • Re: Infinite number of people toss a coin infinite times
    ... >> the assumption of all FINITE sequences having occurred would imply ... >> all sequences (including INfinite ones) have occurred. ...
    (sci.math)