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Re: Continuum Hypothesis Solution Posted
Posted:
Apr 16, 1999 3:06 PM
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Bill Taylor wrote:
> Well I didn't see all this thread, but I read this one, and thought I'd
> add a few cents.
>
> papus_@hotmail.com (Papus) writes:
>
> |> Just to make my point a bit more vivid. Let's consider the set of all
> |> computable real numbers in the [0,1] interval.
> |>
Here they are:
Contrary to the dogma of The Dark Master, a function f: N -> R exists. The
following construction algorithm produces EVERY decimal digit string in the Real
interval [0,1). Note, because this is a NEVER ENDING algorithm that keeps
churning out numbers to greater and greater digit lengths, unbounded (aka
infinite) length decimal strings are in essence constructed.
Number$(1) = "0." ;first number in the interval [0,1).
Reference = 1 ;points to a pre-extended number string.
NewIndex = 2 ;point where new (extended) numbers get appended.
Do
For Digit = 0 to 9
Number$(NewIndex) = Number$(Reference) & Digit
NewIndex = NewIndex + 1
Next Digit
Reference = Reference + 1
Loop
This algorithm is reminiscent of "Dr. Ullrich's Never-ending Journy" and Webster
Kehr's "Hinged Sets" idea. Anyone who is unfamiliar with Webster's ideas should
download his postscript papers from: http://home.sprintmail.com/~websterkehr/
> |> Now. What if I say these are *all* the reals in [0,1] and there are no
> |> others?
>
> You can say it, but not many folk would take you seriously. Mind you,
> that view does excite some immediate sympathy, looking for the *absolute*
> definability of each real, in some sense, but it doesn't really stand scrutiny.
Ok. Scrutinize it. :-) I look forward to your reply.
> |> Can you prove me wrong by *showing* me one I've missed?
>
> Maybe. Depends what you mean by "showing". Obviously I can't,
> if "showing" means giving an algorithm whereby you can be shown the nth
> decimal place for input n, coz that's just the definition of computability.
>
> But I claim that the following real has been perfectly well "shown",
> and is (by the usual math) non-computable.
>
> The real is obtained by listing all possible Turing machines in some standard
> ordering, and writing nth decimal place = 1 if machine n halts, & 0 otherwise.
>
> The resulting real .0010... has thus been "shown". Furthermore, though
> there's no ONE algorithm for determining ALL the digits, there may well be
> a whole bunch of proofs for proving every different place to be 0,
> or to be 1. It just needs different sorts of proof for each place.
>
> Indeed, conversely, it will be IMPOSSIBLE for you to prove
> that any given decimal place cannot be so obtained!
>
You make me laught. :-)
Nathan the Great
Age 11
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