Re: Extrapolate divisor & dividend from quotient?

[Posting from the MathForum. What's happened to Google? Could anyone suggest a *web* alternative: I have lost contact with the discussions in sci.logic at a crucial point...]

Gerry Myerson wrote:
In article
<2acd62ba-2d71-4a6c-b98f-6628f6c093d3@xxxxxxxxxxxxxxxx
groups.com>,
julio@xxxxxxxxxxxxx wrote:

I don't know if there is a standard name for those
numbers for which no rule calculates their decimal
expansion. I suppose these might be called uncomputable
numbers


Thanks, that makes indeed sense.


A decimal expansion is just a sequence of digits.
Standard operating
procedure in mathematics is to accept the existence
of sequences
of digits, even those sequences that cannot be
described by any
finite rule. There are mathematicians who reject this
and insist that
only computable reals have any meaning. If you find
this a congenial
thought, you might look up the work of Errett Bishop.
It's a legitimate
philosophical position. It isn't the way most
mathematicians have chosen
to go.


I wander why! :) Yes, although I am not a finitist in any sense I would accept, I deeply despise any form of "weak thinking" (an indirect reference here to the discussion on post-modernism).


.1234567891011121314151617181920212223242526....
is random and is generated by a very simple rule.

Now this definition of random may strike you as
inadequate.


No, it doesn't, as I am aware of the underlying conceptual problems: that notion cannot but miss the very nature of randomness, as that's its very purpose, in a sense.


Unfortunately, tightening the definition as much as
you propose,
to where randomness is the absence of any rule
whatsoever,
Knuth shows that under a reasonable interpretation
this doesn't
leave any random sequences at all.


Indeed! And so? Why should there be anything missing? Yes, Cantor!


As Arturo has said, this has nothing to do with
Godel, just with
Cantor's notion of different sizes of infinity.


I guess it is just the time for me to have a try and start reading Mr Cantor himself. If anything has gone wrong, I would bet there I'll find a definite indication.



By the way, all three versions of your message
reached my server.
Sometimes one just has to be patient.


Just sometimes? :) Joke apart, thank you very much for all your explanations and references, things start to make sense.

-LV



--
Gerry Myerson (gerry@xxxxxxxxxxxxxxx) (i -> u for
email)

.

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