Re: Epistemology 201: The Science of Science

From: aeo6 (aeo6_at_cornell.edu)
Date: 02/11/05 

Date: Fri, 11 Feb 2005 13:22:17 -0500

stephen@nomail.com said:
> In sci.math Tony Orlow (aeo6) <aeo6@cornell.edu> wrote:
> : stephen@nomail.com said:
> :> In sci.math Tony Orlow (aeo6) <aeo6@cornell.edu> wrote:
> :> : Or, you can simply admit that there are more real numbers between 0 and
> :> : 2 than there are between 0 and 1, even if the "cardinality" is the same,
> :> : for instance, 1.5.
> :> : --
> :> : Smiles,
> :>
> :> : Tony
> :>
> :> Whoever said that there were not numbers in [0,2] that were not in [0,1]?
> :> What is true is that every element in [0,1] can be uniquely mapped to
> :> an element in [0,2]. Why won't you simply admit that? :)
> :>
> :> Stephen
> :>
> : I never denied that. Cantor's cardinality is significant, and
> : distinguishes between classes of infinity. I am making a valid point
> : that there are details of infinity that it does not capture. There is a
> : problem when idiots say there are the same number of reals between 0 and
> : 1 as between 0 and 2, when one obvious has more reals than the other as
> : a proper superset, just because there is the same Cantorian cardinality.
> : Why can't you simply admit that?
>
> Because it is not obvious. Here is an example that shows that
> is not obvious that a subset has fewer elements than its superset.
> As a computer scientist (which is what best describes me as well),
> you should understand it.
>
> Consider two idealized C like computer languages. Both languages
> support arbitrarily large integers. The first language, which I will
> call C10, uses decimal. The second language, C8, uses octal.
>
> Consider programs of the following form:
>
> for (int i=0; i<N; ++i)
> printf("I");
>
> where N is some positive integer constant. For example
>
> for (int i=0; i<100; ++i)
> printf("I");
>
> for (int i=0; i<99; ++i)
> printf("I");
>
> The above two programs are both legal C10 programs. Only
> the first is a legal C8 program, because 99 is not an octal number.
> Every C8 program is a legal C10 program, but not every C10
> program is a legal C8 program.
>
> Let P10 and P8 be the sets of all C10 and C8 programs
> of the above form respectively.
>
> Every element of P8 is also an element of P10, but P10
> contains elements not in P8. By your reasoning, P10
> must have more programs than P8.
>
> Now consider the outputs of those above programs. Every
> program in P8 produces a different output, as does every
> program in P10. Let S8 and S10 be the sets of outputs.
> It is pretty clear that there are exactly as many outputs
> in S8 as there are programs in P8, and exactly as many outputs
> in S10 as there are programs in P10. A little thought makes it
> clear that S8 and S10 are the same set. For every output produced
> by a program in P10, there is a program in P8 that produces that
> exact same output.
>
> So the number of programs in P8 equals the number of outputs in
> S8 which equals the number of outputs in S10 which equals the
> number of programs in P10. In other words |P8|=|S8|=|S10|=|P10|.
> Do you still want to insist that the number of programs in P8
> differs from the number of programs in P10? Or do you think
> that this too is idiotic?
>
> Stephen
>
>
"It is pretty clear that there are exactly as many outputs
in S8 as there are programs in P8"

Yes, that is idiotic. It is not at all clear that one and only one
program can produce any given ouput. It is patently incorrect. 

-- 
Smiles,
Tony

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