Re: Epistemology 201: The Science of Science

stephen_at_nomail.com
Date: 03/07/05 

Date: 7 Mar 2005 15:42:09 GMT

In sci.math Tony Orlow (aeo6) <aeo6@cornell.edu> wrote:
: stephen@nomail.com said:
:> Did you explanation of why the number of elements in (0,2) is
:> larger than the number of elements in (0,1) make use of
:> the fact that (0,1) is a proper subset of (0,2)? That is
:> what I am asking about? Alan is claiming that (0,2) has
:> more elements than (0,1) but the reason is not because (0,2)
:> is a proper superset of (0,1). What the reason is I do not know,
:> because when he tried to explain it the explanation sounded a lot
:> like "(0,1) is a proper subset of (0,2)".
:>
:> And yes, I did kill file you for awhile. You were boring me. :)
:>
:> Stephen
:>
: Being boring deserves a death sentence? Fact is, I've started several of
: the topics that are still going, including this one, so "boring" might
: not be the word you're looking for. Maybe you meant "challenging." :)

No. You definitely were not getting challenging.

: Yes, my argument was based on what I considered a clear example of the
: contradiction between common interpretations of Cantor, and basics of
: set theory and the concepts of size and amount. I did not try to define
: a general method of determining finite differences or ratios between
: infinite sets, based on subsets. It was meant to be an example of where
: Cantorian cardinality falls short, and it was mine, not Allan's.

Then you are not making the same argument that Allan is making.

: If a proper superset of a set is a set that contains ALL the elements of
: the set, PLUS SOME MORE, then it is, by definitons that apply to all
: finite sets, bigger. There is no reason not to apply this concept of
: "bigger" to infinite sets. It makes perfect sense.

But it does not make perfect sense. Here is another example
for you to consider. Every rational number is computable,
meaning that for each rational number r there exists a Turing
machine that when given a number n as an input, the Turing
machine produces the nth digit of r's decimal representation (or
whatever base you desire). Clearly there cannot be more rational
numbers than Turing machines as each Turing machine can only
compute one rational number.

Each Turing machine can be described by a finite string. A Turing
machine has a finite number of states, a finite alphabet and
a finite number of transitions and can be completely described
by a finite string. A string can only describe one Turing machine,
so clearly there cannot be more Turing machines than strings.

Each string can be encoded as an integer. Just consider an
n character string stored on a computer. This is an 8*n bit
number. Again, there are clearly not more strings than integers.

So what do you get? There are not more rationals than Turing machines,
and there are not more Turing machines than strings, and there are
not more strings than integers, which implies there are not more
rationals than integers.

You of course disagree with this conclusion, but what steps
of the above argument do you disagree with? Do you think
there exist uncomputable rationals? Do you think that not
all Turing machines can be described as a string? I am
curious as to what you think is the flaw in the above argument.

: This definition of
: "bigger" is more elementary than any complex ordering and
: counting/correspondence scheme, and should take precedence as being more
: basic. {0,1,2,3...} is one elements bigger than {1,2,3,4...}, obviously,
: and to deny this is ridiculous in my mind. To say there aren't twice as
: many integers as evens or odds is also ridiculous in my mind, since
: exactly HALF the integers are odd, and the other HALF are even. The
: Cantorian counting method distinguishes between two or three different
: CLASSES of infinity, which an important discovery, but it by no means is
: any exact or especially useful quantification of infinite sets. That
: was, is, and will continue to be my position on the subject.

So how can the rationals be computable if there are more rationals
than Turing machines?

Stephen

Relevant Pages

  • Re: Epistemology 201: The Science of Science
    ... infinite sets, based on subsets. ... Each Turing machine can be described by a finite string. ... so clearly there cannot be more Turing machines than strings. ... There are not more rationals than Turing machines, ...
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  • Re: Epistemology 201: The Science of Science
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    (sci.physics)
  • Re: Epistemology 201: The Science of Science
    ... :>: infinite sets, based on subsets. ... (I admit I'm a little rusty on Turing machines so you WILL correct me if I ... the rationals. ... It is not true of all the reals. ...
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  • Re: Epistemology 201: The Science of Science
    ... :>: infinite sets, based on subsets. ... (I admit I'm a little rusty on Turing machines so you WILL correct me if I ... the rationals. ... It is not true of all the reals. ...
    (sci.cognitive)
  • Re: Epistemology 201: The Science of Science
    ... :>: infinite sets, based on subsets. ... (I admit I'm a little rusty on Turing machines so you WILL correct me if I ... the rationals. ... It is not true of all the reals. ...
    (sci.physics)