Re: Epistemology 201: The Science of Science
stephen_at_nomail.com
Date: 03/08/05
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Date: 8 Mar 2005 03:38:16 GMT
In sci.math Tony Orlow (aeo6) <aeo6@cornell.edu> wrote:
: 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:
:> :> : 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.
:> : Fine. So, respond to my argument, and don't be obnoxious about it.
:>
:> I was not obnoxious about it. Again, you did not respond
:> to my argument at all. Are all the rationals computable
:> or not? I gave you a very clear cut argument. If there is
:> something wrong with it, point out the exact error in the logic.
:> Instead you are just responding with mildly insulting remarks.
: You're not? See below.
Well, this is apparently pointless. You are the one who
says things like "regurgitation of the same confounded
nonsense" and I am the insulting one? I think earlier
you claimed I had a forked tongue as well. If you
think these little jibes are part of your logical
argument than I fail to see the logic.
:>
:>
:> :>
:> :> : 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.
:> : Why not? You follow with yet another regurgitation of the same
:> : confounding nonsense that I am arguing against.
:>
:> I just explained why not.
:>
: No, you didn't. I responded below why your repsonse was irrelevant and
: unilluminating. It doesn't answer the question, but further complicates
: things. Respect means responding to the other person, after listening.
And you ignored all my points. Your only argument has been that
a proper superset must have more elements than its proper subset.
To use your terminology, you keep "regurgitating this confounded
nonsense".
:> Is it nonsense to you that the
:> rationals are computable? What is so nonsensical about that?
:> Here are some simple questions for you:
:> are the rationals computable?
:> are there more rationals than Turing machines?
:> are there more Turing machines that integers?
:>
:> <snip>
:>
:> : If you stick to natural orderings, then you don't
:> : introduce arbitrary artificial mappings that obscure more than
:> : illuminate the issue.
:>
:> I think computability really illuminates the issue. It demonstrates
:> that these mappings have very real consequences in some cases. Every
:> rational is computable, so there must be at least one Turing
:> machine for every rational. Every Turing machine can be
:> encoded as a unique integer, so there must be at least one
:> integer for every Turing machine.
: And, what does the introduction of turing machines do to clarify
: anything? It's a further complication, as if we don't already have
: people creating random unnatural mappings between sets and then
: declaring them exactly equal, when it is obvious in cases like [0,3] and
: [1,2] that they are not?
One of the things the TM's do is show that cardinality actually
has some consequences, whereas your definition of "more" apparently
does not? For example, what results follow from your claim
that [0,3] has more elements than [1,2]? Cardinality has
consequences. If you look at the rationals/TM example
again, we have the following:
1) for each rational number there exists a TM that
computes it
2) each Turing machine can be encoded as an integer
If you disagree with 1 then you must believe there
are incomputable rationals. If you disagree with 2,
then you must believe that universal Turing machines
are an impossibility.
Now you believe that there are more rationals than
integers, which means you must think there are either
more rationals than TM's, or more TM's than integers.
However this requires a definition of "more" that has
no consequence. If the cardinality of the rationals
was greater than the cardinality of the TMs, then
it would not be possible for there to exist one TM
for each rational. If the cardinality of the TMs
was greater than the cardinality of integers, it
would not be possible to encode each TM as an integer.
But your definition of "more" does not seem to have
any such consequences. There apparently can be
more rationals than TMs and still have a different TM
for each rational.
: Answer this if you can, without introducing another Cantorian trick
: mapping, why it doesn't make sense for a proper superset to, by
: definition, have more elements than its proper subset.
Because this definition does not seem to have any meaningful
consequences, whereas cardinality does. For example, by
your definition there must be more strings of the form
[1-9][0-9]* (decimals) than of the form [1-7][0-7]* (octals).
However that fact apparently has no consquences. If
the decimals had a larger cardinality than the octals,
then it would follow there were numbers representable in decimal
that were not representable in octal.
: Why does this
: definition hold for finite sets and not infinite sets?
What definition? The definition of cardinality holds
for finite and infinite sets. The definition of proper subset
holds for finite and infinite sets. For finite sets,
a set cannot have the same cardinality as its proper subset.
For infinite sets, a set can have the same cardinality as
its proper subset.
: How do you define
: a proper superset when discussing infinite sets, or do you, ever?
The same way you define any infinite set. The evens are
a proper subset of the integers. The set (0,1) is a proper
subset of [0,2].
Stephen
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