Re: Epistemology 201: The Science of Science

stephen_at_nomail.com
Date: 03/09/05 

Date: 9 Mar 2005 01:44:26 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:
:> :> :> 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.
: I don't use the term "forked tongue", so you must be thinking of someone
: else. When I here the same thing over and over again, as if I didn't
: understand the first time in high school, then I call it regurgitation.

Yes you do. In message
http://groups-beta.google.com/group/comp.ai.philosophy/msg/89b7d63a57d34e04?dmode=source

you said

   "I think both Allan and I have pointed out the forked-tongued
    nature of your example where you treat them as representations of
    integers on the one hand, and meaningless strings on the other."

Apparrently you considered my example 'forked-tongued'. Do
you really consider that a logical rebuttal? In any case,
you should be very careful about what you say and might
later want to deny when google is around.

:>
:>
:> :>
:> :>
:> :> :>
:> :> :> : 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".
: And you still haven't explained why this rule is superceded by
: cardinality for infinite sets, but whatever....

You still haven't explained why your rule should supercede the
rule that among finite sets a bijection cannot exist between
sets of different sizes.

:>
:>
:> :> 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.
: I would imagine that all powersets of finite sets are of the same size,
: namely, the size of the integers, since the power set of a finite set of
: charaacters, plus some ordering, represents the contents of any digital
: number system. There may be more elements in the set of decimal digits,
: and yet they appear equally more rarely in the set of decimal numbers
: than the octal digits do in octal numbers. One might say the same for
: rational numbers, except that the division character can appear only
: once and has a special purpose, so that the set becomes the permutation
: of two sets of integers. The problem I see is that the natural oder of
: numbers as quantities that are each equal to, less than, or greater than
: each other, is ignored.

What do you mean by the "powersets of finite sets"? The power
set of a finite set is a finite set. The powerset of the
set of all finite sets is larger than the integers.

I am not sure what point you are trying to make about the digits
appearing more rarely, or how this has anything to do with the
size of a set of strings.

And yes, order is being ignored. Order is not a property of sets.
Therefore it makes sense to ignore order when trying to describe some
innate quality of sets.

:>
:> : 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.
: The question I asked was why there is this difference between the finite
: and infinite sets as regards the equivalence between a proper superset
: and a larger set. Why not say the same for infinite sets as for finite
: ones?

Because of the definitions chosen. For finite sets two sets
only have the same size if there exists a bijection between them.
Why not say the same for infinite sets as for finite sets?
One of the points of my post was to show you that cardinality
has interesting results whereas your idea of 'more' apparently
does not.

:>
:> : 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].
: I didn't ask for examples. I asked for a definition. Oh well, Stephen,
: we've been through this, and rather than consider any idea of mine you
: are intent on shooting it down, so I won't bother you about it again.
: have a nice day.

I have considered your ideas probably more than you have.
You just want to claim that a superset is larger than its
subset with no thought to what the consequences of that might
be. I have actually thought about those consequences and have
tried to explain why those consequences are not useful or interesting.

If you can think of a consequence that follows from claiming
that there are more rationals than integers I would be
curious to hear it.

Stephen