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Epistemology 201: The Science of Science
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From: Tony Orlow (aeo6) <a...@cornell.edu>
Newsgroups: comp.ai.philosophy,sci.physics,sci.math,sci.cognitive,sci.philosophy.meta
Subject: Re: Epistemology 201: The Science of Science
Date: Mon, 21 Feb 2005 12:31:22 -0500
Organization: Cornell University
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step...@nomail.com said:
> In sci.math Allan C Cybulskie <allan.c.cybuls...@yahoo.ca> wrote:
>
> : <step...@nomail.com> wrote in message
> : news:cut65p$17ho$1@msunews.cl.msu.edu...
> :> In sci.math Allan C Cybulskie <allan.c.cybuls...@yahoo.ca> wrote:
> :>
> :> : <step...@nomail.com> wrote in message
> :> : news:curv26$4b1$1@msunews.cl.msu.edu...
> :> :> In sci.math Allan C Cybulskie <allan.c.cybuls...@yahoo.ca> wrote:
> :> :> : The problem, as I see it, is that it is clear that every single point
> : in
> :> : the
> :> :> : range (0,1) is ALSO in the range (0,2), plus all the points you can
> : get
> :> : by
> :> :> : generating them from (0,1). So how can there NOT be more elements in
> :> : the
> :> :> : range (0,2)?
> :> :>
> :> :> You have to define what "more" means. What is infinity + infinity?
> :>
> :> : I've already dealt clearly with that. It's merely a word game, since
> :> : infinity is the largest number that we can talk about. But that does
> : not
> :> : allow us to go beyond that and draw any inferences beyond "we can't talk
> :> : about the extra elements because we don't have a terminology for it).
> :>
> :> I have not seen you deal clearly with that.
>
> : I dealt clearly with what I meant by "more", which is that in this case the
> : set (0,2) has all of the elements that are in the set (0,1) and the elements
> : in the set (1,2). By any reasonable definition of "more", the set (0,2)
> : will have more elements than the set (0,1). The only way to say otherwise
> : is to insist that they both have an infinite set of elements ... but that's
> : a word game on "infinite".
>
> It is not a word game on "infinite". What does "infinite" mean?
> The mathematical definition of an infinite set is a set that can be put
> in a one to one correspondence with a proper subset of itself. What
> do you think "infinite" means in this context? Do you think that the
> set (0,1) or set (0,2) does not have an infinite number of elements?
Literally, "infinite" means "unending". That's not a very exact concept.
>
>
> :> And it is clearly not
> :> the case the "infinity is the largest number that we can talk about".
> :> The theory of infinite cardinals disproves that.
>
> : Sigh. So what, then, IS "infinity + infinity"? And don't say "infinity",
> : or you will have disproved the above statement.
>
> It depends on what "infinity" you are talking about.
> aleph0+aleph0=aleph0. aleph1>aleph0. Both are infinite cardinals.
>
> :>
> :> :> There is clearly a bijection between [0,1] and [0,2], and in that
> :> :> sense there is exactly one real in [0, 2] for every real in [0 , 1].
> :>
> :> : I wouldn't disagree with that; I just don't think it has meaning.
> :>
> :> What meaning should it have? What meaning do any mathematical
> :> functions have?
>
> : Well, if it has no meaning, then why should anyone other than people playing
> : with mathematics care about cardinality?
>
> Whoever said anyone other than mathematicians should care about
> cardinality? Tony and others were claiming that the mathematicians
> are wrong and their definitions are contradictory.
My claim was that Catorian cardinality is only a very rough measure of
infinity, and that it doesn't equate to any fine definition of the size
of a set. It only distinguishes between sets that are on the order of a
power set above the other. It's like counting 1,infinity, and skipping
all the other integers and saying they're the same as 1. The problem
here is that the mathematicians among us fall back on Cantor and
cardinality and claim his is the final word, and that there is no
improvement to be done in the area of classifying infinities. If they
haven't heard the math from a book or professor, it's automatically
wrong.
>
> :>
> :> :>
> :> :> : It really does seem like just a word game to me; you get stuck
> :> :> : claiming that they are both "infinite" and then give that more
> :> : significance
> :> :> : than it deserves.
> :> :>
> :> :> They are both "infinite". The point is that it is a different
> :> :> type of "infinite" than the "infinite" that describes the set
> :> :> of integers.
> :>
> :> : This REALLY seems to be a word game, right here, when you start talking
> :> : about different kinds of infinity.
> :>
> :> It is not a word game. It is a matter of definitions. When being
> :> technical, the word "infinite" and "infinity" are rarely used.
> :> Technically, a set is infinite if there exists a bijection between
> :> it and a subset of itself. It can be proven that not all infinite
> :> sets have the same cardinality.
>
> : Fine ... but I never said a word about "cardinality".
>
> Well that is what the thread was about.
No it wasn't, really. It was about the limits of cardinality, and the
actual relative sizes of infinite sets.
>
> :> If you stick to the definitions and
> :> do not rely on vague intuitions about what "infinite" should mean
> :> no word games are necessary.
>
> : Then why challenge my comment about "more elements" by appealing to
> : "cardinality"?
>
> Because it is not clear what "more" means. Yes, there are elements
> in (0,2) that are not in (0,1). But "more" also means that
> the number of elements in (0,2) is greater than the number of
> elements in (0,1). This is why cardinality was brought up.
> What does "number of elements in (0,2)" mean if you are
> not talking about cardinality? There are no infinite integers,
> so the "number of elements in (0,2)" cannot be an integer.
> What type of number do you think it is?
In finite sets, a proper superset has more elements than its subset.
There is no reason to change this rule for infinite sets. {0,1,2,3,...}
has one more element than {1,2,3....}, since it contains all the
elements of the second set plus one additional one. In this sense, it
doesn't matter if they are infinite. If your definition of "number of
elements" is restricted to Cantor's cardinality, then you cannot
distinguish the two, and yet there is a difference. Thsi difference is
what we're talking about, not cardinality, already.
>
> :>
> :> <snip>
> :>
> :> : I would not disagree with this, but again this is a word game, since
> : your
> :> : proof depends on the statement that both sets represent, in their
> : entirety
> :> : THE SET OF INTEGERS. So OF COURSE they have the same number of
> : elements.
> :> : Clearly this isn't the case in the example I was dealing with, which was
> :> : either the reals in (0,1) and (0,2) or even integers versus all
> : integers.
> :>
> :> But your whole argument is that because (0,1) is a subset of (0,2)
> :> it cannot have the same number of elements. No justification for
> :> that claim was provided.
>
> : I quite clearly claimed that all of the elements that are in (0,1) are in
> : the set (0,2), plus the elements from (1,2) as well (before this post).
> : This clearly means, relatively speaking, that there are more elements in the
> : set (0,2) than in the set (0,1). You have yet to disprove this reasoning.
> : In addition, I did not make the claim about subsets at all. Besides that,
> : it is clearly false since the set itself is indeed a proper subset of
> : itself, and I would clearly argue that that subset has the same elements as
> : the set itself.
>
> A set is not a proper subset of itself. In any case, all the elements
> in the set of strings that correspond to octal representations of integers
> are in the set of strings that correspond to decimal representations of
> integers. Are there more decimal representations than octal representations?
> I am talking about sets of strings, and every string that represents
> an octal number also represents a decimal number. I can define the
> sets using some Perl like regular expressions (for simplicity lets
> just consider representations of numbers greater than 0).
> octals = [1-7][0-7]*
> decimals = [1-9][0-9]*
> Clearly every element in the first set is an element of the second set.
> Every element in the first set corresponds to an integer, and for
> every integer>0 there is a item in the first set that is its octal
> representation. Likewise for the second set. In both cases the
> sets clearly have exactly as many elements as the set of integers
> greater than 0.
>
> So why are you so sure that (0,2) has more elements than (0,1)?
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. If they are meaningless
strings then they are not ordered in any way and cannot really be
corresponded to each other and counted. This is nto a defense of
Cantor's approach, as I see it, but confirmation that it misses a lot of
detail.
>
> Stephen
>
--
Smiles,
Tony
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