Re: Epistemology 201: The Science of Science

stephen_at_nomail.com
Date: 02/21/05 

Date: 21 Feb 2005 16:21:16 GMT

In sci.math Allan C Cybulskie <allan.c.cybulskie@yahoo.ca> wrote:

: <stephen@nomail.com> wrote in message
<snip>
:> 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?

: I've agreed that they do already. However, that's what I called the word
: game, since relatively speaking my analysis still holds, and that it is only
: that the number of elements is "infinite" that we get into any trouble at
: all.

But we are talking about the case where the number of elements
is "infinite". Your analysis is based on finite numbers and
there is no reason to automatically assume it holds in the
infinite case.

<snip>

:> : Fine ... but I never said a word about "cardinality".
:>
:> Well that is what the thread was about.

: Actually, no it isn't. It's when "cardinality" gets inflated to number of
: elements that the problems all started.

Then you need to define "number of elements" in terms other than
cardinality. I have not seen you present any such definition
and until you do noone can have any idea what you mean.

:>
:> :> 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.

: Except I have no reason to accept that "cardinalty" reflects "number of
: elements".

See above.

:> 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?

: What does it matter? Relatively speaking, there are twice as many elements
: in (0,2) as there are in (0,1). I don't really care what number that
: actually is expressed in, as the relative analysis remains unchallenged
: whether we can give it a number or not.

: For example, if I say that Debbie has twice as many marbles as Zack, it is
: irrelevant whether I know what number that actually is. Whatever number
: that is, Debbie's amount will be greater than Zack's.

Yes it does depend on the number. For a simple example, if Zack
has zero marbles, then Debbie does not have more marbles than Zack.
If Zack has a finite, non-zero number of marbles, than Debbie
will have more marbles than Zack. However we clearly are not talking
about finite, non-zero numbers so you cannot just claim without
any proof that the finite argument applies. In fact, in all the
definitions of infinite numbers that I am aware of, your statement
is false.

For example, see
        http://mathworld.wolfram.com/AffinelyExtendedRealNumbers.html
Are you claiming that equation 5 is wrong?
Or see
        http://mathworld.wolfram.com/Aleph-0.html
Is equation 2 wrong?

Before anyone can make sense of your argument you need to define
what you even mean by "twice as many as infinity". Our naive
notions of multiplication do not apply to the infinite case.

<snip>

:>
:> A set is not a proper subset of itself.

: You might want to check the definition again. It's been a while, but I
: clearly remember the comment that it was, since it was an odd occurence.
: I'll admit that I could be wrong , but ...

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.

: I could challenge that these are actually subsets, since if you stop at an
: integer along the way there will be strings in the set of octals that are
: not in the set of integers. So I fail to see how you can claim that the
: strings representing octals are just a subset of the strings representing
: decimals simply because it doesn't contain all the elements. But this is
: neither here nor there.

If you challenge that they are actual subsets then you do not
understand what subsets are. Any string that satisfies the
definition of an octal number also satisifies the definition
of a decimal number. There are strings that satisfy the definition
of a decimal number that do not satisfy the definition of an octal
number. Therefore the octals are a proper subset of the decimals.

The octals are strings that start with a 1,2,3,4,5,6 or 7 and
are followed by any number of 1,2,3,4,5,6,7 or 0's.

The decimals are strings that start with a 1,2,3,4,5,6,7,8 or 9 and
are followed by any number of 1,2,3,4,5,6,7,8,9 or 0's.

Any string that meets that first definition meets the second
definition. That is what "or" means. Any string that meets
the second definition that contains a '9' does not meet the
first definition. The octals are clearly a proper subset
of the decimals.

I can define (0,1) as all the numbers that are greater than 0
and less than 1.

I can define (0,2) as all the numbers that are greater than 0
and less than 1, or greater than 1 and less than 2, or equal to 1.

The "ors" tell me that (0,1) is a proper subset of (0,2).

:>
:> So why are you so sure that (0,2) has more elements than (0,1)?

: Why do you ask? I've given the reasoning ... and never asserted that just
: because something was a subset of another set that they had to be of
: different sizes.

So you also must conclude that there are more decimal representations
than octal representations, but at the same time believe
that their are exactly as many decimal representations as
integers, and exactly as many octal representations as integers.
Or I suppose you believe that there are integers that can be
represented in decimal that cannot be represented in octal.

Stephen

Relevant Pages

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