Re: Epistemology 201: The Science of Science
Date: 24 Feb 2005 00:51:49 GMT
In sci.math Allan C Cybulskie <email@example.com> wrote:
: <firstname.lastname@example.org> wrote in message
:> In sci.math Allan C Cybulskie <email@example.com> wrote:
:> : <firstname.lastname@example.org> wrote in message
:> :> 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
:> : game, since relatively speaking my analysis still holds, and that it is
:> : that the number of elements is "infinite" that we get into any trouble
:> : 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.
: And there's no reason to assume it DOESN'T. Show me why my analysis doesn't
: work for infinite sets, or the sets in particular.
You have not defined what "twice as many" means when applied to
infinite numbers. Until you do that you cannot make any argument.
: You cannot appeal to "infinite sets act differently" and in the same breath
: base your comments on applying a notion from finite sets and assuming it
: applies to infinite sets as well (ie cardinality and the relation to the
: number of elements in a set).
I am appealing to well defined rules for dealing with certain types
of infinite numbers. You are claiming these rules are wrong because
they are not the same as the rules for finite numbers.
:> :> : 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
:> : 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.
: Personally, I prefer the normal, every-day sense, which in no way appeals of
: "cardinality". It is NOT standard to talk about "number of elements" as
: being the same as "cardinality" except among certain mathematicians.
: For example, if I place a number of things in a bag, and count them, that
: seems to be very similar to what we think we mean by the "number of
: elements" in a set. But this does not in any way appeal to "cardinality".
:> :> 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
:> : 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
:> : 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.
: Yes, you are correct, but then you could also certainly say that Zack had
: twice as many as Debbie, and then all that is is a word game (you have to
: accept that, right?) So let's move on to more reasonable examples, shall
:> 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.
: And you cannot simply argue without proof that it does not.
:> In fact, in all the
:> definitions of infinite numbers that I am aware of, your statement
:> is false.
:> For example, see
:> Are you claiming that equation 5 is wrong?
:> Or see
:> 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.
: Sigh. Of COURSE I don't argue that the equations are wrong, because that
: reasoning was the reason I commented that your analysis was playing a word
: game with the definition of infinity, because what it means is that even if
: the set of integers, say, has twice as many elements as the set of even
: integers since the set of even integers is an infinite set we would get that
: its number of elements is infinity, and then the set of integers would have
: to have 2*infinity, which is infinity since you can't talk about a number
: greater than infinity (all multiplication on it ends up back at infinity).
: But this does not mean that the relative number of elements (throwing the
: "number" of infinity out of the picture for an instant) in the set of
: integers is not twice that of the number of even integers.
Until you define what "relative number of elements" is, it does
not mean anything at all.
:> :> 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.
: Your show of proof is blinding.
That is the definition. You have access to google I assume.
Is it really so hard to check a definition?
:> :> In any case, all the elements
:> :> in the set of strings that correspond to octal representations of
:> :> 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
:> : integer along the way there will be strings in the set of octals that
:> : 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
:> : decimals simply because it doesn't contain all the elements. But this
:> : 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.
: If I challenged the subset idea, it would be like this.
: Take any N where N represents an integer that the strings represent (say, in
: base 10 for simplicity). At any N in the sets of strings representing
: octals and decimals, there will be an string in the set of octal number
: strings that is not in the set of decimal number strings, as long as N is
: greater than 7. This is because to achieve the integer N takes more strings
: in octal than in decimal. So at 10, say, you'll have to go to 12 (?) for
: the octal numbers and 10 for the integers, which means that 11 and 12 will
: not be in the set of integers. Violates the subset rule.
Ah, the old quantifier dylexia problem. I am not talking about
octals and decimals less than some N. I am talking about all of them.
Using that logic you can prove that all sets are finite. Take
the set of integers. At any N there will be a finite number of elements
less than N. Therefore the set of integers is finite.
: But as I said, this is neither here nor there. I never argued that subsets
: could not have the same number of elements as the superset, and so the
: argument was over whether or not they would have the same number of elements
: without taking that into account. Your definition that they included all
: the integers gave a clear and utterly trivial "proof" that they did.
Yes you did. You said:
"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)? "
The only logic I see in there is that there must be more elements in (0,2) than
in (0,1) because (0,1) is a subset of (0,2). That is your argument. If
you meant something else you did a poor job of expressing it.
:> 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.
: And, of course, this seems to be based on an utterly false view of my
: Let me put it clearly in respect to the example you just gave. In the set
: (0,2), there are the elements AND WHAT THEY REPRESENT that make up (0,1),
: PLUS a set of different elements AND REPRESENTATIONS that map from (1,2).
: How can the set (0,2) NOT have more elements?
How is that different than claiming that any superset must have
more elements than a proper subset? I am not sure what your "PLUS"
is supposed to mean. (0,2) contains all the reals greater than 0 and
less than 2 and that is all it contains. (0,1) contains all the reals greater
than 0 and less than 1 and that is all it contains. I am not sure what point
you are trying to make with the capitals.