Re: "Number" of elements; was: Distinct linear orderings on Z

From: Allan C Cybulskie (allan.c.cybulskie_at_yahoo.ca)
Date: 03/23/05 

Date: Tue, 22 Mar 2005 19:15:43 -0500

"Stephen J. Herschkorn" <sjherschko@netscape.net> wrote in message
news:423FDA59.3080300@netscape.net...
> Sorry, I will not let you off the hook here, Alan. There are other
> logical flaws in your argument, but I will concentrate on the ones
> which, unanswered, will reveal you unable to acknowledge the weaknesses
> in your arguments.

I warn you now that I'm going to ignore a lot of the more complicated sets
again because of time and knowledge, but hopefully the simpler examples can
clear everything up.

>
> Allan C Cybulskie wrote:
>
> >"Stephen J. Herschkorn" <sjherschko@netscape.net> wrote in message
> >news:423E01FA.1030705@netscape.net...
> >
> >
> >> i) {0, 2, 4, 6,...} vs. {0, 3, 6, 9, ....}
> >
> >
> >
> >The first is larger.
> >
> i'): {(2,0), (2,1), (2,2),...} = {(2,n): n in N} vs. {(3,0), (3,1),
> (3,2),...} = {(3,n): n in N}
>
> Do you think these two sets have different sizes? I cannot see how
> anyone one would. Yet replace each ordered pair by the product of its
> components, and somehow the relative sizes of the sets change. Is that
> intuitive?

Imagine that these are infinite sets. What you get as the last element is 2
* infinity, and 3 * infinity. Replacing the sets by their product seems to
lead to a problem in itself ... that in a set of numbers that contain all
numbers up to infinity, we add in a large number of numbers greater than
infinity itself. So I'm not even sure the move you're making is an
appropriate move. Certainly it isn't the same set as (0, 2, 4, 6 ...) and
(0, 3, 6, 9 ...), since those don't go to 2 * infinity, 2 * (infinity - 1),
etc ...

>
> >>ii) {1/2, 3/2, 5/2,...} vs. {1/3, 4/3, 7/3, 10/3,...}
> >>
> >
> >The first is larger again.
> >
> >
>
> Please explain *why* the first is larger in example (ii).

I might actually be wrong, but it's not really important. This is why I
want to avoid the more complicated sets to avoid running into the problem
that I make a claim by doing the math wrong.

> >>Another tack:
> >>{0} and {1} have the same number of elements, no?
> >>{0, 1} and {1, 2} have the same number of elements, no?
> >>{0, 1, 2} and {1, 2, 3} have the same number of elements, no?
> >>{0, 1, 2,..., n} and {1, 2, 3,..., n+1} have the same number of
> >>elements for any given n, no?
> >>Yet somehow you insist that {0, 1, 2,...} has a greater number of
> >>elements than {1, 2, 3,...}. Is this not also a "conflicting
> >>conclusion" (ACC's words from another post)?
> >>
> >>
> >
> >If the definition of the second set really does end at "n+1" relative to
the
> >other set, then they would have the same number of elements. But
generally
> >the set I was talking about seems to be defined as {0, 1, 2 ..., n} and
{1,
> >2, 3, ... n}, and so there is clearly one less element in the second set
> >than in the first. So, no contradiction.
> >
> >
>
> Sorry, unconvinvcing argument: You have not answered my presentation.
> Where is the flaw in my argument?

This is where I hope things start to clear up a bit ...

There isn't any. We are just working on different sets. Your set starts as
being {0, 1, 2}and {1, 2, 3}. Which we both agree have the same number of
elements. But the set I was working with would start like this: {0, 1, 2}
and {1, 2}. Which we both would agree indicates that the first set is
larger. Basically, you start the one set at 0, the other at 1, and proceed
the same number of elements leaving the last element in the second set one
number larger than the last element in the first set (n+1). _I_, on the
other hand, start the sets the same way but proceed to the same last element
(n). Which of these best represents the sets {0, 1, 2, 3, ...} and {1, 2,
3, ...}. I'm not certain, but not that your definition has the consequence
that at infinity, the first set has a final element of "infinity" (whatever
that would mean) but that the second would have to have "infinity + 1".
But then it would be said to have "infinity" right? But wouldn't that make
it my definition by default?

> >My claim is that I do not insist that just because something is a proper
> >subset it must have less elements than the superset. It is the
> >mathematicians who have argued that I am merely using that definition
> >without ever refuting the reasoning behind my usage of something that
looks
> >like that definition. If my reasoning is one that must hold for all
proper
> >subsets, the problem is not with me, but with the mathematicians who want
to
> >use BOTH proper subset AND the cardinality theory, and so they must
provide
> >the proof for why the reasoning does not hold.
> >
> >
>
> So you admit that you insist that you think " A is a proper subset of
> of B" implies "A has a smaller number of elements of B."

Nope. Please don't do what the other Stephen did, and insist on this. It
is you who insist on this; I hold no opinion on the matter.

I DO think that the reasoning that "every element that is in A is in B, and
there are additional elements in B that are not in A" pretty much by all
definitions -- even mapping -- implies that A has a smaller number of
elements than B. Since that seems to be what people are saying is meant by
"proper subset", maybe that is the case.

> That is
> certainly true for finite B. Why do you insist this property must
> carry over to infinite sets?

I don't. It is quite possible that it does not, but I need more than just
someone's preference for cardinality to accept that cardinality should carry
over but the other definition should not.

Someone in this thread commented that the definition of an infinite set is
one where some proper subsets can be mapped onto the superset directly.
Since this special quality strikes at the heart of the cardinality and
mapping approach -- this is not true for finite sets, which is why if you
can map two sets onto each other you are certain they have the same number
of elements -- why are you convinced that cardinality can carry over to
infinite sets and still reflect the number of elements accurately?

> >>Huh?? How is there one more seat? None has appeared in this set up.
> >>
> >>
> >
> >If all the seats are taken, in order to have someone else sit down you
need
> >another seat, right?
> >
> >
>
> No, *wrong.* That is exactly the point we are making. My nephew at age
> ten insisted the same thing.

And this means he's wrong?

If all the seats are taken, then to have someone else sit down requires
another seat somewhere, or that we lose a passenger. Perhaps this is too
concrete an example to work for your purposes.

>
> Put it another way: An infinite number of passengers wait for a bus.
> The passengers have the labels 1, 2, 3,... pasted on their jackets, each
> one with a different number. The bus arrives. The seats have the
> labels 1, 2, 3,... on them, each with a different label. Each passenger
> takes the seat that matches their label. Befreo the bus leaves, another
> passenger, with the label 0 on his jacket, runs up and gets on the bus.
> To accomodate him, each passenger moves back one seat, so that passenger
> 1 sits in seat 2, paseenger 2 sits in seat 3, etc. Then passenger 0
> sits in seat 1. NO NEW SEATS HAVE BEEN ADDED TO THE BUS!!!!!

But then you have contradicted your first point -- that all of the seats
have been taken.

I never thought I'd say this but ... can we use a more abstract example
[grin]?

Relevant Pages

  • Re: "Number" of elements; was: Distinct linear orderings on Z
    ... > rationale is that in each case, the latter set is a proper subset of the ... How is there one more seat? ... I claim that that conclusion is a trick of the term infinity (possibly ... relatively meaningless and a rounding error) and so isn't appropriate to be ...
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    ... :> In sci.math Allan C Cybulskie wrote: ... :>: If that is the definition of proper subset, then what it says is that B ... infinity, but then we try to get rid of the infinity which is the only thing ...
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  • Re: Epistemology 201: The Science of Science
    ... :> In sci.math Allan C Cybulskie wrote: ... :>: If that is the definition of proper subset, then what it says is that B ... infinity, but then we try to get rid of the infinity which is the only thing ...
    (sci.math)