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

From: Stephen J. Herschkorn (sjherschko_at_netscape.net)
Date: 03/22/05 

Date: Tue, 22 Mar 2005 03:42:01 -0500

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.

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?

>>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 don't time to work through all the others and fail to see the point at any
>rate/
>
>

Well then read and ponder.

And, please, PLEASE: I really want to see your answer to the ones you
chose to omit, viz.,

iv) A = {0, 1, 2,...} vs. C = {(0,1), (1,2), (2,3),...} = {(n,
n+1): n in N}
v) B = {1, 2, 3, ...} vs. C = {(0,1), (1,2), (2,3),...}

In (iv) and (v), do you say that A and C have the same number of
elements and that B and C have the same number of elements, but
that A has more elements than B?

>>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?

>
>
>>>As for the second, my argument never relied on the proper subset
>>>argument -- despite Stephen's insistence -- and so I clearly don't
>>>
>>>
>confuse
>
>
>>>those concepts.
>>>
>>>
>>>
>>>
>>But you do. You insist that N has more elements than N \ {0} and
>>[0,2] has more elements than [0,1]. As far as I can tell, your
>>rationale is that in each case, the latter set is a proper subset of the
>>former. If not, what is your rationale?
>>
>>
>
>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." That is
certainly true for finite B. Why do you insist this property must
carry over to infinite sets?

>
>
>>>My answer: Yes. There is at least one more seat and one more passenger
>>>
>>>
>than
>
>
>>>you had before, even though we would still have to call is "infinite". I
>>>agree that infinity + 1 = infinity, but that using that term in that way
>>>
>>>
>is
>
>
>>>meaningless since all it really means, I suppose, is an uncountable
>>>
>>>
>number.
>
>
>>>
>>>
>>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.

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!!!!! 

-- 
Stephen J. Herschkorn                        sjherschko@netscape.net

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