Re: Orlow cardinality question

Jan de Vos said:
> In sci.math, Tony Orlow wrote:
> >> But the only "problem" is that it has a property you
> >> don't like. That's a very solipsistic view of "problem".
> > No, it's not a matter of fancy, but of consistency. If a system pretends to
> > measure set sizes, and we agree colloquially that removing an element from a
> > set results in a "smaller" set, but cardinality doesn't reflect this change,
> > then it is not doing the job it claims to be doing. It is not fully
> > distincuishing between sets of different sizes. This is fact, not a matter of
> > opinion.
>
> So your intuition tells you that removing an element from a set makes
> the set smaller. If we think about cardinality as a generalization of
> 'set size', then indeed it does not correspond with your intuition.
>
> However, I have a very different idea about how such a generalization
> should work. Since 'infinity' is not something I recognize as a
> number, a priori, I won't start with thinking about the 'size' of an
> infinite set as a number. However, I do have some idea what it should
> mean that the 'size' of a set A is smaller than the 'size' of another
> infinite set B: if I can embed A in B, but not B in A, then the 'size'
> of B must be bigger than the 'size' of A. If I can embed A in B, and
> B in A (i.e., I can make a bijective map between the two), I don't
> think either of them should be regarded as 'bigger' than the other.
>
> The most important reason to restrict my thinking to this, is that I
> don't think the 'size' of a set should depend on the thinks that are
> in the set. If I have a set, and simultaneously change all the
> elements in the set into other elements, I don't think the size of the
> set should change.
>
> So if I have the set:
>
> N = {1, 2, 3, .... } of all (finite) integers, and then multiply all
> the elements in the set by two, getting N' = {2, 4, 6, 8, ... }, I
> don't think that second set should be 'smaller' than the first one,
> even though the second one is a proper subset of the first.
>
> I could also add 1 to all the elements in the set, and expect the set
> to have the same size as the two above. Notice that I could get the
> same new set by removing the number 1, which, according to your
> intuition, should make the set smaller. See the difference in
> intuitions? Which of the two is 'correct'?
>
>
> Jan
>
>
Well, when proper subsets are always smaller for finite sets, and removing
elements from finite sets always makes them smaller, and when your intuition
says that these two criteria for set size are irrelevant when it comes to
infinite sets, then I must only conclude that your intuition is guided by your
axioms, rather than the other way around, which seems to me like putting the
cart before the horse. It always behooves us to maintain a sort of
fundamentalist approach to our questions, returning periodically to the
original question that spurred the inquiry, so as to stay focused on what we re
doing. Cardinality seems to have strayed far from its roots, now no longer
claiming to be about set size, but unable to define exactly what it IS about.

I believe you have missed much of this discussion. My Theory of Bigulosity
exactly calculates what out intuitions ACTUALLY tell us: that adding an element
increases the size of the set by one, that the even integers are exactly HALF
of the integers, etc. This is not beyond reach, and should be regarded as
progress, not rebellion. The goal is consistnecy, which is sorely lacking in
cardinality.
--
Smiles,

Tony
.

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