Re: Galileo's Paradox

David Marcus wrote:
Six Letters wrote:
I hope it doesn't seem silly or self-important to respond to my own
post, but the thread has grown so large (I haven't checked yet today, but
there are probably numerous posts in it to catch up on), and there isn't
any one place in it where this should go. I just wanted to summarize my own
thinking about this in response to the discussion so far.

The basic paradox is this (I will take as an example E = the even
numbers, to compare with N the natural numbers):

It appears that N > E because N contains E but E does not contain
N; there is a mapping from E to N which leaves the odd numbers out.

It appears that N = E because there is bijective mapping between
the two sets.


It is like one of those ambiguous pictures, the old hag and young
lady comes to mind, in which one can only visualize one interpretation at a
time.
Any argument that focuses on one to the exclusion of the other will
miss the point. In my original post I tended to prefer N > E, but any
postion which doesn't encompass both aspects is going to be wrong.

If it were not for Cantor's proof that R > N, I think the paradox
would be relatively easy to deal with. We would say that it makes no sense
to think of N as having a fixed size. N appears > E from the point of view
of containment. How is it then that we can show a bijective mapping between
the two? Because (we would say), simply, they are infinite sets. Saying
that N = E says no more than that they are both infinite. Since it can
appear trivial that they are infinite, one can understand the bias towards
a containment picture, which at least appears to give some comparative
information about the two sets. But it is not exactly true to say that
their common infinity is trivial. (Not after Cantor anyway)
So are the two sets equal or not? ONLY in the sense that they are
both infinite. Does N have more members than E? Yes, in the sense that N
contains the odd numbers, but since both are infinite it is impossible to
conclude there is a difference in size.
Does infinity have anything to do with size? Yes, to the extent
that the finite does, for the infinite is the not finite. Forget for the
moment about the definition of the finite set as one with no equality to a
proper subset, which may prove to be mathematically convenient, but is as
arse-backward as it is possible for a defintion to be. The finite set is
one with n members where n belongs to N; it has 0 or 1 or 2 or........
members.

Now what difference do Cantor's discoveries make to all this? I
would say none, as far as the reasoning goes, but everything in terms of
what is of interest. Everthing said about the infinity considered (the
denumerably infinite) is still valid. It is as if infinity was a piece of
elastic, which would stretch or shrink to cover all cases. But it turns out
there are hyper-infinities which exceed the elastic's breaking point. Does
this mean our first infinity has a fixed size after all? No. But there is
now a comparison to be made. It is still fallacious to think that a
bijection demonstrates directly equality of size. Just as fallacious as it
is to suppose that containment shows inequality of size. All that is
demonstrated by the bijection is that both are infinite, AND, (since it
turns out there are different orders of infinity) have the same order of
infinity. That there is an infinity greater than the denumerably infinite
sets does not mean all those sets are the same size as each other. It only
means they may be grouped together as a lower order of infinity.

It might still be possible to work with a notion of 'containment
size'. We accept that denumerable sets cannot literally be said to vary in
size. With that said, and starting from containment, would it be possible
to make a sort of evaluation of every denumerable set in terms of N? So
that Z *=* 2N, the squares *=* sq.rt of N, E *=* N/2 etc.? If that could be
done in a thoroughgoing and consistent way, it would seem to me to be well
worth doing. (It might be possible to handle divergent series in a similar
way.) It would not, however, have any bearing on the theory of transfinite
numbers, and possibly no bearing on anything currently of interest to
mathematicians.

I'm not too clear on why you don't like the standard solution to this paradox. To wit:

Once you specify how you wish to compare sets, then you get a specific answer as to how E and N compare. Different ways of comparing give different answers. Some of the ways of comparing sets are containment, cardinality, and density.

For finite sets, you generally get the same answer regardless of which way you look at it. For infinite sets, you don't. This makes infinite sets paradoxical (if you think they should behave like finite sets), but also more interesting.

So, what is wrong with this solution?


There is nothing wrong with saying E and N have the same cardinality. It's a fact. Six Letters is essentially suggesting IFR, my Inverse Function Rule, which indeed can parametrically compare sets mapped onto the real line, using real valued functions, as a generalization to set density. It works for finite and infinite sets. So, what is wrong with trying to form a more cohesive theory of infinite set size, which distinguishes set sizes that cardinality cannot? There certainly seems to be intuitive impetus for such a theory.

Tony
.

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