Re: Galileo's Paradox

On Fri, 24 Nov 2006 11:35:31 +0000, Six Letters wrote:


GALILEO'S PARADOX

1 2 3 4 5 .................
1 4 9 16 25 ...............

There is a paradox because the 1:1 Correspondence suggests the sets
are equal in size, by extension from the finite case, and yet clearly the
second set is contained in the first set. That an infinite set can be put
into 1:1 C with a proper subset is not by itself paradoxical. That is only
the beginning, the facts of the case. The paradox is that the squares seem
to be both smaller than N and the same size as N.

I want to suggest there are only two sensible ways to resolve the
paradox:

1) So- called denumerable sets may be of different size.

2) It makes no sense to compare infinite sets for size, neither to say one
is bigger than the other, nor to say one is the same size as another. The
infinite is just infinite.


My line of thought is that the 1:1C is a sacred cow. That there is
no extension from the finite case.

If we want to compare the two sets for size we would write, not the
above, but:

1 2 3 4 5 6 7 8 9 ...............
1 2 3 4 5 6 7 8 9................
^ ^ ^

(The intention here is to highlight the squares in the second row of
integers.)

Then we would notice that the relative size of the squares set
becomes ever smaller as n increases, that increasingly large numbers of
integers are missed out. In fact, if we wanted to find a plausible
candidate for a set eual in size to N, then we would choose not the
squares, but the non-squares.

The contrived nature of the 1:1C becomes more obvious when we
compare with N, sets that appear to be larger than N. The clearest example
is Z.

We have:

1 2 3 4 5 6 7 .........
1 -1 2 -2 3 -3 4 .........

which is mildly clever, but again if we wanted to compare the two sets for
size we would write:

0 1 2 3 4....
...-4 -3 -2 -1 0 1 2 3 4....

with a perfect 1:2 Correspondence.

Here one would like to say, since not only is there 1:1C between Z
and a proper subset, but an identity (1 2 3 4 ......), that however you
define infinity there has got to be more in Z than in N. Not, of course, if
you make all countable sets equal in size by definition. But for me, that
doesn't relieve the paradox at all. On the contrary it builds it into the
foundation of the mathematics.
I would like to suggest that the existence of 1:1C between the two
sets is a CONSEQUENCE of the fact that they are both infinite. The
infinities are what gives one room to manoeuvre, to manufacture a 1:1C. It
has no bearing on their relative size.

Can one make sense of Z = 2N, of Q = N^2, etc.? (Incidentally the
number of squares would be sq.rt. of N, since after n^2 integers there are
n squares.) Maybe it's complete rubbish, but my argument is that the
alternative is the ineffable infinity. If it does make sense, there is no
place for a diagonal argument, or a power set argument, since it would
already be conceded that 10^N > N, that 2^N > N, or in general that k^N >
N, just as Z > N and Q > N.

There remains of course Cantor's proof that R cannot be put into a
1:1C with N, which is very interesting. But what does it mean?
Maybe something like this:

So-called denumerable sets can be represented on a
finite-dimensional lattice, so that a self-avoiding walk can be shown to
systematically cover the entire line, are, volume or hyper-volume. For R
understood as a set of decimals (to choose that -- perfectly good --
representation), by contrast, every decimal place can be construed as an
axis.

In any case what I don't understand is how this affects the simple
paradox with which we began.
However, it may very well be that my insufficiently tutored brain
has flown its coop again, in which case I would be very grateful for any
illumination.


Six Letters 24/11/06



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.

Six Letters
.

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