>>> Bob Sewell on Infinity                                                 
 BS> It seems to be the same problem many people have with the idea of
 BS> God.  They cannot or will not believe in what they cannot empirically
 BS> perceive.
Gnomes, fairies, elves, etc.  Another realm of reality. -)
 BS> I think God is limited to doing things that not logical
 BS> contradictions.  In other words, divine omnipotence doesn't mean
 BS> there are no limits to what an omnipotent being can do, but as Thomas
 BS> Aquinas said, "God can do all things that are possible."  That means
 BS> he cannot make a rock too big for him to lift, or make a round square,
 BS> etc. Indeed, why the hell would he want to?
Logic is greater than god. -)  Another resolution of this paradox is 'he 
could if he would'.  So he won't because he wants to establish logic as a 
standard.  Interesting paradox as his history shows, he can be illogical as 
humans.
 
 BS> From about 1884 until he died Cantor suffered occasional mental
 BS> illness. Persons familiar with his life have suggested that these
 BS> attacks were brought on by the difficulty of his research and by
 BS> the unwillingness of other mathematicians (Kronecker, in
 BS> particular) to accept his unusual results. Cantor died on Jan. 6,
 BS> 1918, in the psychiatric institute at Halle.
Hm, rejection by peers seems to be the root of his problem.  Contemplation of 
infinity has it's mental hazards as mystics can attest. 
 
 WE> I read his original thesis
 WE> upon transfinite numbers when a kid.  Thanks to you, -) I'm borrowing
 WE> a library copy of his original work.  Maybe it will have some insights
 WE> to help you.  I'll look.  It should make interesting discussion. 
 WE> Maybe even philosophical.
 BS> I'm looking forward to it.
Well, it covered basics skipping some crucial material.
Modern reruns are smoother and more comprehensive.
May have to look for one of them to find a general proof of C + C = C for 
infinite C > A1.  Rather weak and awkward is the following:  set A' = 2^Ai.  
Thus A' + A' = 2A' = 2*2^Ai = 2^(Ai+1) = 2^Ai = A'
 WE> The interesting thing about proving cardinality of P(S) > cardinality
 WE> of S is it's similarity with the Russell paradox:  Can the set of all
 WE> sets not belonging to itself, belong to itself?
 BS> Does the barber shave himself?  ;^)
I cut my own hair.
 BS> I know.  It still doesn't make it easier on a common sense level.
 BS> And it makes my brain hurt to try to make sense of it on that level.
Wrong approach.  Mystics have no concern for common sense.  Intuition 
instead.  That's a better approach.
 BS> I understand it on a purely logical level, but I don't understand
 BS> it on a practical level, I think is what I'm trying to say.  I can't
 BS> see it empirically, therefore it doesn't make sense on a more basic
 BS> level. I don't grok it, if you'll pardon the Heinleinism.
Infinity isn't practical, pragmatic.  From a hands on view, infinity is the 
potential for additions.  There are infinitely many real numbers.  That means 
that we can always have a supply of numbers at our disposal that will be 
sufficient for the magnitudes and accuracies we wish to or can express.  A 
computer uses but a large finite number.  If I'm unsatisfied with the 
numerical capacity of the computer, I can expand the software to handle 
larger numbers with a greater number of significant digits.  
Eventually, I will have to get a bigger computer.  When I was a student, four 
digits accuracy was the very most needed, six for accurate professional work. 
 Now, 10 place hand calculators are common place.  Since there's an infinity 
of numbers, this expansion of usage was no conceptual problem.  That's 
tangible infinity, no logical limitation to a process.  To zero add one 
giving one.  To one add one giving two.  To two add one giving three, and so 
on infinitum.  You'll never need them all, only the potential to make more 
whenever needed.  A million digits of pi, but never them all, except for a 
formula that describes a continual process.
 WE> 'From fullness take fullness and fullness remains.'  
 BS> That's interesting!  So, you think the Hindus already had a good
 BS> grasp on infinity a few millennia back?
Yes, Christian mystics, and mystics of other cultures also.
Hindu philosophy is vast in its scope.  As for nothing, read Zen.
 
 WE> The infinite is beyond comprehension is what Christian mystics say.
 WE> What mathematicians say is there are some infinities so big that they
 WE> are inexpressible by any formulas.  They are called inaccessible
 WE> cardinals.  That is a profound and philosophically insightful
 WE> discussion for another post.
 BS> Wouldn't that be Absolute Infinity?  Or is that the beginnings of
 BS> the aleph-n, for all n > 1?
Nope, they are hypothetical infinities, that may not exist.
Ai for finite and transfinite i are described by formulas.
Inaccessible infinities, cannot be described by a formula.
They are beyond the reach of symbols.  It's been proven that their existence 
cannot be proven, that they're absolutely hypothetical.  But's that's just a 
beginning.  More to follow once you understand this start.
As for the absolute, that is a parochial religious notion, no longer 
philosophically popular.  Eastern philosophy does think much of it, 'all is 
flux', 'The Tao that can be told is not the eternal Tao.'
 BS> Yeah, but the limit *is* infinity.  Another way of describing
 BS> infinity with some mathematical operations is "undefined," as in
 BS> division by zero.  Infinity *is* undefinable.
You just defined it. -)  This is a Russell paradox rerun.
If the undefinable is the undefinable, then it's defined as the undefinable, 
so it's not the undefinable.  If the undefinable is not the undefinable, then 
it can be defined (as the undefinable), so the undefinable is the 
definable.
 BS> But really, all this is just word and math games, for even if one
 BS> infinite set is shown to be larger than another, say the set of all
 BS> integers vs the set of even integers, how can you really tell?  
So is economics, accounting, banking, balancing a check book, making change.  
You can tell by the basic definition of 1-1 correspondence, that how.  It 
works as well, the same, for finite counting as it does for infinite.
Who
 BS> has seen the end of either series to know?  I can logically see that
 BS> the latter should be half the size of the former, but my base
 BS> instincts argue against this, since neither set has an end?  Again,
 BS> how can you really tell?
Infinity is in the mind of the beholder.  Yes, the infinite is paradoxical, 
this the mystics of the ages a test to.  Indeed, even a set that's 
equinumerous to a proper subset, is paradoxical.  Didn't it surprise you to 
discover that the real line is equinumerous to the real plane, and to three 
space.  That one infinity is greater than another is more that just for every 
one you count I count another infinity.  It's more like for every one you AND 
I count count, I count another infinity.  The first isn't hard to visualize, 
as for the later, that is a different problem, a much greater problem.
---
---------------