You are making the grave error of confusing Mathematics with common
sense and reality. Both are irrelevant to Mathematics. Math is
just the manipulation of symbols according to the rules and definitions
given. Occasionally, we can 'map' some part of reality onto some
mathematical model, use it to produce a theorem, and then map that
theorem back onto reality to predict something; but such /applied/
mathematics is just guesswork. We really have no way of knowing how
accurate our mappings are, and 'pure' mathematicians don't care
anyway; they just care whether or not the results properly follow
from the givens, nothing more.
In the case of transfinites, no mapping to reality is possible, so
all common-sense arguments like "something can't be equal to helf of
itself" are irrelevant. Transfinites /are/ equal to half of themselves,
/by definition/. Why have mathematicians chose such a bizarre
definition? Because it makes other mathematics easier, simpler, more
consistent, and perhaps more practical. Why is 0^0 equal to 1?
Because /defining/ it as such simplifies a lot of calculations that
would otherwise have ugly special cases (the graph of y = x^x is
made continuous at [0,1], for example). Not because it 'seems right'
or 'makes sense' or something.
Bertrand Russel said it best: Mathematics is unconcerned with whether
a proposition is 'true', only whether certain propositions logically
follow from other (given) propositions. And ideally, they are not
propositions about specific things, but must be generalized to hold
for anything we care to fill into the variables. Mathematics, then,
is the science in which we never know what we are talking about, or
whether what we are saying is true.
-- Lee Daniel Crocker <lee@piclab.com> <http://www.piclab.com/lcrocker.html> "All inventions or works of authorship original to me, herein and past, are placed irrevocably in the public domain, and may be used or modified for any purpose, without permission, attribution, or notification."--LDC