Calculations with transfinite ordinals
Before to read the post, I would like to show a list of the most import rules of calculations. Have a look the list below:
The reader is invited to verify, by easy direct calculation or easy induction, the following identities:
,
,
the binomial theorem is not true any more !!
In particular:
, no double product !!!
, classical decompositions don’t work any more !!
substitution of
by
here is true !!
,
,
Recall that but observe that
=
is not the same as before.
More generally:
,
As a consequence of these results we obtain:
Decomposition into irreducible (unzerlegbar) multiplicative factors, an example explains better than unreadable formules:
Squares and square roots:
given , we distinguish two cases:
that is
successor or
that is
limit.
In the first case
and since this “double form” is in canonical form it is the canonical decomposition of : let’s call it form a)
In the second case it’s easily seen that . that we call form b).
In summary: a successor is a square iff of form a) a limit number is a square iff of form b)
examples:
, the radicand is a successor not in double form a)
we have no root.
, we search
such that
that gives
and hence the solution
Powers and roots:
Similarly has a form a):
or a form b):
For the following exercises the following rules are useful:
definitory property of
-number
:
:
in general a tower of n is a tower of n
with on the top
and in general
if
because every
is a delta number!
se
in fact
in fact
Put in normal Cantor form the following numbers:
sol:
sol: by construction of the first fixed point after
Solve the following equations:
: sol.
,
,
(right multiplicative injectivity)
Solve the following sistems:
sol: the first equation bounds the possible solution to x=a+b and y=
c+d with a,b,c,d in
and with
, substituting these expressions and equating coefficients we have
Divisions with rest:
divided by
sol: we reproduce elementary division: ,
; if we subtract this from the dividend we obtain the rest
,
so we have the natural 2 as second term of the quotient; we multiply
by this second term quotient and obtain
; we subtract this from
and obtain
; this is the rest because it’s less than
. Summing up:
References
H. Bachmann Transfinite Zahlen Springer Verlag 1967
J. Donald Monk Introduction to set theory Mc Graw Hill New York 1969
Lectures on Set Theory euclid.colorado.edu/~monkd/setth.pdf
M. Di Nasso people.dm.umipi.it/dinasso/~Mauro Di Nasso