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 !!
, 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.
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)
, 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
Put in normal Cantor form the following numbers:
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:
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:
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