WEEK 5
M: Successor ordinals, limit ordinals, transfinite recursion
on ordinals, rank, the V\alpha hierarchy, the well-founded core.
1.42. Proposition. (a) If x is an ordinal then
S(x) is an ordinal and is the least ordinal larger than x.
(b) If A
is a set of ordinals then UA is an ordinal and is the least upper bound
on all ordinals in A.
1.43. Definition. (a) If x is an ordinal then
S(x) is the successor x; we write x + 1.
(b) If A is a set of ordinals
then UA is the supremum of A; we write sup(A).
(c) Ordinals of the form
S(x) are called successor ordinals;
ordinals
not of that form are called limit ordinals.
1.44. Proposition. Existence of rank function
rankR: V --> On for well-founded set-like
relations.
1.45. Definition. The rank function rankR
: V --> On.
1.46. Definition. The V\alpha hierarchy.
The well-founded core WF.
1.47. Proposition. (a) V\alpha is transitive
for all \alpha \in On.
(b) \alpha
< \beta ==> V\alpha \in V\beta.
(c) \alpha
< \beta ==> V\alpha \subset V\beta.
(d) WF
is transitive.
W: V and the axiom of foundation. Ordinal addition,
multiplication and exponentiation.
1.48. Theorem. (ZF)
WF = V.
1.49. Definition. Lexicographical
ordering.
1.50. Proposition. The lexicographical ordering
incuced by two well-orderings is a well-ordering.
1.51. Proposition. (a) \alpha + \beta = otp(
({0} × \alpha) \cup ({1} × \beta), <lex)
(b) \alpha
· \beta = otp( \beta × \alpha, <lex)
F: Basic facts from ordinal arithmetic.
1.52. Proposition. (a) associativity of ordinal
addition
(b) associativity
of ordinal multiplication
(c) left distributivity
of ordinal multiplication with respect to ordinal addition
(d) monotonicity
of ordinal addition (multiplication) with respect to the right summant/factor
(e) If \alpha
\le \xi < \alpha + \beta then there is \rho < \beta such that \xi =
\alpha + \rho
(f) \alpha\beta
+ \gamma = \alpha\beta · \alpha\gamma
(g) \alpha(\beta
· \gamma) = (\alpha\beta)\gamma
1.53. Proposition. Division algorithm.