We have seen that ordinals are either zero, a successor, or a limit, and are based on which is well-founded and has induction and recursion principles. Therefore there are induction and recursion principles on ordinals, which are essentially just special cases of -induction and recursion. This web page states these principles: the proofs are the same as for -induction and recursion. Examples will be given later.
As elsewhere, etc. and range over ordinals with being reserved for limit ordinals.
Suppose is a first order formula and
Then for all ordinals we have .
Remark: in the above, and everywhere in these notes, is thought of as a limit ordinal. However the third case includes the first for and the second for so strictly speaking the first two cases are redundant.
Suppose are class functions defined on ordinals alpha and is defined by
Then there is a class function defined as above, uniform in any parameters in .
There are similar induction and recursion theorems on any well-founded relation. A well-order is a particular kind of well-founded relation closely related to ordinals.
Note: some of the following will be moved to somewhere more appropriate in the future.
A binary relation, , is well-founded if . A set is well-founded if the relation is well-founded. The relation is local if .
A set is called a well-founded set if the set membership relation is well-founded on the transitive closure of .
A well-order on a set is a linear order such that for all nonempty there is a -least element of .
If is a well-ordered set there is a unique ordinal such that is order-isomorphic to .
Consider the class of set-functions
Initial part of and ranges over ordinals.
is a class i.e. defined by a first order formula. (We shall show eventually it is a set.)
It is nonempty because it contains the empty cunction .
Given in , if the domain of is not the whole of we can take to be -least and define and for hence extend . Furthermore we can prove that for any either or (since both are initial parts of ) and that agree on their common doman. (Else take to be least such that disagree at .) It follows that the union of a set of fucntions in is a function in .
Now for each let be the unique element of with domain proving (by considering the least such such that doesn't exis) that such always exists. Then is a set by replacement and is a set function mapping to some ordinal, as may be checked.