Ordinal Number - Some “large” Countable Ordinals

Some “large” Countable Ordinals

We have already mentioned (see Cantor normal form) the ordinal ε₀, which is the smallest satisfying the equation, so it is the limit of the sequence 0, 1, etc. Many ordinals can be defined in such a manner as fixed points of certain ordinal functions (the -th ordinal such that is called, then we could go on trying to find the -th ordinal such that, “and so on”, but all the subtlety lies in the “and so on”). We can try to do this systematically, but no matter what system is used to define and construct ordinals, there is always an ordinal that lies just above all the ordinals constructed by the system. Perhaps the most important ordinal that limits a system of construction in this manner is the Church–Kleene ordinal, (despite the in the name, this ordinal is countable), which is the smallest ordinal that cannot in any way be represented by a computable function (this can be made rigorous, of course). Considerably large ordinals can be defined below, however, which measure the “proof-theoretic strength” of certain formal systems (for example, measures the strength of Peano arithmetic). Large ordinals can also be defined above the Church-Kleene ordinal, which are of interest in various parts of logic.