What is the ordinal of one?
Admissible ordinal Admissible ordinal In set theory, an ordinal number α is an admissible ordinal if L is an admissible set (that is, a transitive model of Kripke–Platek set theory); in other words, α is admissible when α is a limit ordinal and L⊧Σ-collection. The first two admissible ordinals are ω and formula_1 (the least non-recursive ordinal, also called the Church–Kleene ordinal). Any regular uncountable cardinal is an admissible ordinal. By a theorem of Sacks, the countable admissible ordinals are exactly those constructed in a manner similar to the Church-Kleene ordinal, but for Turing machines with oracles. One sometimes writes formula_2 for
Ordinal number operation is idempotent. As mentioned above (see Cantor normal form) the ordinal ε, which is the smallest satisfying the equation formula_20, so it is the limit of the sequence 0, 1, formula_65, formula_73, formula_74, etc. Many ordinals can be defined in such a manner as fixed points of certain ordinal functions (the formula_75-th ordinal such that formula_20 is called formula_77, then one could go on trying to find the formula_75-th ordinal such that formula_79, "and so on", but all the subtlety lies in the "and so on"). One could try to do this systematically, but no matter what system is