Constructing Ordinals

Algebraic Ordinals

An algebraic tree T is one determined by a finite system of fixed point equations. The frontier Fr(T) of an algebraic tree T is linearly ordered by the lexicographic order < ℓ. If (Fr(T) < ℓ) is well-ordered, its order type is an algebraic ordinal. We prove that the algebraic ordinals are exactly the ordinals less than ω ω ω .

Large Ordinals

Let j be an elementary embedding of V into V that is not the identity, and let be the critical point of j. Let A be the closure of fjg under the operation a(b) of application, and let be the closure of f g under the operation minf : a( ) b( )g. We give a complete description of the set under an assumption (Threshold Hypothesis) on cyclic left distributive algebras.

Countable Ordinals

This development defines a well-ordered type of countable ordinals. It includes notions of continuous and normal functions, recursively defined functions over ordinals, least fixed-points, and derivatives. Much of ordinal arithmetic is formalized, including exponentials and logarithms. The development concludes with formalizations of Cantor Normal Form and Veblen hierarchies over normal functions.

Automatic Ordinals

We prove that the injectively ω-tree-automatic ordinals are the ordinals smaller than ω ω . Then we show that the injectively ωautomatic ordinals, where n ≥ 1 is an integer, are the ordinals smaller than ω n . This strengthens a recent result of Schlicht and Stephan who considered in [SS11] the subclasses of finite word ω-automatic ordinals. As a by-product we obtain that the hierarchy of injec...

An Introduction to Ordinals