Title:

Ordinal time Turing computation

This thesis develops the theory of Ordinal Time Turing Machines (OTTMs) and explores connections between
this theory, inner model theory, αrecursion and Turing computation. We first provide a rigorous
definition of an OTTM. We define how such machines may be taken to operate on sets, we prove that the
class of OTTMs has a universal machine, we prove that the class 0 of OTTM computable sets is equal to L,
we prove an analogue of the condensation lemma and we prove that the Generalized Continuum hypothesis
& ◊ωl hold in L using lemmas concerning OTTMs.
We also define several variants of computer limited to α time. We expose weaknesses in all bar one of the
variants (uniformαcomputation) and then we use this remaining variant to develop a degree theory. Vie
show this theory is isomorphic to the theory of αrecursion, we show that αrecursion is not equivalent to αcomputation
and we give a proof of a form of the SACKS SIMPSON theorem stated for the uniformαcomputer.
We then prove results about halting computations, universal and metaversal programs ('metaversal program'
is defined in this thesis) for the uniformαcomputer. We define a (B,α)computer which is closely related
to inner model theory. We prove an analogue of the Density theorem for the (B,α)computer and find a
metaversal program for the (B,α)computer.
Finally we compare the αcomputers and OTTMs with Turing machines.
The introduction consists entirely of preexisting results and definitions which provide a necessary background
for the rest of the thesis. The proof of SACKSSIMPSON for uniformacomputers is adapted from Benjamin
Seyfferth's proof for nonuniformacomputers. The Density theorem for αrecursion of Sacks is modified and
adapted for uniform(B,α)computers. All other results are entirely my own work unless otherwise stated.
