A god is a cosmic designer-creator. Atheism says the number of gods is 0. But it is hard to defeat the minimal thesis that some possible universe is actualized by some possible god. Monotheists say the number of gods is 1. Yet no degree of perfection can be coherently assigned to any unique god. Lewis says the number of gods is at least the second beth number. Yet polytheists cannot defend an arbitrary plural number of gods. An alternative is that, (...) for every ordinal, there is a god whose perfection is proportional to it. The n -th god actualizes the best universe(s) in the n -th level of an axiological hierarchy of possible universes. Despite its unorthodoxy, ordinal polytheism has many metaphysically attractive features and merits more serious study. (shrink)

Ordinal polytheism is motivated by the cosmological and design arguments. It is also motivated by Leibnizian–Lewisian modal realism. Just as there are many universes, so there are many gods. Gods are necessary concrete grounds of universes. The god-universe relation is one-to-one. Ordinal polytheism argues for a hierarchy of ranks of ever more perfect gods, one rank for every ordinal number. Since there are no maximally perfect gods, ordinal polytheism avoids many of the familiar problems of monotheism. It links theology with (...) counterpart theory, mathematics and computer science. And it entails that the system of universes has an attractive axiological structure. (shrink)

In this paper we isolate a notion that we call “formalism freeness” from Gödel's 1946 Princeton Bicentennial Lecture, which asks for a transfer of the Turing analysis of computability to the cases of definability and provability. We suggest an implementation of Gödel's idea in the case of definability, via versions of the constructible hierarchy based on fragments of second order logic. We also trace the notion of formalism freeness in the very wide context of developments in mathematical logic in the (...) 20th century. (shrink)

Infinite time register machines (ITRMs) are register machines which act on natural numbers and which are allowed to run for arbitrarily many ordinal steps. Successor steps are determined by standard register machine commands. At limit times register contents are defined by appropriate limit operations. In this paper, we examine the ITRMs introduced by the third and fourth author (Koepke and Miller in Logic and Theory of Algorithms LNCS, pp. 306–315, 2008), where a register content at a limit time is set (...) to the lim inf of previous register contents if that limit is finite; otherwise the register is reset to 0. The theory of these machines has several similarities to the infinite time Turing machines (ITTMs) of Hamkins and Lewis. The machines can decide all ${\Pi^1_1}$ sets, yet are strictly weaker than ITTMs. As in the ITTM situation, we introduce a notion of ITRM-clockable ordinals corresponding to the running times of computations. These form a transitive initial segment of the ordinals. Furthermore we prove a Lost Melody theorem: there is a real r such that there is a program P that halts on the empty input for all oracle contents and outputs 1 iff the oracle number is r, but no program can decide for every natural number n whether or not ${n \in r}$ with the empty oracle. In an earlier paper, the third author considered another type of machines where registers were not reset at infinite lim inf’s and he called them infinite time register machines. Because the resetting machines correspond much better to ITTMs we hold that in future the resetting register machines should be called ITRMs. (shrink)

We consider the following problem for various infinite-time machines. If a real is computable relative to a large set of oracles such as a set of full measure or just of positive measure, a comeager set, or a nonmeager Borel set, is it already computable? We show that the answer is independent of ZFC for ordinal Turing machines with and without ordinal parameters and give a positive answer for most other machines. For instance, we consider infinite-time Turing machines, unresetting and (...) resetting infinite-time register machines, and α-Turing machines for countable admissible ordinals α. (shrink)

In this note, we consider constraints on the physical possibility of transfinite Turing machines that arise from how one models the continuous structure of space and time in one's best physical theories. We conclude by suggesting a version of Church's thesis appropriate as an upper bound for physical computation given how space and time are modeled on our current physical theories.