## Notre Dame Journal of Formal Logic

### Infinite Computations with Random Oracles

#### Abstract

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 $\mathrm{ZFC}$ 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 $\alpha$-Turing machines for countable admissible ordinals $\alpha$.

#### Article information

Source
Notre Dame J. Formal Logic, Volume 58, Number 2 (2017), 249-270.

Dates
Accepted: 21 August 2014
First available in Project Euclid: 21 February 2017

https://projecteuclid.org/euclid.ndjfl/1487646410

Digital Object Identifier
doi:10.1215/00294527-3832619

Mathematical Reviews number (MathSciNet)
MR3634980

Zentralblatt MATH identifier
06751302

#### Citation

Carl, Merlin; Schlicht, Philipp. Infinite Computations with Random Oracles. Notre Dame J. Formal Logic 58 (2017), no. 2, 249--270. doi:10.1215/00294527-3832619. https://projecteuclid.org/euclid.ndjfl/1487646410

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