TY - JOUR
T1 - Recognizable sets and Woodin cardinals
T2 - computation beyond the constructible universe
AU - Carl, Merlin
AU - Schlicht, Philipp
AU - Welch, Philip D
PY - 2018/4
Y1 - 2018/4
N2 - We call a subset of an ordinal λ recognizable if it is the unique subset x of λ for which some Turing machine with ordinal time and tape, which halts for all subsets of λ as input, halts with the final state 0. Equivalently, such a set is the unique subset x which satisfies a given Σ1 formula in L[x]. We prove several results about sets of ordinals recognizable from ordinal parameters by ordinal time Turing machines. Notably we show the following results from large cardinals.(1) Computable sets are elements of L, while recognizable objects with infinite time computations appear up to the level of Woodin cardinals.(2) A subset of a countable ordinal λ is in the recognizable closure for subsets of λ if and only if it is an element of M∞, where M∞ denotes the inner model obtained by iterating the least measure of M1 through the ordinals, and where the recognizable closure for subsets of λ is defined by closing under relative recognizability for subsets of λ.
AB - We call a subset of an ordinal λ recognizable if it is the unique subset x of λ for which some Turing machine with ordinal time and tape, which halts for all subsets of λ as input, halts with the final state 0. Equivalently, such a set is the unique subset x which satisfies a given Σ1 formula in L[x]. We prove several results about sets of ordinals recognizable from ordinal parameters by ordinal time Turing machines. Notably we show the following results from large cardinals.(1) Computable sets are elements of L, while recognizable objects with infinite time computations appear up to the level of Woodin cardinals.(2) A subset of a countable ordinal λ is in the recognizable closure for subsets of λ if and only if it is an element of M∞, where M∞ denotes the inner model obtained by iterating the least measure of M1 through the ordinals, and where the recognizable closure for subsets of λ is defined by closing under relative recognizability for subsets of λ.
KW - Infinite time
KW - Turing machines
KW - Algorithmic randomness
KW - Effective descriptive set theory
KW - Woodin cardinals
KW - Inner models
UR - https://arxiv.org/abs/1512.06101
U2 - 10.1016/j.apal.2017.12.007
DO - 10.1016/j.apal.2017.12.007
M3 - Article
VL - 169
SP - 312
EP - 332
JO - Annals of Pure and Applied Logic
JF - Annals of Pure and Applied Logic
SN - 0168-0072
IS - 4
ER -