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The consistency strength of long projective determinacy

(2020) JOURNAL OF SYMBOLIC LOGIC. 85(1). p.338-366
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Abstract
We determine the consistency strength of determinacy for projective games of length omega(2). Our main theorem is that Pi(1)(n+1)-determinacy for games of length omega(2) implies the existence of a model of set theory with omega + n Woodin cardinals. In a first step, we show that this hypothesis implies that there is a countable set of reals A such that M-n(A), the canonical inner model for n Woodin cardinals constructed over A, satisfies A = R and the Axiom of Determinacy. Then we argue how to obtain a model with omega + n Woodin cardinal from this. We also show how the proof can be adapted to investigate the consistency strength of determinacy for games of length omega(2) with payoff in (sic)(R)Pi(1)(1) or with sigma-projective payoff.
Keywords
Philosophy, Logic, infinite game, determinacy, inner model theory, large cardinal, long game, mouse

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MLA
Aguilera, J. P., and Sandra Müller. “The Consistency Strength of Long Projective Determinacy.” JOURNAL OF SYMBOLIC LOGIC, vol. 85, no. 1, 2020, pp. 338–66, doi:10.1017/jsl.2019.78.
APA
Aguilera, J. P., & Müller, S. (2020). The consistency strength of long projective determinacy. JOURNAL OF SYMBOLIC LOGIC, 85(1), 338–366. https://doi.org/10.1017/jsl.2019.78
Chicago author-date
Aguilera, J. P., and Sandra Müller. 2020. “The Consistency Strength of Long Projective Determinacy.” JOURNAL OF SYMBOLIC LOGIC 85 (1): 338–66. https://doi.org/10.1017/jsl.2019.78.
Chicago author-date (all authors)
Aguilera, J. P., and Sandra Müller. 2020. “The Consistency Strength of Long Projective Determinacy.” JOURNAL OF SYMBOLIC LOGIC 85 (1): 338–366. doi:10.1017/jsl.2019.78.
Vancouver
1.
Aguilera JP, Müller S. The consistency strength of long projective determinacy. JOURNAL OF SYMBOLIC LOGIC. 2020;85(1):338–66.
IEEE
[1]
J. P. Aguilera and S. Müller, “The consistency strength of long projective determinacy,” JOURNAL OF SYMBOLIC LOGIC, vol. 85, no. 1, pp. 338–366, 2020.
@article{8669150,
  abstract     = {We determine the consistency strength of determinacy for projective games of length omega(2). Our main theorem is that Pi(1)(n+1)-determinacy for games of length omega(2) implies the existence of a model of set theory with omega + n Woodin cardinals. In a first step, we show that this hypothesis implies that there is a countable set of reals A such that M-n(A), the canonical inner model for n Woodin cardinals constructed over A, satisfies A = R and the Axiom of Determinacy. Then we argue how to obtain a model with omega + n Woodin cardinal from this.

We also show how the proof can be adapted to investigate the consistency strength of determinacy for games of length omega(2) with payoff in (sic)(R)Pi(1)(1) or with sigma-projective payoff.},
  articleno    = {PII S0022481219000781},
  author       = {Aguilera, J. P. and Müller, Sandra},
  issn         = {0022-4812},
  journal      = {JOURNAL OF SYMBOLIC LOGIC},
  keywords     = {Philosophy,Logic,infinite game,determinacy,inner model theory,large cardinal,long game,mouse},
  language     = {eng},
  number       = {1},
  pages        = {PII S0022481219000781:338--PII S0022481219000781:366},
  title        = {The consistency strength of long projective determinacy},
  url          = {http://dx.doi.org/10.1017/jsl.2019.78},
  volume       = {85},
  year         = {2020},
}

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