## Ultimate truth vis à vis stable truth (2003)

Venue: | Journal of Philosophical Logic |

Citations: | 8 - 6 self |

### BibTeX

@ARTICLE{Welch03ultimatetruth,

author = {P. D. Welch},

title = {Ultimate truth vis à vis stable truth},

journal = {Journal of Philosophical Logic},

year = {2003},

volume = {1}

}

### OpenURL

### Abstract

Abstract. We show that the set of ultimately true sentences in Hartry Field’s Revenge-immune solution to the semantic paradoxes is recursively isomorphic to the set of stably true sentences obtained in Hans Herzberger’s revision sequence starting from the null hypothesis. We further remark that this shows that a substantial subsystem of second order number theory is needed to establish the semantic values of sentences over the ground model of the standard natural numbers: ¢¡-Comprehension Axiom scheme) is insufficient. £-¤¦¥¨ § (second order number theory with a ©��

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Citation Context ...rmed in ����� ) can be effected in a sufficiently strong fragment of second .s4 P.D.Welch order number theory, but not in ��� (Kripke-Platek set theory) nor ��� � -����� . (For the latter theory see (=-=Simpson, 1999-=-).) � Similar results will hold for countable ground models � that are acceptable (this time in the sense of (Moschovakis, 1974)), that is, the least � � � � � ����� � (Fieldian) acceptable point for ... |

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Citation Context ...�� (Kripke-Platek set theory) nor ��� � -����� . (For the latter theory see (Simpson, 1999).) � Similar results will hold for countable ground models � that are acceptable (this time in the sense of (=-=Moschovakis, 1974-=-)), that is, the least � � � � � ����� � (Fieldian) acceptable point for will be the least -extendible ordinal relative to , and hence will be less than the least cardinal greater than ��� that of We ... |

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Citation Context ...ning to Field’s construction under discussion, we might imagine after following it through, that we no longer have the feeling, as Kripke seemingly does at the paragraph towards the end of his paper (=-=Kripke, 1975-=-), that the ‘ghost of the Tarski hierarchy is still with us.’ Kripke might have felt that Liar sentences are not true in the object language, ‘... but we are precluded from saying this is the object l... |

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Citation Context ... look at the set of stable sentences, that is, those that from some point on have a set truth-value. (We emphasise that we are not considering categorical truth sets in the style of Gupta and Belnap (=-=Gupta and Belnap, 1993-=-), where a global averaging is obtained by considering all possible starting hypotheses for the Herzberger revision process.) The � ������� � values are continuous at limits (cf. his Continuity Lemma ... |

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Citation Context ...ing from a quasi-inductive definition but (b) that the first repeat point, that is the first acceptable point, � is - this latter ordinal has been characterised as the � � first -extendible ordinal ((=-=Burgess, 1986-=-) Sect.14), that is the first � ordinal so ��� that has a � � proper -elementary end Equivalently it is also the first place where all Infinite Time Turing extension.� machine (ITTM) computations have... |

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Citation Context ...g distribution of truth values would serve equally well). For the notion and properties of Herzberger’s revision sequence the reader may consult that author’s papers already cited, or the account in (=-=McGee, 1991-=-). THEOREM 1. � is recursively isomorphic to the set � This set � has been noted to be equivalent to other sets of integers, defined independently. We shall comment on this further in Section 3, but b... |

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Citation Context ...� ¡ of � � � � � ��� � ¨ � ��� ���¡ ����� ����� � ��� ����� � � ��� � . � � � ���� The proposition is essentially obtained from the relativisations of the theorems of Kleene and Spector, (see, e.g., (=-=Rogers, 1967-=-), Theorems XLI & XLIV) that classify � � � the predicates on integers as � � those -definable over (the latter the class of hypearithmetic reals, together with the fact that � the reals ��� ¡���� of ... |

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