Eléments de logique Π¹_{𝑛}

Girard, Jean-Yves; Ressayre, Jean-Pierre

doi:10.1090/pspum/042/791069

Cited by 16 publications

(17 citation statements)

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“…Hence for any H ~ PT '~-2, g[H] is a well-order. So G~ [H] [3]. In fact we have defined n-functors and ptykes to be what in [3] are shown to be the codings of these functors and ptykes.…”

Section: Lemma (I) Any Substructure Of An N-functor (Or N-ptyx) Is Amentioning

confidence: 99%

“…So from (1), (4), (2), (3) /3,c~1,...,an,Pl,"" 9 #~ If i = m then Q = w u~, and if p(29) ~ x i < (<=)yj then for some k _-> j, #i < (=<)ak, and 6j => co ak, so Q < (=<)6j.…”

Section: (< Co) ~-~Lo(x) Ro(fx)mentioning

confidence: 99%

“…The goal of this paper is to formalise the notion of "ptykes" (introduced in [3]) in second-order arithmetic, and to show how they can be analysed using proof-theoretic techniques. We derive bounds for the 1-ptykes (i.e.…”

Section: Introductionmentioning

confidence: 99%

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A proof-theoretical analysis of ptykes

Catlow

1994

Arch Math Logic

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Summary. The notion of a "ptyx" is formalised in second-order arithmetic, and, using proof-theoretic techniques based on sequent-calculus, bounds are obtained for the ptykes of type 1 and 2 which can be proved to be ptykes using arithmetic comprehension. IntroductionThe goal of this paper is to formalise the notion of "ptykes" (introduced in [3]) in second-order arithmetic, and to show how they can be analysed using proof-theoretic techniques. We derive bounds for the 1-ptykes (i.e. dilators) and 2-ptykes provable in the theory ACA o of arithmetic comprehension (together with, respectively, the true //11 and//21 sentences). This extends the traditional proof-theoretic objective of finding a bound on the provable well-orderings of a theory (the well-orderings are the ptykes of the lowest level, the "0-ptykes"). Statements of "being an n-ptyx" are canonical 1 H~'+I sentences in the same way that well-ordering statements are (as proved by Kleene) canonical//~ sentences -see Sect. 3.The same ideas used here can be applied to the finding of bounds on the provable dilators and 2-ptykes of other theories of second-order arithmetic. For example, drawing on the notation and methods of [1] (where the ordinals of certain theories are established) the author has found bounds on the dilators and 2-ptykes provable in H~CA (H~ comprehension). ConventionsWe shall often use letters with bars to represent tuples, for example g may denote al, ..., a~, in which case l(~) denotes k (the length of the tuple). We denote by O~b the concatenation of the tuples O~ and b.Mathematics Subject Classification (1991): 03FI0, 0JF15

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“…Hence for any H ~ PT '~-2, g[H] is a well-order. So G~ [H] [3]. In fact we have defined n-functors and ptykes to be what in [3] are shown to be the codings of these functors and ptykes.…”

Section: Lemma (I) Any Substructure Of An N-functor (Or N-ptyx) Is Amentioning

confidence: 99%

“…So from (1), (4), (2), (3) /3,c~1,...,an,Pl,"" 9 #~ If i = m then Q = w u~, and if p(29) ~ x i < (<=)yj then for some k _-> j, #i < (=<)ak, and 6j => co ak, so Q < (=<)6j.…”

Section: (< Co) ~-~Lo(x) Ro(fx)mentioning

confidence: 99%

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A proof-theoretical analysis of ptykes

Catlow

1994

Arch Math Logic

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“…In [2], bounds were found for those ptykes (see [3]) of level 1 and 2 which can be proved to be ptykes in the second-order theory of arithmetic ACAo. A question which was frequently asked in relation to that paper was, 'are there corresponding results in first-order arithmetic?'…”

Section: Introductionmentioning

confidence: 98%

Functoroids and ptykoids

Catlow¹

1995

Arch Math Logic

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“…The two papers [8] and [9] by Pappinghaus give a useful extra background, in particular concerning the applicability of the theory of dilators and ptykes. In [4] ptykes are viewed as logical structures. This formalism is not followed here.…”

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confidence: 99%

Embeddability of ptykes

Girard¹,

Normann

1992

J. symb. log.

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The notion of a ptyx (plural, ptykes) is obtained by generalising the ordinals On and the dilators Dil introduced by Girard [1] into a typed hierarchy. The notion is due to Girard, and a detailed treatment will appear in Girard [2].In this Introduction we will give a summary of the theory of ptykes. It will be an advantage to be familiar with the theory of dilators as introduced in Girard [1] or as presented in Girard and Normann [3].The class On of ordinals is organised into a category ON by using strictly increasing maps f: x → y as morphisms. A dilator will be a functor F: ON → ON commuting with pullbacks and direct limits. Associated with each dilator F there is a denotation system DF, obtained as follows: For each x ∈ On and y ∈ F(x), y can be given a unique denotation (c; x0,…, xn−1;x)F, where c∈F(n) (n = {0,…, n − 1}), y = F(ϕ)(c) (where ϕ is defined by ϕ(i) = xi for i < n), and for no m < n and ψ: m → n do we have c ∈ im(ψ).We let the trace Tr(F) be the set of pairs (c, n) occurring in a denotation.

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Eléments de logique Π¹_{𝑛}

Cited by 16 publications

References 0 publications

A proof-theoretical analysis of ptykes

A proof-theoretical analysis of ptykes

Functoroids and ptykoids

Embeddability of ptykes

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