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Partial realizations of Hilbert’s program
 JOURNAL OF SYMBOLIC LOGIC
, 1988
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Higher Order Logic
 In Handbook of Logic in Artificial Intelligence and Logic Programming
, 1994
"... Contents 1 Introduction : : : : : : : : : : : : : : : : : : : : : : : : : : : : 2 2 The expressive power of second order Logic : : : : : : : : : : : 3 2.1 The language of second order logic : : : : : : : : : : : : : 3 2.2 Expressing size : : : : : : : : : : : : : : : : : : : : : : : : 4 2.3 Definin ..."
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Contents 1 Introduction : : : : : : : : : : : : : : : : : : : : : : : : : : : : 2 2 The expressive power of second order Logic : : : : : : : : : : : 3 2.1 The language of second order logic : : : : : : : : : : : : : 3 2.2 Expressing size : : : : : : : : : : : : : : : : : : : : : : : : 4 2.3 Defining data types : : : : : : : : : : : : : : : : : : : : : 6 2.4 Describing processes : : : : : : : : : : : : : : : : : : : : : 8 2.5 Expressing convergence using second order validity : : : : : : : : : : : : : : : : : : : : : : : : : 9 2.6 Truth definitions: the analytical hierarchy : : : : : : : : 10 2.7 Inductive definitions : : : : : : : : : : : : : : : : : : : : : 13 3 Canonical semantics of higher order logic : : : : : : : : : : : : 15 3.1 Tarskian semantics of second order logic : : : : : : : : : 15 3.2 Function and re
Challenges to Predicative Foundations of Arithmetic
 in Between Logic and Intuition Essays in Honor of Charles Parsons
, 1996
"... This paper was written while the first author was a Fellow at the Center for Advanced Study in the Behavioral Sciences (Stanford, CA) whose facilities and support, under grants from the Andrew W. Mellon Foundation and the National Science Foundation, have been greatly appreciated. ..."
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This paper was written while the first author was a Fellow at the Center for Advanced Study in the Behavioral Sciences (Stanford, CA) whose facilities and support, under grants from the Andrew W. Mellon Foundation and the National Science Foundation, have been greatly appreciated.
Finite Trees And The Necessary Use Of Large Cardinals
, 1998
"... this paper, a tree T = (V,) is a partial ordering with a minimum element, where V is finite, and the ancestors of any x V are linearly ordered under . The minimum element of T is called the root of T, and is written r(T). A tree is said to be trivial if and only if it has exactly one vertex, which m ..."
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Cited by 3 (1 self)
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this paper, a tree T = (V,) is a partial ordering with a minimum element, where V is finite, and the ancestors of any x V are linearly ordered under . The minimum element of T is called the root of T, and is written r(T). A tree is said to be trivial if and only if it has exactly one vertex, which must be its root. V = V(T) represents the set of all vertices of the tree T = (V,). In a tree T, if x < y and for no z is x < z < y, then we say that y is a child of x and x is the parent of y. Every vertex has at most one parent. However, vertices may have zero or more children. We write p(x,T) for the parent of x in T. We use Ch(T) = V(T)\{r(T)} for the set of all children of T. We write T 1 T 2 if and only if i) r(T 1 ) = r(T 2 ); ii) for all x Ch(T 1 ), p(x,T 1 ) = p(x,T 2 ). This is a partial ordering on trees. Note that if T 1 T 2
Predicative functionals and an interpretation of c ID<ω
 Ann. Pure Appl. Logic
, 1998
"... In 1958 Gödel published his Dialectica interpretation, which reduces classical arithmetic to a quantifierfree theory T axiomatizing the primitive recursive functionals of finite type. Here we extend Gödel’s T to theories Pn of “predicative ” functionals, which are defined using MartinLöf’s univers ..."
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In 1958 Gödel published his Dialectica interpretation, which reduces classical arithmetic to a quantifierfree theory T axiomatizing the primitive recursive functionals of finite type. Here we extend Gödel’s T to theories Pn of “predicative ” functionals, which are defined using MartinLöf’s universes of transfinite types. We then extend Gödel’s interpretation to the theories of arithmetic inductive definitions ÎDn, so that each ÎDn is interpreted in the corresponding Pn. Since the strengths of the theories ÎDn are cofinal in the ordinal Γ0, as a corollary this analysis provides an ordinalfree characterization of the <Γ0recursive functions.
An Effective Proof that Open Sets are Ramsey
, 1996
"... Solovay has shown that if O is an open subset of P (ω) withcodeS and no infinite set avoids O, then there is an infinite set hyperarithmetic in S that lands in O. We provide a direct proof of this theorem that is easily formalizable in AT R0. 1 ..."
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Solovay has shown that if O is an open subset of P (ω) withcodeS and no infinite set avoids O, then there is an infinite set hyperarithmetic in S that lands in O. We provide a direct proof of this theorem that is easily formalizable in AT R0. 1
Finitization Procedures and Finite Model Property.
"... Investigations into the Relevant and Paraconsistent model theory of rstorder arithmetic have provided interesting new methods and results which have revived the interest in Hilbert's program. The attempt to develop Strict Finitist Mathematics using G. Priest's Collapsing lemma to nitize ..."
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Investigations into the Relevant and Paraconsistent model theory of rstorder arithmetic have provided interesting new methods and results which have revived the interest in Hilbert's program. The attempt to develop Strict Finitist Mathematics using G. Priest's Collapsing lemma to nitize innite models is an example. In the investigation of some systems of Relevant Logics, another nitization procedure is used to solve positively their decision problem and to prove the nite model property for these systems. Some results related to the procedure used in these investigations show that Hilbert's ideal cannot be entirely fullled or that it must be reinterpreted. 1
Published In Predicative Functionals and an Interpretation of ÎD<ω∗
, 1997
"... In 1958 Gödel published his Dialectica interpretation, which reduces classical arithmetic to a quantifierfree theory T axiomatizing the primitive recursive functionals of finite type. Here we extend Gödel’s T to theories Pn of “predicative ” functionals, which are defined using MartinLöf’s uni ..."
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In 1958 Gödel published his Dialectica interpretation, which reduces classical arithmetic to a quantifierfree theory T axiomatizing the primitive recursive functionals of finite type. Here we extend Gödel’s T to theories Pn of “predicative ” functionals, which are defined using MartinLöf’s universes of transfinite types. We then extend Gödel’s interpretation to the theories of arithmetic inductive definitions ÎDn, so that each ÎDn is interpreted in the corresponding Pn. Since the strengths of the theories ÎDn are cofinal in the ordinal Γ0, as a corollary this analysis provides an ordinalfree characterization of the <Γ0recursive functions. 1
Published In An Effective Proof that Open Sets are Ramsey
, 1996
"... Solovay has shown that if O is an open subset of P (ω) with code S and no infinite set avoids O, then there is an infinite set hyperarithmetic in S that lands in O. We provide a direct proof of this theorem that is easily formalizable in ATR0. 1 ..."
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Solovay has shown that if O is an open subset of P (ω) with code S and no infinite set avoids O, then there is an infinite set hyperarithmetic in S that lands in O. We provide a direct proof of this theorem that is easily formalizable in ATR0. 1