Ordinal analysis of simple cases of bar recursion

1981 ◽  
Vol 46 (1) ◽  
pp. 17-30 ◽  
Author(s):  
W. A. Howard
Keyword(s):  
Free Variable ◽  
Recursion Operator ◽  
Ordinal Analysis ◽  
Functional Interpretation ◽  
Free Variables ◽  
Ordinal Measure ◽  
Level 3 ◽  
Transfinite Recursion ◽  
Primitive Recursive

The purpose of the following is to give an ordinal analysis of bar recursion of type 0 for the cases in which the bar recursion operator has type level 3 or 4. We show that the associated ordinals are the ε0th epsilon number and the first ε0-critical number, respectively. Also we analyze a restricted bar recursion operator of level 3 which provides the Gödel functional interpretation of an intuitionistic form of König's lemma. The ordinal in this case is ε0.Let + BR0 be Gödel's theory of primitive recursive functional of finite type extended by functors Φ for bar recursion of type zero. To make bar recursion resemble transfinite recursion more closely, and thereby facilitate the ordinal analysis, we extend + BR0 to a system of terms . By Theorem 1.1a term of + BR0 which is computable by the rules of is already computable by the rules of + BR0, so it is sufficient to give an ordinal analysis of the terms of .A term H of is said to be semi-closed if its free variables have level not exceeding 1. If, in addition, H has level not exceeding 2, then an ordinal measure for H is given by a length for one of its computation trees. We call this a computation size for H. Let F(X) be a term whose free variables have level not exceeding 2, and suppose the variable X of type 2 is the only free variable of level 2.

scholarly journals Term extraction and Ramsey's theorem for pairs

2012 ◽  
Vol 77 (3) ◽  
pp. 853-895 ◽  
Author(s):  
Alexander P. Kreuzer ◽  
Ulrich Kohlenbach
Keyword(s):  
Uniform Method ◽  
Second Step ◽  
Functional Interpretation ◽  
Ramsey's Theorem ◽  
Term Extraction ◽  
Ramsey’S Theorem ◽  
Skolem Functions ◽  
Primitive Recursive

AbstractIn this paper we study with proof-theoretic methods the function(al)s provably recursive relative to Ramsey's theorem for pairs and the cohesive principle (COH).Our main result on COH is that the type 2 functional provably recursive fromare primitive recursive. This also provides a uniform method to extract bounds from proofs that use these principles. As a consequence we obtain a new proof of the fact thatis-conservative over PRA.Recent work of the first author showed thatis equivalent to a weak variant of the Bolzano-Weierstraß principle. This makes it possible to use our results to analyze not only combinatorial but also analytical proofs.For Ramsey's theorem for pairs and two colorswe obtain the upper bounded that the type 2 functional provable recursive relative toare inT1. This is the fragment of Gödel's systemTcontaining only type 1 recursion—roughly speaking it consists of functions of Ackermann type. With this we also obtain a uniform method for the extraction ofT1-bounds from proofs that use. Moreover, this yields a new proof of the fact thatis-conservative over.The results are obtained in two steps: in the first step a term including Skolem functions for the above principles is extracted from a given proof. This is done using Gödel's functional interpretation. After this the term is normalized, such that only specific instances of the Skolem functions are used. In the second step this term is interpreted using-comprehension. The comprehension is then eliminated in favor of induction using either elimination of monotone Skolem functions (for COH) or Howard's ordinal analysis of bar recursion (for).

A system of abstract constructive ordinals

1972 ◽  
Vol 37 (2) ◽  
pp. 355-374 ◽  
Author(s):  
W. A. Howard
Keyword(s):  
Finite Type ◽  
Free Variable ◽  
Transfinite Induction ◽  
Functional Interpretation ◽  
Variable Theory ◽  
Inductive Definitions ◽  
Heyting Arithmetic ◽  
Abstract Notion ◽  
Primitive Recursive ◽  
Primitive Recursive Functionals

As Gödel [6] has pointed out, there is a certain interchangeability between the intuitionistic notion of proof and the notion of constructive functional of finite type. He achieves this interchange in the direction from logic to functionals by his functional interpretation of Heyting arithmetic H in a free variable theory T of primitive recursive functionals of finite type. In the present paper we shall extend Gödel's functional interpretation to the case in which H and T are extended by adding an abstract notion of constructive ordinal. In other words, we obtain the Gödel functional interpretation of an intuitionistic theory U of numbers (i.e., nonnegative integers) and constructive ordinals in a free variable theory V of finite type over both numbers and constructive ordinals. This allows us to obtain an analysis of noniterated positive inductive definitions [8].The notion of constructive ordinal to be treated is as follows. There is given a function J which embeds the nonnegative integers in the constructive ordinals. A constructive ordinal of the form Jn is said to be minimal. There is also given a function δ which associates to each constructive ordinal Z and number n a constructive ordinal δZn which we denote by Zn. When Z is nonminimal, each Zn is called an immediate predecessor of Z. The basic principle for forming constructive ordinals says: for every function f from numbers n to constructive ordinals, there exists a constructive ordinal Z such that Zn = fn for all n. The principle of transfinite induction with respect to constructive ordinals says: if a property Q(Z) of constructive ordinals Z holds for minimal Z, and if ∀nQ(Zn) → Q(Z) holds for all Z, then Q(Z) holds for all Z.

Solution of a problem of Leon Henkin

1955 ◽  
Vol 20 (2) ◽  
pp. 115-118 ◽  
Author(s):  
M. H. Löb
Keyword(s):  
Number Theory ◽  
Recursive Function ◽  
Formal System ◽  
Free Variable ◽  
Formal Proof ◽  
Predicate Calculus ◽  
First Order ◽  
Free Variables ◽  
Order Predicate Calculus

If Σ is any standard formal system adequate for recursive number theory, a formula (having a certain integer q as its Gödel number) can be constructed which expresses the proposition that the formula with Gödel number q is provable in Σ. Is this formula provable or independent in Σ? [2].One approach to this problem is discussed by Kreisel in [4]. However, he still leaves open the question whether the formula (Ex)(x, a), with Gödel-number a, is provable or not. Here (x, y) is the number-theoretic predicate which expresses the proposition that x is the number of a formal proof of the formula with Gödel-number y.In this note we present a solution of the previous problem with respect to the system Zμ [3] pp. 289–294, and, more generally, with respect to any system whose set of theorems is closed under the rules of inference of the first order predicate calculus, and satisfies the subsequent five conditions, and in which the function (k, l) used below is definable.The notation and terminology is in the main that of [3] pp. 306–326, viz. if is a formula of Zμ containing no free variables, whose Gödel number is a, then ({}) stands for (Ex)(x, a) (read: the formula with Gödel number a is provable in Zμ); if is a formula of Zμ containing a free variable, y say, ({}) stands for (Ex)(x, g(y)}, where g(y) is a recursive function such that for an arbitrary numeral the value of g() is the Gödel number of the formula obtained from by substituting for y in throughout. We shall, however, depart trivially from [3] in writing (), where is an arbitrary numeral, for (Ex){x, ).

On the Gödel class with identity

1981 ◽  
Vol 46 (2) ◽  
pp. 354-364 ◽  
Author(s):  
Warren D. Goldfarb
Keyword(s):  
Recursive Function ◽  
Decision Problem ◽  
Decision Procedure ◽  
Finite Model ◽  
Quantification Theory ◽  
Free Variables ◽  
Primitive Recursive

The Gödel Class is the class of prenex formulas of pure quantification theory whose prefixes have the form ∀y1∀y2∃x1 … ∃xn. The Gödel Class with Identity, or GCI, is the corresponding class of formulas of quantification theory extended by inclusion of the identity-sign “ = ”. Although the Gödel Class has long been kndwn to be solvable, the decision problem for the Gödel Class with Identity is open. In this paper we prove that there is no primitive recursive decision procedure for the GCI, or, indeed, for the subclass of the GCI containing just those formulas with prefixes ∀y1∀y2∃x.Throughout this paper we take quantification theory to include, aside from logical signs, infinitely many k-place predicate letters for each k > 0, but no function signs or constants. Moreover, by “prenex formula” we include only those without free variables. A decision procedure for a class of formulas is a recursive function that carries a formula in the class to 0 if the formula is satisfiable and to 1 if not. A class is solvable iff there exists a decision procedure for it. A class is finitely controllable iff every satisfiable formula in the class has a finite model. Since we speak only of effectively specified classes, finite controllability implies solvability (but not conversely).The GCI has a curious history. Gödel showed the Gödel Class (without identity) solvable in 1932 [4] and finitely controllable in 1933 [5].

Functionals defined by transfinite recursion

1965 ◽  
Vol 30 (2) ◽  
pp. 155-174 ◽  
Author(s):  
W. W. Tait
Keyword(s):  
Functional Equations ◽  
Partial Ordering ◽  
Closure Properties ◽  
Primitive Recursive Arithmetic ◽  
Primitive Recursion ◽  
Definition Of ◽  
Second Order Systems ◽  
Transfinite Recursion ◽  
Primitive Recursive ◽  
Elementary Closure

This paper deals mainly with quantifier-free second order systems (i.e., with free variables for numbers and functions, and constants for numbers, functions, and functionals) whose basic rules are those of primitive recursive arithmetic together with definition of functionals by primitive recursion and explicit definition. Precise descriptions are given in §2. The additional rules have the form of definition by transfinite recursion up to some ordinal ξ (where ξ is represented by a primitive recursive (p.r.) ordering). In §3 we discuss some elementary closure properties (under rules of inference and definition) of systems with recursion up to ξ. Let Rξ denote (temporarily) the system with recursion up to ξ. The main results of this paper are of two sorts:Sections 5–7 are concerned with less elementary closure properties of the systems Rξ. Namely, we show that certain classes of functional equations in Rη can be solved in Rη for some explicitly determined η < ε(η) (the least ε-number > ξ). The classes of functional equations considered all have roughly the form of definition by recursion on the partial ordering of unsecured sequences of a given functional F, or on some ordering which is obtained from this by simple ordinal operations. The key lemma (Theorem 1) needed for the reduction of these equations to transfinite recursion is simply a sharpening of the Brouwer-Kleene idea.

Choice principles, the bar rule and autonomously iterated comprehension schemes in analysis

1983 ◽  
Vol 48 (1) ◽  
pp. 63-70 ◽  
Author(s):  
S. Feferman ◽  
G. Jäger
Keyword(s):  
General Result ◽  
Recursive Relation ◽  
Transfinite Induction ◽  
Conservative Extension ◽  
Main Result Theorem ◽  
Functional Interpretation ◽  
Result Theorem ◽  
Primitive Recursive ◽  
Comprehension Axiom

In [10] Friedman showed that (-AC) is a conservative extension of (-CA)<ε0for-sentences wherei= min(n+ 2, 4), i.e.,i= 2, 3, 4 forn= 0, 1, 2 +m. Feferman [5], [7] and Tait [11], [12] reobtained this result forn= 0, 1 and even with (-DC) instead of (-AC). Feferman and Sieg established in [9] the conservativeness of (-DC) over (-CA)<ε0for-sentences (ias above) for alln. In each paper, different methods of proof have been used. In particular, Feferman and Sieg showed how to apply familiar proof-theoretical techniques by passing through languages with Skolem functionals.In this paper we study the same choice principles in the presence of theBar Rule(BR), which permits one to infer the scheme of transfinite induction on a primitive recursive relation ≺ when it has been proved that ≺ is wellfounded. The main result (Theorem 1 below) characterizes (-DC) + (BR) as a conservative extension of a system of the autonomously iterated-comprehension axiom for-sentences (idepending onnas above). Forn= 0 this has been proved by Feferman in the form that (-DC) + (BR) is a conservative extension of (-CA)<Γ0; this was first done in [8] by use of the Gödel functional interpretation for the stronger systemZω+μ+ (QF-AC) + (BR) and then more recently by the simpler methods of [9]. Jäger showed how the latter methods could also be used to obtain the general result of Theorem 1 below.

scholarly journals Functional interpretation and inductive definitions

2009 ◽  
Vol 74 (4) ◽  
pp. 1100-1120 ◽  
Author(s):  
Jeremy Avigad ◽  
Henry Towsner
Keyword(s):  
Finite Type ◽  
Functional Interpretation ◽  
Inductive Definitions ◽  
Transfinite Recursion

AbstractExtending Gödel's Dialectica interpretation, we provide a functional interpretation of classical theories of positive arithmetic inductive definitions, reducing them to theories of finite-type functionals defined using transfinite recursion on well-founded trees.

scholarly journals Intuitionistic Choice and Restricted Classical Logic

2000 ◽  
Vol 7 (12) ◽  
Author(s):  
Ulrich Kohlenbach
Keyword(s):  
Classical Logic ◽  
Classical System ◽  
Axiom Of Choice ◽  
Strong Form ◽  
Functional Interpretation ◽  
Primitive Recursive Arithmetic ◽  
The Axiom Of Choice ◽  
Primitive Recursive ◽  
Intuitionistic Arithmetic

Recently, Coquand and Palmgren considered systems of intuitionistic arithmetic<br />in all finite types together with various forms of the axiom of choice and<br />a numerical omniscience schema (NOS) which implies classical logic for arithmetical<br />formulas. Feferman subsequently observed that the proof theoretic<br />strength of such systems can be determined by functional interpretation based<br />on a non-constructive mu-operator and his well-known results on the strength<br />of this operator from the 70's.<br />In this note we consider a weaker form LNOS (lesser numerical omniscience<br />schema) of NOS which suffices to derive the strong form of binary K¨onig's<br />lemma studied by Coquand/Palmgren and gives rise to a new and mathematically<br />strong semi-classical system which, nevertheless, can proof theoretically<br />be reduced to primitive recursive arithmetic PRA. The proof of this fact relies<br />on functional interpretation and a majorization technique developed in a<br />previous paper.

The substitution method

1965 ◽  
Vol 30 (2) ◽  
pp. 175-192 ◽  
Author(s):  
W. W. Tait
Keyword(s):  
Free Variable ◽  
Order System ◽  
Conservative Extension ◽  
Substitution Method ◽  
Recursive Definition ◽  
First Order ◽  
Variable Elimination ◽  
Primitive Recursion ◽  
Definition Of ◽  
Primitive Recursive

This paper deals with Hilbert's substitution method for eliminating bound variables from first order proofs. With a first order system S framed in the ε-calculus [2] the problem is to associate a system S' without bound variables and an effective procedure for transforming derivations in S into derivations in S′. The transform of a formula A derived in S is to be an “ε-substitution instance” of A, i.e. it is obtained by replacing terms εxB(x) in A by terms of S′. In general the choice of these terms will depend on the particular derivation of A, and not on A alone. Cf. [4]. The present formulation sharpens Hilbert's original statement of the problem, i.e. that the transform of A should be finitistically verifiable, by making explicit the methods of verification used, namely those formalized in S′; on the other hand, it generalizes Hilbert's formulation since S′ need not be restricted to finitist systems.The bound variable elimination procedure can always be taken to be primitive recursive in (the Gödel number of) the derivation of A. Constructions which transcend primitive recursion can simply be built into S′.In this paper we show that if S′ is taken to be a second order system with constants for functionals, then the existence of suitable ε-substitution instances can be expressed by the solvability of certain functional equations in S′. We deal with two cases here. If S is number theory without induction, i.e. essentially predicate calculus with identity, then we can solve the equations in question by taking for S′ the free variable part S* of S with an added rule of definition of functionals by cases (recursive definition on finite ordinals), which is a conservative extension of S*.

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