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.

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*.

A hierarchy for the 1-section of any type two object

1974 ◽  
Vol 39 (1) ◽  
pp. 88-94 ◽  
Author(s):  
S. S. Wainer
Keyword(s):  
Affirmative Answer ◽  
The Other ◽  
Uniqueness Property ◽  
Hyperarithmetical Hierarchy ◽  
Other Hand ◽  
Primitive Recursion ◽  
Ordinal Notations ◽  
Definition Of ◽  
Image Position ◽  
Primitive Recursive

Shoenfield [6] constructed a hierarchy for any type two object F in which 2E is recursive by generalizing the hyperarithmetical hierarchy, using a jump operation jF defined by(A similar hierarchy was constructed independently by Hinman [3].) In order to construct a hierarchy for an arbitrary type two F we must first associate with F an operator which, on one hand will always be recursive in F, and on the other hand will generate all the functions recursive in F when iterated over a simultaneously generated set of ordinal notations OF. Clearly the use of 2E in the above definition of jF(f) can be avoided if, instead of diagonalizing F over all functions recursive in f, we only diagonalize F over, say, the functions primitive recursive in f. If furthermore we code in a function which enumerates all functions primitive recursive in f then the resulting operation will certainly generate a primitive recursively expanding hierarchy of functions recursive in F. The problem that remains is whether this hierarchy will exhaust the 1-section of F. But this reduces to an effectiyized version of the following problem: If g is recursive in some level of the hierarchy, is g primitive recursive in some higher level ? An affirmative answer is suggested by the completeness results of Feferman [2], and our main theorem below will be proved by combining his ideas with those of Shoenfield [6]. The result is a hierarchy which applies to all type two objects, and which replaces the notion of recursion in F by the simpler notion of primitive recursion in certain functions generated by F. Unfortunately, in contrast with the Shoenfield hierarchy, the hierarchy developed here cannot always be expected to have the uniqueness property (even w.r.t. ≤T), and for this reason the proof of our main theorem is rather more complicated than the corresponding proof in [6].

Finitism, imperative programs and primitive recursion

2021 ◽  
Vol 31 (1) ◽  
pp. 179-192
Author(s):  
Daniel Leivant
Keyword(s):  
Recursive Function ◽  
Primitive Recursive Function ◽  
Generic Framework ◽  
Primitive Recursive Arithmetic ◽  
Primitive Recursion ◽  
Explicit Reference ◽  
Primitive Recursive

Abstract Following the Crisis of Foundations Hilbert proposed to consider a finitistic form of arithmetic as mathematics’ safe core. This approach to finitism has often admitted primitive recursive function definitions as obviously finitistic, but some have advocated the inclusion of additional variants of recurrence, while others argued that, to the contrary, primitive recursion exceeds finitism. In a landmark essay, William Tait contested the finitistic nature of these extensions, due to their impredicativity, and advocated identifying finitism with primitive recursive arithmetic, a stance often referred to as Tait’s Thesis. However, a problem with Tait’s argument is that the recurrence schema has itself impredicative and non-finitistic facets, starting with an explicit reference to the functions being defined, which are after all infinite objects. It is therefore desirable to buttress Tait’s Thesis on grounds that avoid altogether any trace of concrete infinities or impredicativity. We propose here to do just that, building on the generic framework of [ 13]. We provide further evidence for Tait’s Thesis by outlining a proof of a purely finitistic version of Parsons’ theorem, whose intuitive gist is that finitistic reasoning is equivalent to finitistic computing.

On the comparison of waiting times in tandem queues

1981 ◽  
Vol 18 (03) ◽  
pp. 707-714 ◽  
Author(s):  
Shun-Chen Niu
Keyword(s):  
Waiting Time ◽  
Queueing System ◽  
Waiting Times ◽  
Distribution Functions ◽  
Partial Ordering ◽  
Tandem Queues ◽  
Order Relations ◽  
Service Distribution ◽  
In Series ◽  
Definition Of

Using a definition of partial ordering of distribution functions, it is proven that for a tandem queueing system with many stations in series, where each station can have either one server with an arbitrary service distribution or a number of constant servers in parallel, the expected total waiting time in system of every customer decreases as the interarrival and service distributions becomes smaller with respect to that ordering. Some stronger conclusions are also given under stronger order relations. Using these results, bounds for the expected total waiting time in system are then readily obtained for wide classes of tandem queues.

On finite lattices of degrees of constructibility of reals

1976 ◽  
Vol 41 (2) ◽  
pp. 313-322 ◽  
Author(s):  
Zofia Adamowicz
Keyword(s):  
Standard Model ◽  
Finite Lattice ◽  
Partial Ordering ◽  
Turing Degrees ◽  
Upper Semilattice ◽  
Finite Lattices ◽  
Definition Of ◽  
The Universe

Theorem. Assume that there exists a standard model of ZFC + V = L. Then there is a model of ZFC in which the partial ordering of the degrees of constructibility of reals is isomorphic with a given finite lattice.The proof of the theorem uses forcing. The definition of the forcing conditions and the proofs of some of the lemmas are connected with Lerman's paper on initial segments of the upper semilattice of the Turing degrees [2]. As an auxiliary notion we shall introduce the notion of a sequential representation of a lattice, which slightly differs from Lerman's representation.Let K be a given finite lattice. Assume that the universe of K is an integer l. Let ≤K be the ordering in K. A sequential representation of K is a sequence Ui ⊆ Ui+1 of finite subsets of ωi such that the following holds:(1) For any s, s′ Є Ui, i Є ω, k, m Є l, k ≤Km & s(m) = s′(m) → s(k) = s′(k).(2) For any s Є Ui, i Є ω, s(0) = 0 where 0 is the least element of K.(3) For any s, s′ Є i Є ω, k,j Є l, if k y Kj = m and s(k) = s′(k) & s(j) = s′(j) → s(m) = s′(m), where vK denotes the join in K.

The complexity of ascendant sequences in locally nilpotent groups

2014 ◽  
Vol 24 (02) ◽  
pp. 189-205 ◽  
Author(s):  
Chris J. Conidis ◽  
Richard A. Shore
Keyword(s):  
Nilpotent Groups ◽  
Low Complexity ◽  
Order Type ◽  
Usual Definition ◽  
Locally Nilpotent Groups ◽  
Locally Nilpotent ◽  
Normal Form Theorem ◽  
Algebraic Normal Form ◽  
Definition Of ◽  
Transfinite Recursion

We analyze the complexity of ascendant sequences in locally nilpotent groups, showing that if G is a computable locally nilpotent group and x0, x1, …, xN ∈ G, N ∈ ℕ, then one can always find a uniformly computably enumerable (i.e. uniformly [Formula: see text]) ascendant sequence of order type ω + 1 of subgroups in G beginning with 〈x0, x1, …, xN〉G, the subgroup generated by x0, x1, …, xN in G. This complexity is surprisingly low in light of the fact that the usual definition of ascendant sequence involves arbitrarily large ordinals that index sequences of subgroups defined via a transfinite recursion in which each step is incomputable. We produce this surprisingly low complexity sequence via the effective algebraic commutator collection process of P. Hall, and a related purely algebraic Normal Form Theorem of M. Hall for nilpotent groups.

On finite lattices of degrees of constructibility

1977 ◽  
Vol 42 (3) ◽  
pp. 349-371 ◽  
Author(s):  
Zofia Adamowicz
Keyword(s):  
Finite Lattice ◽  
Partial Ordering ◽  
Generic Model ◽  
Crucial Point ◽  
Generic Models ◽  
Ordinal Numbers ◽  
Finite Lattices ◽  
Definition Of ◽  
Two Stages ◽  
The Given

We shall prove the following theorem:Theorem. For any finite lattice there is a model of ZF in which the partial ordering of the degrees of constructibility is isomorphic with the given lattice.Let M be a standard countable model of ZF satisfying V = L. Let K be the given finite lattice. We shall extend M by forcing.The paper is divided into two parts. The first part concerns the definition of the set of forcing conditions and some properties of this set expressible without the use of generic filters.We define first a representation of a lattice and then the set of conditions. In Lemmas 1, 2 we show that there are some canonical isomorphisms between some conditions and that a single condition has some canonical automorphisms.In Lemma 3 and Definition 7 we show some methods of defining conditions. We shall use those methods in the second part to define certain conditions with special properties.Lemma 4 gives a connection between the sets P and Pk (see Definitions 4 and 5). It is next employed in the second part in Lemma 10 in an essential way.Indeed, Lemma 10 is necessary for Lemma 13, which is the crucial point of the whole construction. Lemma 5 is also employed in Lemma 13 (exactly in its Corollary).The second part of the paper is devoted to the examination of the structure of degrees of constructibility in a generic model. First, we show that degrees of some “sections” of a generic real (Definition 9) form a lattice isomorphic with K. Secondly, we show that there are no other degrees in the generic model; this is the most difficult property to obtain by forcing. We prove, in two stages, that it holds in our generic models. We first show, by using special properties of the forcing conditions, that sets of ordinal numbers have no other degrees. Then we show that the degrees of sets of ordinals already determine the degrees of other sets.

scholarly journals Primitive recursive algebraic theories and program schemes

1977 ◽  
Vol 17 (2) ◽  
pp. 207-233 ◽  
Author(s):  
W. Kühnel ◽  
J. Meseguer ◽  
M. Pfender ◽  
I. Sols
Keyword(s):  
Programming Language ◽  
Generation Process ◽  
Recursive Functions ◽  
Algebraic Theories ◽  
Primitive Recursion ◽  
Recursive Theory ◽  
Carry Over ◽  
Primitive Recursive

We introduce primitive recursion as a generation process for arrows of algebraic theories in the sense of Lawvere and carry over important results on algebraic theories and functorial semantics to the enriched setting of “primitive recursive algebra”: existence of free primitive recursive theories and of theories presented by operations and equations on primitive recursive functions; existence of free models presented by generators and equations. Finally semantical correctness of translations is reduced to correctness for the basic operations. There is a connection to the theory of program schemes: program schemes involving primitive recursion correspond to arrows of a primitive recursive theory freely generated over a graph of basic operations. This theory T can be viewed as a programming language with “arithmetics” given by the basic operations and with DO-loops. A machine loaded with a compiler for T can be interpreted as a T-model in Lawvere's sense, preserving primitive recursion.

VON NEUMANN’S CONSISTENCY PROOF

2016 ◽  
Vol 9 (3) ◽  
pp. 429-455
Author(s):  
LUCA BELLOTTI
Keyword(s):  
Order Theory ◽  
Successor Function ◽  
Consistency Proof ◽  
Von Neumann ◽  
First Order Theory ◽  
Primitive Recursive Arithmetic ◽  
History Of ◽  
Elementary Calculus ◽  
Primitive Recursive ◽  
Consistency Proofs

AbstractWe consider the consistency proof for a weak fragment of arithmetic published by von Neumann in 1927. This proof is rather neglected in the literature on the history of consistency proofs in the Hilbert school. We explain von Neumann’s proof and argue that it fills a gap between Hilbert’s consistency proofs for the so-called elementary calculus of free variables with a successor and a predecessor function and Ackermann’s consistency proof for second-order primitive recursive arithmetic. In particular, von Neumann’s proof is the first rigorous proof of the consistency of an axiomatization of the first-order theory of a successor function.

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