Primitive Recursive Arithmetic and Its Role in the Foundations of Arithmetic: Historical and Philosophical Reflections

2012 ◽  
pp. 161-180 ◽  
Author(s):  
William W. Tait
Keyword(s):  
Primitive Recursive Arithmetic ◽  
Primitive Recursive ◽  
Foundations Of Arithmetic

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.

scholarly journals Things that can and things that can’t be done in PRA

1998 ◽  
Vol 5 (18) ◽  
Author(s):  
Ulrich Kohlenbach
Keyword(s):  
Mathematical Reasoning ◽  
The Other ◽  
Function Parameter ◽  
Recursive Functions ◽  
Other Hand ◽  
Elementary Analysis ◽  
Primitive Recursive Arithmetic ◽  
Primitive Recursive ◽  
Different Levels ◽  
Bounded Sequences

It is well-known by now that large parts of (non-constructive) mathematical reasoning can be carried out in systems T which are conservative over primitive recursive arithmetic PRA (and even much weaker systems). On the other hand there are principles S of elementary analysis (like the Bolzano-Weierstrass principle, the existence of a limit superior for bounded sequences etc.) which are known to be equivalent to arithmetical<br />comprehension (relative to T ) and therefore go far beyond the strength of PRA (when added to T ). <br />In this paper we determine precisely the arithmetical and computational strength (in terms of optimal conservation results and subrecursive characterizations of provably recursive functions) of weaker function parameter-free schematic versions S− of S, thereby<br />exhibiting different levels of strength between these principles as well as a sharp borderline between fragments of analysis which are still conservative over PRA and extensions which just go beyond the strength of PRA.

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.

scholarly journals An application of graphical enumeration to PA

2003 ◽  
Vol 68 (1) ◽  
pp. 5-16
Author(s):  
Andreas Weiermann
Keyword(s):  
Normal Form ◽  
Asymptotic Analysis ◽  
Natural Number ◽  
Proof Theory ◽  
Peano Arithmetic ◽  
First Order ◽  
Primitive Recursive Arithmetic ◽  
Independence Result ◽  
Primitive Recursive

AbstractFor α less than ε0 let Nα be the number of occurrences of ω in the Cantor normal form of α. Further let ∣n∣ denote the binary length of a natural number n, let ∣n∣h denote the h-times iterated binary length of n and let inv(n) be the least h such that ∣n∣h ≤ 2. We show that for any natural number h first order Peano arithmetic, PA, does not prove the following sentence: For all K there exists an M which bounds the lengths n of all strictly descending sequences 〈α0, …, αn〉 of ordinals less than ε0 which satisfy the condition that the Norm Nαi of the i-th term αi is bounded by K + ∣i∣ · ∣i∣i.As a supplement to this (refined Friedman style) independence result we further show that e.g., primitive recursive arithmetic, PRA, proves that for all K there is an M which bounds the length n of any strictly descending sequence 〈α0, …, αn〉 of ordinals less than ε0 which satisfies the condition that the Norm Nαi of the i-th term αi is bounded by K + ∣i∣ · inv(i). The proofs are based on results from proof theory and techniques from asymptotic analysis of Polya-style enumerations.Using results from Otter and from Matoušek and Loebl we obtain similar characterizations for finite bad sequences of finite trees in terms of Otter's tree constant 2.9557652856.…

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.

scholarly journals Foundational and Mathematical Uses of Higher Types

1999 ◽  
Vol 6 (31) ◽  
Author(s):  
Ulrich Kohlenbach
Keyword(s):  
Reverse Mathematics ◽  
Continuous Functions ◽  
Higher Order ◽  
Second Order ◽  
Polish Spaces ◽  
Primitive Recursive Arithmetic ◽  
Higher Types ◽  
A Priority ◽  
Primitive Recursive

In this paper we develop mathematically strong systems of analysis in<br />higher types which, nevertheless, are proof-theoretically weak, i.e. conservative<br />over elementary resp. primitive recursive arithmetic. These systems<br />are based on non-collapsing hierarchies (Phi_n-WKL+, Psi_n-WKL+) of principles<br />which generalize (and for n = 0 coincide with) the so-called `weak' K¨onig's<br />lemma WKL (which has been studied extensively in the context of second order<br />arithmetic) to logically more complex tree predicates. Whereas the second<br />order context used in the program of reverse mathematics requires an encoding<br />of higher analytical concepts like continuous functions F : X -> Y between<br />Polish spaces X, Y , the more flexible language of our systems allows to treat<br />such objects directly. This is of relevance as the encoding of F used in reverse<br />mathematics tacitly yields a constructively enriched notion of continuous functions<br />which e.g. for F : N^N -> N can be seen (in our higher order context) to be equivalent<br /> to the existence of a continuous modulus of pointwise continuity.<br />For the direct representation of F the existence of such a modulus is<br />independent even of full arithmetic in all finite types E-PA^omega plus quantifier-free<br />choice, as we show using a priority construction due to L. Harrington.<br />The usual WKL-based proofs of properties of F given in reverse mathematics<br />make use of the enrichment provided by codes of F, and WKL does not seem<br />to be sufficient to obtain similar results for the direct representation of F in<br />our setting. However, it turns out that   Psi_1-WKL+ is sufficient.<br />Our conservation results for (Phi_n-WKL+,  Psi_n-WKL+) are proved via a new<br />elimination result for a strong non-standard principle of uniform Sigma^0_1-<br />boundedness<br />which we introduced in 1996 and which implies the WKL-extensions studied<br />in this paper.

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