Primitive Recursive Arithmetic Research Articles

Modern perspectives in Proof Theory.

Modern perspectives in Proof Theory.

FINITIST AXIOMATIC TRUTH

AbstractFollowing the finitist’s rejection of the complete totality of the natural numbers, a finitist language allows only propositional connectives and bounded quantifiers in the formula-construction but not unbounded quantifiers. This is opposed to the currently standard framework, a first-order language. We conduct axiomatic studies on the notion of truth in the framework of finitist arithmetic in which at least smash function $\#$ is available. We propose finitist variants of Tarski ramified truth theories up to rank $\omega $ , of Kripke–Feferman truth theory and of Friedman–Sheard truth theory, and show that all of these have the same strength as the finitist arithmetic of one higher level along Grzegorczyk hierarchy. On the other hand, we also show that adding Burgess-style groundedness schema, adjusted to the finitist setting, makes Kripke–Feferman truth theory as strong as primitive recursive arithmetic. Meanwhile, we obtain some basic results on finitist theories of (full and hat) inductive definitions and on the second order axiom of hat inductive definitions for positive operators.

Finitism, imperative programs and primitive recursion

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.

The Consistency of Arithmetic

In 2010, Vladimir Voevodsky gave a lecture on "What If Current Foundations of Mathematics Are Inconsistent?" Among other things he said that he was seriously suspicious that an inconsistency in PA (first-order Peano arithmetic) might someday be found. About a year later, Edward Nelson announced that he had discovered an inconsistency not just in PA, but in a small fragment of primitive recursive arithmetic. Soon, Daniel Tausk and Terence Tao independently found a fatal error, and Nelson withdrew his claim, stating that consistency of PA was an "open problem." Many mathematicians may find such claims bewildering. Is the consistency of PA really an open problem? If so, would the discovery of an inconsistency in PA cause all of mathematics to come crashing down like a house of cards? This expository article attempts to address these questions, by sketching and discussing existing proofs of the consistency of PA (including Gentzen's proof and Friedman's relative consistency proof that appeals to the Bolzano-Weierstrass theorem). Since Nelson was a self-avowed formalist, the article also examines the implications of formalism.

The Implicit Commitment of Arithmetical Theories and Its Semantic Core

According to the implicit commitment thesis, once accepting a mathematical formal system S, one is implicitly committed to additional resources not immediately available in S. Traditionally, this thesis has been understood as entailing that, in accepting S, we are bound to accept reflection principles for S and therefore claims in the language of S that are not derivable in S itself. It has recently become clear, however, that such reading of the implicit commitment thesis cannot be compatible with well-established positions in the foundations of mathematics which consider a specific theory S as self-justifying and doubt the legitimacy of any principle that is not derivable in S: examples are Tait’s finitism and the role played in it by Primitive Recursive Arithmetic, Isaacson’s thesis and Peano Arithmetic, Nelson’s ultrafinitism and sub-exponential arithmetical systems. This casts doubts on the very adequacy of the implicit commitment thesis for arithmetical theories. In the paper we show that such foundational standpoints are nonetheless compatible with the implicit commitment thesis. We also show that they can even be compatible with genuine soundness extensions of S with suitable form of reflection. The analysis we propose is as follows: when accepting a system S, we are bound to accept a fixed set of principles extending S and expressing minimal soundness requirements for S, such as the fact that the non-logical axioms of S are true. We call this invariant component the semantic core of implicit commitment. But there is also a variable component of implicit commitment that crucially depends on the justification given for our acceptance of S in which, for instance, may or may not appear (proof-theoretic) reflection principles for S. We claim that the proposed framework regulates in a natural and uniform way our acceptance of different arithmetical theories.

VON NEUMANN’S CONSISTENCY PROOF

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.

Provability and Interpretability Logics with Restricted Realizations

The provability logic of a theory $T$ is the set of modal formulas, which under any arithmetical realization are provable in $T$. We slightly modify this notion by requiring the arithmetical realizations to come from a specified set $\Gamma$. We make an analogous modification for interpretability logics. We first study provability logics with restricted realizations and show that for various natural candidates of $T$ and restriction set $\Gamma$, the result is the logic of linear frames. However, for the theory Primitive Recursive Arithmetic (PRA), we define a fragment that gives rise to a more interesting provability logic by capitalizing on the well-studied relationship between PRA and I$\Sigma_1$. We then study interpretability logics, obtaining upper bounds for IL(PRA), whose characterization remains a major open question in interpretability logic. Again this upper bound is closely related to linear frames. The technique is also applied to yield the nontrivial result that IL(PRA) $\subset$ ILM.

UNFOLDING FINITIST ARITHMETIC

The concept of the (full) unfolding $\user1{{\cal U}}(S)$ of a schematic system $S$ is used to answer the following question: Which operations and predicates, and which principles concerning them, ought to be accepted if one has accepted $S$? The program to determine $\user1{{\cal U}}(S)$ for various systems $S$ of foundational significance was previously carried out for a system of nonfinitist arithmetic, $NFA$; it was shown that $\user1{{\cal U}}(NFA)$ is proof-theoretically equivalent to predicative analysis. In the present paper we work out the unfolding notions for a basic schematic system of finitist arithmetic, $FA$, and for an extension of that by a form $BR$ of the so-called Bar Rule. It is shown that $\user1{{\cal U}}(FA)$ and $\user1{{\cal U}}(FA + BR)$ are proof-theoretically equivalent, respectively, to Primitive Recursive Arithmetic, $PRA$, and to Peano Arithmetic, $PA$.

Godel's interpretation of intuitionism

Godel regarded the Dialectica interpretation as giving constructive content to intuitionism, which otherwise failed to meet reasonable conditions of constructivity. He founded his theory of primitive recursive functions, in which the interpretation is given, on the concept of computable function of finite type. I will 1) criticize this foundation, 2) propose a quite different one, and 3) note that essentially the latter foundation also underlies the Curry-Howard type theory and hence Heyting’s intuitionistic conception of logic. Thus the Dialectica interpretation (in so far as its aim was to give constructive content to intuitionism) is superfluous. In the set of notes [1938a] for an informal lecture, Godel refers to a hierarchy of constructive or finitary systems, the lowest level of which he called finitary number theory and is, in fact, primitive recursive arithmetic (PRA). He mentions three constructive extensions of this system: the intuitionism of Brouwer and Heyting (the “modal-logical route”), concerning which he is quite critical, the use of transfinite induction, inspired by Gentzen’s consistency proof for PA, and the use of functions of finite or even transfinite type over the type N of natural numbers. The latter extension seems to be mentioned here for the first time in connection with constructive extensions of finitary number theory. He ranks it the most satisfactory of these extensions ∗Versions of this paper were presented at a joint workshop on Hilbert at CarnegieMellon University and the University of Pittsburgh in June 1998, at a workshop on foundations of mathematics at UCLA in May 2005 and at the IMLA ’05 workshop [Intuitionistic Modal Logic and Applications, of course] in Chicago. I received helpful comments on all of these occasions. As for the history of the idea of a hierarchy of types, I know only of these examples: The hierarchy of transfinite types in which the natural numbers constitute the lowest type

On mathematical instrumentalism

AbstractIn this paper we devise some technical tools for dealing with problems connected with the philosophical view usually called mathematical instrumentalism. These tools are interesting in their own right, independently of their philosophical consequences. For example, we show that even though the fragment of Peanos Arithmetic known as IΣ1 is a conservative extension of the equational theory of Primitive Recursive Arithmetic (PRA). IΣ1 has a super-exponential speed-up over PRA. On the other hand, theories studied in the Program of Reverse Mathematics that formalize powerful mathematical principles have only polynomial speed-up over IΣ1.

On some generalizations of the primitive recursive arithmetic

Formal arithmetical system PRAU is defined as an extension of R.L. Goodstein's system PRA of the primitive recursive arithmetic; it is based on the consideration of functions similar to primitive recursive functions but in general not everywhere defined. It is proved that PRAU is a conservative extension of PRA. Some classes of the program schemes (PRA-schemes and PRAU-schemes) are introduced; it is proved that the classes of functions computable by such schemes coincide with the classes of functions taking part, correspondingly, in PRA and PRAU.

The Practice of Finitism: Epsilon Calculus and Consistency Proofs in Hilbert's Program

After a brief flirtation with logicism around 1917, David Hilbertproposed his own program in the foundations of mathematics in 1920 and developed it, in concert with collaborators such as Paul Bernays andWilhelm Ackermann, throughout the 1920s. The two technical pillars of the project were the development of axiomatic systems for everstronger and more comprehensive areas of mathematics, and finitisticproofs of consistency of these systems. Early advances in these areaswere made by Hilbert (and Bernays) in a series of lecture courses atthe University of Gottingen between 1917 and 1923, and notably in Ackermann's dissertation of 1924. The main innovation was theinvention of the ∈-calculus, on which Hilbert's axiom systemswere based, and the development of the ∈-substitution methodas a basis for consistency proofs. The paper traces the developmentof the ``simultaneous development of logic and mathematics'' throughthe ∈-notation and provides an analysis of Ackermann'sconsistency proofs for primitive recursive arithmetic and for thefirst comprehensive mathematical system, the latter using thesubstitution method. It is striking that these proofs use transfiniteinduction not dissimilar to that used in Gentzen's later consistencyproof as well as non-primitive recursive definitions, and that thesemethods were accepted as finitistic at the time.

L i D Z ? as a basis for PRA

This paper is a sequel to my [7]. It focuses on the notion of natural number as introduced in section 11 of that paper with regard to forms of induction and recursive definitions. One point is that this notion of natural number is somewhat weaker than the classical one in so far as it is defined in terms of a weak implication. The other point is the lack of even a weak form of extensionality. As a main result of the present paper it will turn out that the means provided in [7] are sufficient to account for an interpretation of primitive recursive arithmetic.

Transfer principles in nonstandard intuitionistic arithmetic

Using a slight generalization, due to Palmgren, of sheaf semantics, we present a term-model construction that assigns a model to any first-order intuitionistic theory. A modification of this construction then assigns a nonstandard model to any theory of arithmetic, enabling us to reproduce conservation results of Moerdijk and Palmgren for nonstandard Heyting arithmetic. Internalizing the construction allows us to strengthen these results with additional transfer rules; we then show that even trivial transfer axioms or minor strengthenings of these rules destroy conservativity over HA. The analysis also shows that nonstandard HA has neither the disjunction property nor the explicit definability property. Finally, careful attention to the complexity of our definitions allows us to show that a certain weak fragment of intuitionistic nonstandard arithmetic is conservative over primitive recursive arithmetic.

Proof by computation in the [formula omitted] system

In informal mathematics, statements involving computations are seldom proved. Instead, it is assumed that readers of the proof can carry out the computations on their own. However, when using an automated proof development system based on type theory, the user is forced to find proofs for all claimed propositions, including computational statements. This paper presents a method to automatically prove statements from primitive recursive arithmetic. The method replaces logical formulas by boolean expressions. A correctness proof is constructed, which states that the original formula is derivable, if and only if the boolean expression equals true. Because the boolean expression reduces to true, the conversion rule yields a trivial proof of the equality. By combining this proof with the correctness proof, we get a proof for the original statement.

The prime number theorem is PRA-provable

We introduce several theories the language of which is rich enough to talk about usual objects of real and complex analysis, and the axioms of which allow to recover a lot of classical results (sometimes with slight modifications). We prove that these theories are conservative extensions of primitive recursive arithmetic (PRA) and develop analysis within them. Moreover, as an example of their efficiency, we shall prove that the prime number theorem is PRA-provable.

Foundational and Mathematical Uses of Higher Types

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.

Partition theorems for factorisations of ascending parameter words

Abstract We prove, in primitive recursive arithmetic, an extension of a theorem established by Milliken and independently by Nesetřil and Rodl on partitions of fixed rank partitions of a finite set. In this way we depict the preservation of structure under partitions of the outcomes of substitution systems, which can also undergo various stage transitions.

Partition theorems for factorisations of ascending parameter words

We prove, in primitive recursive arithmetic, an extension of a theorem established by Milliken and independently by Nešetřil and Rödl on partitions of fixed rank partitions of a finite set. In this way we depict the preservation of structure under partitions of the outcomes of substitution systems, which can also undergo various stage transitions.

Nonconstructive Computational Mathematics

We describe a nonconstructive extension to primitive recursive arithmetic, both abstractly and as implemented on the Boyer-Moore prover. Abstractly, this extension is obtained by adding the unbounded µ operator applied to primitive recursive functions; doing so, one can define the Ackermann function and prove the consistency of primitive recursive arithmetic. The implementation does not mention the µ operator explicitly but has the strength to define the µ operator through the built-in functions EVALd and VC.