Metamathematical Results Research Articles

A Step Towards Absolute Versions of Metamathematical Results

There is a well-known gap between metamathematical theorems and their philosophical interpretations. Take Tarski’s Theorem. According to its prevalent interpretation, the collection of all arithmetical truths is not arithmetically definable. However, the underlying metamathematical theorem merely establishes the arithmetical undefinability of a set of specific Gödel codes of certain artefactual entities, such as infix strings, which are true in the standard model. That is, as opposed to its philosophical reading, the metamathematical theorem is formulated (and proved) relative to a specific choice of the Gödel numbering and the notation system. A similar observation applies to Gödel’s and Church’s theorems, which are commonly taken to impose severe limitations on what can be proved and computed using the resources of certain formalisms. The philosophical force of these limitative results heavily relies on the belief that these theorems do not depend on contingencies regarding the underlying formalisation choices. The main aim of this paper is to provide metamathematical facts which support this belief. While employing a fixed notation system, I showed in previous work (Review of Symbolic Logic, 2021, 14(1):51–84) how to abstract away from the choice of the Gödel numbering. In the present paper, I extend this work by establishing versions of Tarski’s, Gödel’s and Church’s theorems which are invariant regarding both the notation system and the numbering. This paper thus provides a further step towards absolute versions of metamathematical results which do not rely on contingent formalisation choices.

KURT GÖDEL ON LOGICAL, THEOLOGICAL, AND PHYSICAL ANTINOMIES

AbstractThis paper presents hitherto unpublished writings of Kurt Gödel concerning logical, epistemological, theological, and physical antinomies, which he generally considered as “the most interesting facts in modern logic,” and which he used as a basis for his famous metamathematical results. After investigating different perspectives on the notion of the logical structure of the antinomies and presenting two “antinomies of the intensional,” a new kind of paradox closely related to Gödel’s ontological proof for the existence of God is introduced and completed by a compilation of further theological antinomies. Finally, after a presentation of unpublished general philosophical remarks concerning the antinomies, Gödel’s type-theoretic variant of Leibniz’Monadology, discovered in his notes on the foundations of quantum mechanics, is examined. Most of the material presented here has been transcribed from the Gabelsberger shorthand system for the first time.

LIFSCHITZ REALIZABILITY AS A TOPOLOGICAL CONSTRUCTION

AbstractWe develop a number of variants of Lifschitz realizability for $\mathbf {CZF}$ by building topological models internally in certain realizability models. We use this to show some interesting metamathematical results about constructive set theory with variants of the lesser limited principle of omniscience including consistency with unique Church’s thesis, consistency with some Brouwerian principles and variants of the numerical existence property.

A constructive version of Tarski's geometry

Constructivity, in this context, refers to a theory of geometry whose axioms and language are closely related to ruler and compass constructions. It may also refer to the use of intuitionistic (or constructive) logic, but the reader who is interested in ruler and compass geometry but not in constructive logic will still find this work of interest. Euclid's reasoning is essentially constructive (in both senses). Tarski's elegant and concise first-order theory of Euclidean geometry, on the other hand, is essentially non-constructive (in both senses), even if we restrict attention (as we do here) to the theory with line–circle continuity in place of first-order Dedekind completeness. Hilbert's axiomatization has a much more elaborate language and many more axioms, but it contains no essential non-constructivities. Here we exhibit three constructive versions of Tarski's theory. One, like Tarski's theory, has existential axioms and no function symbols. We then consider a version in which function symbols are used instead of existential quantifiers. This theory is quantifier-free and proves the continuous dependence on parameters of the terms giving the intersections of lines and circles, and of circles and circles. The third version has a function symbol for the intersection point of two non-parallel, non-coincident lines, instead of only for intersection points produced by Pasch's axiom and the parallel axiom; this choice of function symbols connects directly to ruler and compass constructions. All three versions have this in common: the axioms have been modified so that the points they assert to exist are unique and depend continuously on parameters. This modification of Tarski's axioms, with classical logic, has the same theorems as Tarski's theory, but we obtain results connecting it with ruler and compass constructions as well. In particular, we show that constructions involving the intersection points of two circles are justified, even though only line–circle continuity is included as an axiom. We obtain metamathematical results based on the Gödel double-negation interpretation, which permit the wholesale importation of proofs of negative theorems from classical to constructive geometry, and of proofs of existential theorems where the object asserted to exist is constructed by a single construction (as opposed to several constructions applying in different cases). In particular, this enables us to import the proofs of correctness of the geometric definitions of addition and multiplication, once these can be given by a uniform construction.We also show, using cut-elimination, that objects proved to exist can be constructed by ruler and compass. (This was proved in [3] for a version of constructive geometry based on Hilbert's axioms.) Since these theories are interpretable in the theory of Euclidean fields, the independence results about different versions of the parallel postulate given in [5] apply to them; and since addition and multiplication can be defined geometrically, their models are exactly the planes over (constructive) Euclidean fields.

On the regular extension axiom and its variants

AbstractThe regular extension axiom, REA, was first considered by Peter Aczel in the context of Constructive Zermelo‐Fraenkel Set Theory as an axiom that ensures the existence of many inductively defined sets. REA has several natural variants. In this note we gather together metamathematical results about these variants from the point of view of both classical and constructive set theory.

Pure Proof Theory Aims, Methods and Results: Extended Version of Talks Given at Oberwolfach and Haifa

Apologies. The purpose of the following talk is to give an overview of the present state of aims, methods and results in Pure Proof Theory. Shortage of time forces me to concentrate on my very personal views. This entails that I will emphasize the work which I know best, i.e., work that has been done in the triangle Stanford, Munich and Münster. I am of course well aware that there are as important results coming from outside this triangle and I apologize for not displaying these results as well.Moreover the audience should be aware that in some points I have to oversimplify matters. Those who complain about that are invited to consult the original papers.1.1. General. Proof theory startedwithHilbert's Programme which aimed at a finitistic consistency proof for mathematics.By Gödel's Theorems, however, we know that we can neither formalize all mathematics nor even prove the consistency of formalized fragments by finitistic means. Inspite of this fact I want to give some reasons why I consider proof theory in the style of Gentzen's work still as an important and exciting field of Mathematical Logic. I will not go into applications of Gentzen's cut-elimination technique to computer science problems—this may be considered as applied proof theory—but want to concentrate on metamathematical problems and results. In this sense I am talking about Pure Proof Theory.Mathematicians are interested in structures. There is only one way to find the theorems of a structure. Start with an axiom system for the structure and deduce the theorems logically. These axiom systems are the objects of proof-theoretical research. Studying axiom systems there is a series of more or less obvious questions.

How to glue analysis models

Among the more traditional semantics for intuitionistic logic the Beth and the Kripke semantics seem well-suited for direct manipulations required for the derivation of metamathematical results. In particular Smoryński demonstrated the usefulness of Kripke models for the purpose of obtaining closure properties for first-order arithmetic, [S], and second-order arithmetic, [J-S]. Weinstein used similar techniques to handle intuitionistic analysis, [W]. Since, however, Beth-models seem to lend themselves better for dealing with analysis, cf. [D], we have developed a somewhat more liberal semantics, that shares the features of both Kripke and Beth semantics, in order to obtain analogues of Smoryński's collecting operations, which we will call Smoryński-glueing, in line with the categorical tradition.

Recursive models for constructive set theories

We define recursive models of Martin-Löf's (type or) set theories. These models are a sort of recursive realizability; in fact, we show that for implication-free formulae of HA ω, satisfaction in the model coincides with mr-HEO realizability. Using an idea of Aczel, we extend the model to a recursive model of the constructive set theories of Myhill and Friedman. Our models can be described without presupposing any knowledge of Martin-Löf's theories, and may make them seem less mysterious. We use our models to obtain several metamathematical results, for example consistency and independence results concerning continuity of functions on compact metric spaces. On the other hand, Martin-Löfs (latest) theories refute continuity of functions from N N to N, as well as Church's thesis, although a show that all provably well-defined functions are continuous.

How to Godel a Frege-Russell: Godel's Incompleteness Theorems and Logicism

Any viable philosophy of mathematics must square with incontrovertible metamathematical results. Thus, as is well known, any version of formalism committed to identifying mathematical with provability in some formal system is to be rejected in light of Godel's first incompleteness theorem, according to which any consistent formal system capable of representing the primitive recursive numbertheoretic functions has an undecidable sentence. In fact, Godel showed how, given any such system, one could construct such a sentence which, on the standard interpretation of its symbols, makes a definite claim about natural numbers. Thus, either such a Godel sentence or its negation must be an truth, but neither is provable in the system for which it was constructed. Any formalist philosophy which respects arithmetical truth runs afoul of the first incompleteness theorem.' The impact of Godel's incompleteness theorems on logicism, however, is not so straightforwardly assessed. While there are independent grounds for regarding logicism (as conceived by Frege and Russell) as a lost cause, it remains instructive to examine the force of G6del's theorems in this quarter. In fact, a negative result (i.e., incompatibility of logicism with a metatheorem) could be quite powerful, especially if it does not depend on a particularly restrictive way of drawing the controversial line between logic and non-logic. The purpose of this note is to present precisely this kind of an argument: so long as logic is taken to be formalizable (i.e., logically valid formulas form a recursively enumerable set), a natural thesis of epistemological logicism is, modulo quite elementary assumptions, incompatible with Godel's second incompleteness theorem.2 However, for this argument, it needs to be assumed that the logicist system is to be finitely axiomatized. Extending the argument to the infinite case turns out to depend on a crucial step whose

On the proof theory of the modal logic for arithmetic provability

The modal logic GL has been found by Solovay [13] to formalize the provable propositional properties of the provability-predicate for Peano's Arithmetic PA (cf. §1 below). We give several sequential calculi for GL, compare their merits, and use one calculus to syntactically derive several metamathematical results about GL.Some of our results have been proved model theoretically, and similar proofs are fairly straightforward for several of the remaining ones (G. Boolos and the referee have provided such proofs for 4.1, 4.3 and 5.1 below). However, our syntactic techniques often yield more concise and obviously constructive proofs, they offer additional insight into the nature of the systems considered, and are easily adaptable to systems for which semantical analysis is problematic.I am indebted to G. Boolos and to the referee for their valuable advice. The referee has suggested the rule GL of §3 below as an axiomatization of GL; the resulting sequential calculus has allowed a definite improvement of our original presentation.

Avoiding self-referential statements

Recursion-theoretic proofs of metamathematical results tend to rely on a pair of effectively inseparable r.e. sets and its properties. We establish a special property for a small configuration of such pairs and derive from it some metamathematical results not previously accessible to recursion-theoretic techniques.

Model-interpretability into trees and applications

It will be shown thatn-separated graphs are model-interpretable into trees, in particular this holds for unary function graphs. Hence metamathematical results for trees carry over to more general graphs. We show that trees are stable in some infinite cardinality, hencen-separated graphs are stable, in particular this holds for unary functions. This generalizes results in [4]. Other examples concern decidability and the finite valency satisfiability property.

Andrzej Mostowski. Recent results in set theory. Problems in the philosophy of mathematics, Proceedings of the International Colloquium in the Philosophy of Science, London, 1965, volume 1, edited by Imre Lakatos, Studies in logic and the foundations of mathematics, North-Holland Publishing Company, Amsterdam1967, pp. 82–96. - G. Kreisel, A. Robinson, L. Kalmár, and A. Mostowski. Discussion. Problems in the philosophy

Andrzej Mostowski. Recent results in set theory. Problems in the philosophy of mathematics, Proceedings of the International Colloquium in the Philosophy of Science, London, 1965, volume 1, edited by Imre Lakatos, Studies in logic and the foundations of mathematics, North-Holland Publishing Company, Amsterdam1967, pp. 82–96. - G. Kreisel, A. Robinson, L. Kalmár, and A. Mostowski. Discussion. Problems in the philosophy of mathematics, Proceedings of the International Colloquium in the Philosophy of Science, London, 1965, volume 1, edited by Imre Lakatos, Studies in logic and the foundations of mathematics, North-Holland Publishing Company, Amsterdam1967, pp. 97–108. - Andrzej Mostowski. O niektórych nowych wynikach meta-matematycznych dotyczących teorii mnogości (On some new metamathematical results concerning set theory). Polish with Russian and English summaries. Studia logica, vol. 20 (1967), pp. 99–116. - Volume 37 Issue 4

Jean van Heijenoort. Introductory note. From Frege to Gödel, A source book in mathematical logic, 1879–1931, edited by Jean van Heijenoort, Harvard University Press, Cambridge, Mass., 1967, pp. 1–5. Reprinted in Frege and Gödel, Two fundamental texts in mathematical logic, edited by Jean van Heijenoort, Harvard University Press, Cambridge, Mass., 1970, pp. 1–5. - Gottlob Frege. Begriffsschrift, a formula language,

Jean van Heijenoort. Introductory note. From Frege to Gödel, A source book in mathematical logic, 1879–1931, edited by Jean van Heijenoort, Harvard University Press, Cambridge, Mass., 1967, pp. 1–5. Reprinted in Frege and Gödel, Two fundamental texts in mathematical logic, edited by Jean van Heijenoort, Harvard University Press, Cambridge, Mass., 1970, pp. 1–5. - Gottlob Frege. Begriffsschrift, a formula language, modeled upon that of arithmetic, for pure thought. English translation of 491 by Stefan Bauer-Mengelberg. From Frege to Gödel, A source book in mathematical logic, 1879–1931, edited by Jean van Heijenoort, Harvard University Press, Cambridge, Mass., 1967, pp. 5–82. Reprinted in Frege and Gödel, Two fundamental texts in mathematical logic, edited by Jean van Heijenoort, Harvard University Press, Cambridge, Mass., 1970, pp. 5–82. - Jean van Heijenoort. Introductory note. From Frege to Gödel, A source book in mathematical logic, 1879–1931, edited by Jean van Heijenoort, Harvard University Press, Cambridge, Mass., 1967, pp. 592–595. Reprinted in Frege and Gödel, Two fundamental texts in mathematical logic, edited by Jean van Heijenoort, Harvard University Press, Cambridge, Mass., 1970, pp. 83–86. - Kurt Gödel. Some metamathematical results on completeness and consistency. English translation of 4181 by Stefan Bauer-Mengelberg. From Frege to Gödel, A source book in mathematical logic, 1879–1931, edited by Jean van Heijenoort, Harvard University Press, Cambridge, Mass., 1967, pp. 595–596. Reprinted in Frege and Gödel, Two fundamental texts in mathematical logic, edited by Jean van Heijenoort, Harvard University Press, Cambridge, Mass., 1970, pp. 86–87. - Kurt Gödel. On formally undecidable propositions of Principia mathematica and related systems I. English translation of 4183 by Jean van Heijenoort. From Frege to Gödel, A source book in mathematical logic, 1879–1931, edited by Jean van Heijenoort, Harvard University Press, Cambridge, Mass., 1967, pp. 596–616. Reprinted in Frege and Gödel, Two fundamental texts in mathematical logic, edited by Jean van Heijenoort, Harvard University Press, Cambridge, Mass., 1970, pp. 87–107. - Kurt Gödel. On completeness and consistency. English translation of 4188 by Jean van Heijenoort. From Frege to Gödel, A source book in mathematical logic, 1879–1931, edited by Jean van Heijenoort, Harvard University Press, Cambridge, Mass., 1967, pp. 616–617. Reprinted in Frege and Gödel, Two fundamental texts in mathematical logic, edited by Jean van Heijenoort, Harvard University Press, Cambridge, Mass., 1970, pp. 107–108. - Volume 37 Issue 2

On some metamathematical results as properties of general systems

On some metamathematical results as properties of general systems

On some new metamathematical results concerning set theory

The aim of the article is to present some of the newest meta-mathematical results concerning set theory and to discuss on this basis the problem whether and to what extent set theory can serve as a basis for mathematics.

Categoricity in power

Introduction. A theory, 1, (formalized in the first order predicate calculus) is categorical in power K if it has exactly one type of models of power K. This notion was introduced by Los [ 9] and Vaught [ 16] in 1954. At that time they pointed out that a theory (e.g., the theory of dense linearly ordered sets without end points) may be categorical in power N0 and fail to be categorical in any higher power. Conversely, a theory may be categorical in every uncountable power and fail to be categorical in power N0 (e.g., the theory of algebraically closed fields of characteristic 0). Los' then raised the following question. Is a theory categorical in one uncountable power necessarily categorical in every uncountable power? The principal result of this paper is an affirmative answer to that question. We actually prove a stronger result, namely: If a theory is categorical in some uncountable power then every uncountable model of that theory is saturated. (Terminology used in the Introduction will be defined in the body of the paper; roughly speaking, a model is saturated, or universalhomogeneous, if it contains an element of every possible elementary type relative to its subsystems of strictly smaller power.) It is known(2) that a theory can have (up to isomorphism) at most one saturated model in each power. It is interesting to note that our results depend essentially on an analogue of the usual analysis of topological spaces in terms of their derived spaces and the Cantor-Bendixson theorem. The paper is divided into five sections. In ?1 terminology and some meta-mathematical results are summarized. In particular, for each theory, 1, there is described a theory, *, which has essentially the same models as z but is neater to work with. In ?2 is defined a topological space, S(A), corresponding to each subsystem, A, of a model of a theory, 1; the points of S(A) being the isomorphism types of elements with respect to A. With each monomorphism (= isomorphic imbedding), f: A -+ B, is associated a dual continuous map, f*: S(B) -+ S(A). Then there is defined for each S(A) a decreasing sequence ISa(A) I of subspaces which is analogous to (but different from)

Representation of modular lattices and of relation algebras

Introduction. This paper gives an axiomatic characterization of the class of all those (modular) lattices which are isomorphic to lattices of commuting equivalence relations. As might be expected, this problem turns out to be closely related to the representation problem for relation algebras, and we are able to borrow some basic ideas from the work of R. Lyndon [7; 8]. It turns out to be convenient to consider first the class of all those algebras SI = (A, ;, •,, 1') which can be represented isomorphically by means of binary relations over some set U in such a way that the operations ;, •, correspond to relation-theoretic multiplication, set-theoretic multiplication and relationtheoretic conversion, respectively, and 1' corresponds to the identity relation over U. This class of algebras is characterized in Theorem 1. In Theorem 2 this result is applied to lattices. The key observation here is that a lattice St = (A, +, •) with a zero element 0 is isomorphic to a lattice of commuting equivalence relations if and only if the algebra 31 = (A, +, ■,, 0), where 2c~ = x for every xEA, can be represented isomorphically by means of binary relations in the manner discussed above with -fand 0 taking the place of ; and 1'. We can give a more direct, although more involved, proof of Theorem 2. The added complications are due mostly to the fact that we are then unable to take advantage of the metamathematical results of Henkin [2] and Tarski [9]. The trouble is that we see no direct way of proving that the class of lattices under consideration coincides with the class of all subalgebras of algebras from an arithmetic class in the wider sense. The method employed here also has the advantage that it brings out the close connection with the representation problem for relation algebras, and in this context Theorem 1 is of independent interest. Theorems 3 and 4 concern the existence of what we call weak representations for relation algebras, and the paper concludes with a brief discussion of some open questions related to our work. 1. Preliminaries. We adopt the notation and terminology of Tarski [9], in particular, we refer the reader to that paper for the definitions of an arithmetic class and an arithmetic class in the wider sense. The relationtheoretic product of two binary relations and 5 will be denoted by R S, their set-theoretic sum and product by RKJS and RC\S, and the relationtheoretic converse of by R~x. By an algebra of relations we mean a system (a, |, C\, ~ I) where ft is a set of binary relations which is closed under the