Axiom Of Infinity Research Articles

Intuitionistic sets and numbers: small set theory and Heyting arithmetic

AbstractIt has long been known that (classical) Peano arithmetic is, in some strong sense, “equivalent” to the variant of (classical) Zermelo–Fraenkel set theory (including choice) in which the axiom of infinity is replaced by its negation. The intended model of the latter is the set of hereditarily finite sets. The connection between the theories is so tight that they may be taken as notational variants of each other. Our purpose here is to develop and establish a constructive version of this. We present an intuitionistic theory of the hereditarily finite sets, and show that it is definitionally equivalent to Heyting Arithmetic , in a sense to be made precise. Our main target theory, the intuitionistic small set theory is remarkably simple, and intuitive. It has just one non-logical primitive, for membership, and three straightforward axioms plus one axiom scheme. We locate our theory within intuitionistic mathematics generally.

FINITARY UPPER LOGICISM

Abstract This paper proposes and partially defends a novel philosophy of arithmetic—finitary upper logicism. According to it, the natural numbers are finite cardinalities—conceived of as properties of properties—and arithmetic is nothing but higher-order modal logic. Finitary upper logicism is furthermore essentially committed to the logicality of finitary plenitude, the principle according to which every finite cardinality could have been instantiated. Among other things, it is proved in the paper that second-order Peano arithmetic is interpretable, on the basis of the finite cardinalities’ conception of the natural numbers, in a weak modal type theory consisting of the modal logic $\mathsf {K}$ , negative free quantified logic, a contingentist-friendly comprehension principle, and finitary plenitude. By replacing finitary plenitude for the axiom of infinity this result constitutes a significant improvement on Russell and Whitehead’s interpretation of second-order Peano arithmetic, itself based on the finite cardinalities’ conception of the natural numbers.

Independence Results for Finite Set Theories in Well-Founded Locally Finite Graphs

We consider all combinatorially possible systems corresponding to subsets of finite set theory (i.e., Zermelo-Fraenkel set theory without the axiom of infinity) and for each of them either provide a well-founded locally finite graph that is a model of that theory or show that this is impossible. To that end, we develop the technique of axiom closure of graphs.

The Cardinal Squaring Principle and an Alternative Axiomatization of NFU

In this paper, we rigorously prove the existence of type-level ordered pairs in Quine’s New Foundations with atoms, augmented by the axiom of infinity and the axiom of choice (NFU + Inf + AC). The proof uses the cardinal squaring principle; more precisely, its instance for the (infinite) universe (VCSP), which is a theorem of NFU + Inf + AC. Therefore, we have a justification for proposing a new axiomatic extension of NFU, in order to obtain type-level ordered pairs almost from the beginning. This axiomatic extension is NFU + Inf + AC + VCSP, which is equivalent to NFU + Inf + AC, but easier to reason about.

PLURAL ANCESTRAL LOGIC AS THE LOGIC OF ARITHMETIC

AbstractNeo-Fregeanism aims to provide a possible route to knowledge of arithmetic via Hume’s principle, but this is of only limited significance if it cannot account for how the vast majority of arithmetic knowledge, accrued by ordinary people, is obtained. I argue that Hume’s principle does not capture what is ordinarily meant by numerical identity, but that we can do much better by buttressing plural logic with plural versions of the ancestral operator, obtaining natural and plausible characterizations of various key arithmetic concepts, including finiteness, equinumerosity and addition and multiplication of cardinality—revealing these to be logical concepts, and obtaining much of ordinary arithmetic knowledge as logical knowledge. Supplementing this with an abstraction principle and a simple axiom of infinity (known either empirically or modally) we obtain a full interpretation of arithmetic.

On the relative strengths of fragments of collection

AbstractLet be the basic set theory that consists of the axioms of extensionality, emptyset, pair, union, powerset, infinity, transitive containment, Δ0‐separation and set foundation. This paper studies the relative strength of set theories obtained by adding fragments of the set‐theoretic collection scheme to . We focus on two common parameterisations of the collection: ‐collection, which is the usual collection scheme restricted to ‐formulae, and strong ‐collection, which is equivalent to ‐collection plus ‐separation. The main result of this paper shows that for all , proves that there exists a transitive model of Zermelo Set Theory plus ‐collection, the theory is ‐conservative over the theory . It is also shown that (2) holds for when the Axiom of Choice is included in the base theory. The final section indicates how the proofs of (1) and (2) can be modified to obtain analogues of these results for theories obtained by adding fragments of collection to a base theory (Kripke‐Platek Set Theory with Infinity plus ) that does not include the powerset axiom.

The Fall and Original Sin of Set Theory

Hermann Weyl published a brief survey as preface to a review of The Philosophy of Bertrand Russell in 1946. In this survey he used the phrase, “The Fall and Original Sin of Set Theory.” Investigating the background of this remark will require that we pay attention to a number of issues within the foundations of mathematics. For example: Did God make the integers—as Kronecker alleged? Is mathematics set theory? Attention will also be given to axiomatic set theory and relevant ontic pre-conditions, such as the difference between number and number symbols, to number as “an aspect of objective reality” (Gödel), integers and induction (Skolem) as well as to the question if infinity—as endlessness—could be completed. In 1831 Gauss objected to viewing the infinite as something completed, which is not allowed in mathematics. It will be argued that the actual infinite is rather connected to what is present “at once,” as an infinite totality. By the year 1900 mathematicians believed that mathematics had reached absolute rigour, but unfortunately the rest of the twentieth century witnessed the opposite. The axiom of infinity ruined the expectations of logicism—mathematics cannot be reduced to logic. The intuitionism of Brouwer, Weyl and others launched a devastating attack on classical analysis, further inspired by the outcome of Gödel’s famous proof of 1931, in which he has shown that a formal mathematical system is inconsistent or incomplete. Intuitionism created a whole new mathematics, which finds no counter-part in classical mathematics. Slater remarked that within this logical paradise of Russell lurked a serpent, hidden behind the unjustified employment of the at once infinite. According to Weyl, “This is the Fall and original sin of set theory for which it is justly punished by the antinomies.” In conclusion, a few systematic distinctions are introduced.

THE RAMIFIED ANALYTICAL HIERARCHY USING EXTENDED LOGICS

AbstractThe use of Extended Logics to replace ordinary second order definability in Kleene’s Ramified Analytical Hierarchy is investigated. This mirrors a similar investigation of Kennedy, Magidor and Väänänen [11] where Gödel’s universe L of constructible sets is subjected to similar variance. Enhancing second order definability allows models to be defined which may or may not coincide with the original Kleene hierarchy in domain. Extending the logic with game quantifiers, and assuming strong axioms of infinity, we obtain minimal correct models of analysis. A wide spectrum of models can be so generated from abstract definability notions: one may take an abstract Spector Class and extract an extended logic for it. The resultant structure is then a minimal model of the given kind of definability.

Bi-Modal Naive Set Theory

This paper describes a modal conception of sets, according to which sets are 'potential' with respect to their members. A modal theory is developed, which invokes a naive comprehension axiom schema, modified by adding 'forward looking' and 'backward looking' modal operators. We show that this 'bi-modal' naive set theory can prove modalized interpretations of several ZFC axioms, including the axiom of infinity. We also show that the theory is consistent by providing an S5 Kripke model. The paper concludes with some discussion of the nature of the modalities involved, drawing comparisons with noneism, the view that there are some non-existent objects.

Why Believe Infinite Sets Exist?

The axiom of infinity states that infinite sets exist. I will argue that this axiom lacks justification. I start by showing that the axiom is not self-evident, so it needs separate justification. Following Maddy’s (J Symb Log 53(2):481–511, 1988) distinction, I argue that the axiom of infinity lacks both intrinsic and extrinsic justification. Crucial to my project is Skolem’s (in: van Heijnoort (ed) From Frege to Godel: a source book in mathematical logic, 1879–1931, Cambridge, Harvard University Press, pp. 290–301, 1922) distinction between a theory of real sets, and a theory of objects that theory calls “sets”. While Dedekind’s (in: Essays on the theory of numbers, pp. 14–58, 1888. http://www.gutenberg.org/ebooks/21016 ) argument fails, his approach was correct: the axiom of infinity needs a justification it currently lacks. This epistemic situation is at variance with everyday mathematical practice. A dilemma ensues: should we relax epistemic standards or insist, in a skeptical vein, that a foundational problem has been ignored?

The treewidth of proofs

So-called ordered variants of the classical notions of pathwidth and treewidth are introduced and proposed as proof theoretically meaningful complexity measures for the directed acyclic graphs underlying proofs. Ordered pathwidth is roughly the same as proof space and the ordered treewidth of a proof is meant to serve as a measure of how far it is from being treelike. Length-space lower bounds for k-DNF refutations are generalized to arbitrary infinity axioms and strengthened in that the space measure is relaxed to ordered treewidth.

Finite Model Reasoning in Expressive Fragments of First-Order Logic

Over the past two decades several fragments of first-order logic have been identified and shown to have good computational and algorithmic properties, to a great extent as a result of appropriately describing the image of the standard translation of modal logic to first-order logic. This applies most notably to the guarded fragment, where quantifiers are appropriately relativized by atoms, and the fragment defined by restricting the number of variables to two. The aim of this talk is to review recent work concerning these fragments and their popular extensions. When presenting the material special attention is given to decision procedures for the finite satisfiability problems, as many of the fragments discussed contain infinity axioms. We highlight most effective techniques used in this context, their advantages and limitations. We also mention a few open directions of study.

Consequences of Vopěnka’s Principle over weak set theories

It is shown that Vop\v{e}nka's Principle (VP) can restore almost the entire ZF over a weak fragment of it. Namely, if EST is the theory consisting of the axioms of Extensionality, Empty Set, Pairing, Union, Cartesian Product, $\Delta_0$-Separation and Induction along $\omega$, then ${\rm EST+VP}$ proves the axioms of Infinity, Replacement (thus also Separation) and Powerset. The result was motivated by previous results in \cite{Tz14}, as well as by H. Friedman's \cite{Fr05}, where a distinction is made among various forms of VP. As a corollary, ${\rm EST}+$Foundation$+{\rm VP}$=${\rm ZF+VP}$, and ${\rm EST}+$Foundation$+{\rm AC+VP}={\rm ZFC+VP}$. Also it is shown that the Foundation axiom is independent from ZF--\{Foundation\}+${\rm VP}$. It is open whether the Axiom of Choice is independent from ${\rm ZF+VP}$. A very weak form of choice follows from VP and some similar other forms of choice are introduced.

Carnap’s surprising views on the axiom of infinity

Carnap’s surprising views on the axiom of infinity

Reconsidering Pairs and Functions as Sets

We give representations for ordered pairs and functions in set theory with the property that ordered pairs are functions from the finite ordinal 2. We conjecture that these representations are useful for formalized mathematics since certain isomorphic sets are identified. The definitions, theorems and proofs have been formalized in the proof assistant Coq using only the simply typed features of Coq. We describe the development within the context of an intuitionistic simply typed (higher-order) version of (well-founded) Zermelo-Fraenkel set theory without the axiom of infinity.

More Wit in PM

78 Discussion c:\users\kenneth\documents\type3401\rj 3401 193 red.docx 2014-05-14 8:54 PM MORE WIT IN PM Michael E. Berumen 37155 Dickerson Run Windsor, co 80550, usa opinealot@gmail.com eldom does one find a piece by Bertrand Russell without a morsel of humour or wryness. Sometimes the Russellian wit is obvious. Other times it is more subtle, maybe even requiring a second look just to be sure. Headlong into some of his most technical and difficult works, a delightful bon mot might suddenly punctuate the philosophical solemnity. One doesn’t ordinarily think of Principia Mathematica as a sourcebook for Russellian waggishness , but it is there, to be sure—albeit one will negotiate many pages of symbols and scant prose before finding a specimen. Perhaps the best-known witticism amongst cognoscenti is, after having proved 1 + 1 = 2 (PM ∗110 .643, 2: 83), the authors observe, “The above proposition is occasionally useful .” The most complete directory for these jewels that I know is by Kenneth Blackwell.1 I thought it unlikely another would be found, so imagine my surprise when, after having dusted off my seldom-used Volume 3 to look something up, I came across a droll nugget that (to the best of my knowledge) has not been highlighted before. Something tells me Russell was biting down on his pipe stem and had a twinkle in his eye when he (I assume Russell, but they, anyway) wrote in the summary of Section A of Part vi, “Quantity”: Great difficulties are caused, in this section, by the existence-theorems and the question of types. These difficulties disappear if the axiom of infinity is assumed, but it seems improper to make the theory of (say) 2/3 depend upon the assumption that the number of objects in the universe is not finite. (PM 3: 234; italics mine) “Improper” indeed, the erstwhile Victorian says! By way of explanation, he goes on to say, We have, accordingly, taken pains not to make this assumption, except where, as in the theory of real numbers, it is really essential, and not merely convenient. Even that has a bit of Russellian sarcasm. So, accept the challenge and find another. In the meantime, take comfort in the knowledge that the sometimes controversial Axiom of Infinity is an unnecessary assumption in mixing one part soda and two parts Red Hackle. 1 “The Wit and Humour of Principia Mathematica”, Russell 31 (2011): 151–60. S ...

Uncovering the Mathematics Behind Russell’s Philosophy of Mathematics

c:\users\kenneth\documents\type3401\rj 3401 193 red.docx 2014-05-14 8:54 PM UNCOVERING THE MATHEMATICS BEHIND RUSSELL’S PHILOSOPHY OF MATHEMATICS Graham Stevens Philosophy / U. of Manchester Manchester m13 9pl, uk graham.p.stevens@manchester.ac.uk Sébastien Gandon. Russell’s Unknown Logicism. (History of Analytic Philosophy series.) Basingstoke, uk, and New York: Palgrave Macmillan, 2012. Pp. xiii + 263. isbn: 978-0-230-57699-5. £55; us$90. n My Philosophical Development, Russell recorded the disappointment that both he and Whitehead felt about the reception of Principia Mathematica by the mathematical community. While the philosophical parts of the book, including those parts dealing with philosophical logic, were of course widely discussed and of tremendous importance in the subsequent development of philosophical logic, the purely mathematical aspects of the work went largely ignored. Russell was perhaps exaggerating when he claimed he used to know of “only six people who had read the later parts of the book” (MPD, p. 86) but it was certainly true that the impact of the book on mathematicians working in areas outside of the philosophical foundations of their subject was minimal . The situation has hardly changed since the publication of those remarks in 1959. Principia Mathematica remains the target of philosophical, not mathematical , attention. In this outstanding new book on Russell’s logicism, however , Sébastien Gandon offers a welcome exception to the rule. While the book is still very much a book on Russell’s philosophy, its central claim is that new light can be shed on that philosophy by examining the hitherto neglected mathematical parts of both Principia Mathematica and The Principles of Mathematics . What parts of Russell’s philosophy are to be better understood in light of these forays into the “terra incognita” (p. 2) of Russell’s treatment of advanced mathematics in Principia and the Principles? Gandon’s insightful suggestion is that the very notion of analysis at the heart of Russell’s logicist project can be grasped in a new and more complete way by reflection on Russell’s development of areas of mathematics such as his theories of geometry and quantity. The suggestion is a compelling one, not least because Gandon does a superb job of arguing the case through his own painstaking analysis of Russell ’s mathematics in the book. To understand why these seemingly remote parts of Russell’s philosophical writings carry such significance, Gandon argues that Russell’s logicist project must be understood both in terms of its contribution to philosophy and to f= Reviews 93 c:\users\kenneth\documents\type3401\rj 3401 193 red.docx 2014-05-14 8:54 PM mathematics. Logicism makes a bold assertion about mathematics—that mathematical truths (or, perhaps, some subset of them—e.g. those that exclude geometry in the case of Frege’s version of logicism) are nothing more than logical truths. As we know, the logicist project does not rest content with bold assertion, it seeks to prove the assertion. The proof will be a demonstration that every mathematical truth can be translated into a logical truth. Frege’s attempt at that demonstration famously failed because it overlooked Russell’s paradox. Russell and Whitehead’s attempted demonstration may have avoided that pitfall, but it had problems of its own (the axioms of reducibility and infinity) that left many unconvinced of its success. Gandon’s project here is not to provide a fresh argument for accepting Russell and Whitehead ’s demonstration, nor even for accepting the truth of logicism. Rather, it is to examine more closely what the claim made by the logicist is and to subsequently draw a subtle but important distinction between the projects of Frege and Russell, as well as the project which Gandon, more tentatively, ascribes to Wittgenstein. To illustrate the differences between the three approaches, Gandon invites us to reflect on what kind of analysis of mathematical reasoning is being proposed by the logicist, and suggests that the three positions mentioned above present themselves as three possible outcomes of that reflection. The logicist is faced with an analysandum—pre-logicized mathematics—to which the analysans must be related in a certain way. One option...

Strong Axioms of Infinity and the Debate About Realism

Strong Axioms of Infinity and the Debate About Realism

Neo-Logicism and Russell's Logicism

Certain advocates of the so-called "neo-logicist" movement in the philosophy of mathematics identify themselves as "neo-Fregeans" (e.g., Hale and Wright), presenting an updated and revised version of Frege’s form of logicism. Russell’s form of logicism is scarcely discussed in this literature and, when it is, often dismissed as not really logicism at all (in light of its assumption of axioms of infinity, reducibility and so on). In this paper I have three aims: firstly, to identify more clearly the primary meta-ontological and methodological differences between Russell’s logicism and the more recent forms; secondly, to argue that Russell’s form of logicism offers more elegant and satisfactory solutions to a variety of problems that continue to plague the neo-logicist movement (the bad company objection, the embarrassment of richness objection, worries about a bloated ontology, etc.); thirdly, to argue that neo-Russellian forms of logicism remain viable positions for current philosophers of mathematics.

Neo-Logicism and Russell’s Logicism

Certain advocates of the so-called “neo-logicist” movement in the philosophy of mathematics identify themselves as “neo-Fregeans” (e.g., Hale and Wright), presenting an updated and revised version of Frege’s form of logicism. Russell’s form of logicism is scarcely discussed in this literature and, when it is, often dismissed as not really logicism at all (in light of its assumption of axioms of infinity, reducibility and so on). In this paper I have three aims: firstly, to identify more clearly the primary meta-ontological and methodological differences between Russell’s logicism and the more recent forms; secondly, to argue that Russell’s form of logicism offers more elegant and satisfactory solutions to a variety of problems that continue to plague the neo-logicist movement (the bad company objection, the embarrassment of richness objection, worries about a bloated ontology, etc.); thirdly, to argue that neo-Russellian forms of logicism remain viable positions for current philosophers of mathematics.