Predicative Fragments of Frege Arithmetic

2004 ◽  
Vol 10 (2) ◽  
pp. 153-174 ◽  
Author(s):  
Øystein Linnebo
Keyword(s):  
Natural Number ◽  
Order Theory ◽  
Peano Arithmetic ◽  
Two Dimensions ◽  
Second Order ◽  
Order Logic ◽  
The Other ◽  
Hume’S Principle ◽  
Important Exception ◽  
Second Order Logic

AbstractFrege Arithmetic (FA) is the second-order theory whose sole non-logical axiom is Hume's Principle, which says that the number of Fs is identical to the number of Gs if and only if the Fs and the Gs can be one-to-one correlated. According to Frege's Theorem, FA and some natural definitions imply all of second-order Peano Arithmetic. This paper distinguishes two dimensions of impredicativity involved in FA—one having to do with Hume's Principle, the other, with the underlying second-order logic—and investigates how much of Frege's Theorem goes through in various partially predicative fragments of FA. Theorem 1 shows that almost everything goes through, the most important exception being the axiom that every natural number has a successor. Theorem 2 shows that the Successor Axiom cannot be proved in the theories that are predicative in either dimension.

Logicism in the Twenty‐first Century

2009 ◽  
Author(s):  
Bob Hale ◽  
Crispin Wright
Keyword(s):  
Higher Order ◽  
Second Order ◽  
Order Logic ◽  
First Century ◽  
The Other ◽  
Hume’S Principle ◽  
Main Assumption ◽  
Twenty First Century ◽  
Second Order Logic ◽  
Abstraction Principles

This article focuses on issues which neo-Fregeanism must address, even if the scope of its leading claims is restricted to elementary arithmetic. Many of these concern the capacity of abstraction principles—centrally, but not only, Hume's Principle itself—to discharge the implicitly definitional role in which the neo-Fregean casts them, and thereby to subserve a satisfactory apriorist epistemology for (at least part of) mathematics. Others concern the other main assumption that undergirds the specifically logicist aspect of the neo-Fregean project (and equally, of course, Frege's original project): that the logic to which abstraction principles are to be adjoined may legitimately be taken to include higher-order—at the very least, second-order—logic without compromise of the epistemological purposes of the project.

scholarly journals THE STRENGTH OF ABSTRACTION WITH PREDICATIVE COMPREHENSION

2016 ◽  
Vol 22 (1) ◽  
pp. 105-120 ◽  
Author(s):  
SEAN WALSH
Keyword(s):  
Peano Arithmetic ◽  
Second Order ◽  
Order Logic ◽  
Equivalence Relations ◽  
Hume’S Principle ◽  
Abstraction Principle ◽  
Basic Law ◽  
Second Order Logic ◽  
Abstraction Principles

AbstractFrege’s theorem says that second-order Peano arithmetic is interpretable in Hume’s Principle and full impredicative comprehension. Hume’s Principle is one example of anabstraction principle, while another paradigmatic example is Basic Law V from Frege’sGrundgesetze. In this paper we study the strength of abstraction principles in the presence of predicative restrictions on the comprehension schema, and in particular we study a predicative Fregean theory which contains all the abstraction principles whose underlying equivalence relations can be proven to be equivalence relations in a weak background second-order logic. We show that this predicative Fregean theory interprets second-order Peano arithmetic (cf. Theorem 3.2).

The development of arithmetic in Frege's Grundgesetze der arithmetik

1993 ◽  
Vol 58 (2) ◽  
pp. 579-601 ◽  
Author(s):  
Richard G. Heck
Keyword(s):  
Cardinal Number ◽  
Formal Theory ◽  
Second Order ◽  
Order Logic ◽  
Hume’S Principle ◽  
Philosophical Significance ◽  
Philosophy Of Arithmetic ◽  
Second Order Logic

AbstractFrege's development of the theory of arithmetic in his Grundgesetze der Arithmetik has long been ignored, since the formal theory of the Grundgesetze is inconsistent. His derivations of the axioms of arithmetic from what is known as Hume's Principle do not, however, depend upon that axiom of the system—Axiom V—which is responsible for the inconsistency. On the contrary, Frege's proofs constitute a derivation of axioms for arithmetic from Hume's Principle, in (axiomatic) second-order logic. Moreover, though Frege does prove each of the now standard Dedekind-Peano axioms, his proofs are devoted primarily to the derivation of his own axioms for arithmetic, which are somewhat different (though of course equivalent). These axioms, which may be yet more intuitive than the Dedekind-Peano axioms, may be taken to be “The Basic Laws of Cardinal Number”, as Frege understood them.Though the axioms of arithmetic have been known to be derivable from Hume's Principle for about ten years now, it has not been widely recognized that Frege himself showed them so to be; nor has it been known that Frege made use of any axiomatization for arithmetic whatsoever. Grundgesetze is thus a work of much greater significance than has often been thought. First, Frege's use of the inconsistent Axiom V may invalidate certain of his claims regarding the philosophical significance of his work (viz., the establish may invalidate certain of his claims regarding the philosophical significance of his work (viz., the establishment of Logicism), but it should not be allowed to obscure his mathematical accomplishments and his contribution to our understanding of arithmetic. Second, Frege's knowledge that arithmetic is derivable from Hume's Principle raises important sorts of questions about his philosophy of arithmetic. For example, “Why did Frege not simply abandon Axiom V and take Hume's Principle as an axiom?”

Polynomial rings and weak second-order logic

1985 ◽  
Vol 50 (4) ◽  
pp. 953-972 ◽  
Author(s):  
Anne Bauval
Keyword(s):  
Zero Divisor ◽  
Order Theory ◽  
Second Order ◽  
Order Logic ◽  
Polynomial Rings ◽  
Commutative Field ◽  
First Order ◽  
First Order Theory ◽  
Order Language ◽  
Second Order Logic

This article is a rewriting of my Ph.D. Thesis, supervised by Professor G. Sabbagh, and incorporates a suggestion from Professor B. Poizat. My main result can be crudely summarized (but see below for detailed statements) by the equality: first-order theory of F[Xi]i∈I = weak second-order theory of F.§I.1. Conventions. The letter F will always denote a commutative field, and I a nonempty set. A field or a ring (A; +, ·) will often be written A for short. We shall use symbols which are definable in all our models, and in the structure of natural numbers (N; +, ·):— the constant 0, defined by the formula Z(x): ∀y (x + y = y);— the constant 1, defined by the formula U(x): ∀y (x · y = y);— the operation ∹ x − y = z ↔ x = y + z;— the relation of division: x ∣ y ↔ ∃ z(x · z = y).A domain is a commutative ring with unity and without any zero divisor.By “… → …” we mean “… is definable in …, uniformly in any model M of L”.All our constructions will be uniform, unless otherwise mentioned.§I.2. Weak second-order models and languages. First of all, we have to define the models Pf(M), Sf(M), Sf′(M) and HF(M) associated to a model M = {A; ℐ) of a first-order language L [CK, pp. 18–20]. Let L1 be the extension of L obtained by adjunction of a second list of variables (denoted by capital letters), and of a membership symbol ∈. Pf(M) is the model (A, Pf(A); ℐ, ∈) of L1, (where Pf(A) is the set of finite subsets of A. Let L2 be the extension of L obtained by adjunction of a second list of variables, a membership symbol ∈, and a concatenation symbol ◠.

scholarly journals A Multivariate Interlace Polynomial and its Computation for Graphs of Bounded Clique-Width

10.37236/793 ◽  
2008 ◽  
Vol 15 (1) ◽  
Author(s):  
Bruno Courcelle
Keyword(s):  
Second Order ◽  
Order Logic ◽  
The Other ◽  
Multivariate Polynomial ◽  
Recursive Definition ◽  
Local Complementation ◽  
The One ◽  
Recursive Definitions ◽  
Second Order Logic ◽  
Monadic Second Order Logic

We define a multivariate polynomial that generalizes in a unified way the two-variable interlace polynomial defined by Arratia, Bollobás and Sorkin on the one hand, and a one-variable variant of it defined by Aigner and van der Holst on the other. We determine a recursive definition for our polynomial that is based on local complementation and pivoting like the recursive definitions of Tutte's polynomial and of its multivariate generalizations are based on edge deletions and contractions. We also show that bounded portions of our polynomial can be evaluated in polynomial time for graphs of bounded clique-width. Our proof uses an expression of the interlace polynomial in monadic second-order logic, and works actually for every polynomial expressed in monadic second-order logic in a similar way.

The relative expressive power of some logics extending first-order logic

1979 ◽  
Vol 44 (2) ◽  
pp. 129-146 ◽  
Author(s):  
John Cowles
Keyword(s):  
Model Theory ◽  
Expressive Power ◽  
Second Order ◽  
Order Logic ◽  
The Other ◽  
First Order Logic ◽  
Finite Sets ◽  
Limited Ability ◽  
First Order ◽  
Second Order Logic

In recent years there has been a proliferation of logics which extend first-order logic, e.g., logics with infinite sentences, logics with cardinal quantifiers such as “there exist infinitely many…” and “there exist uncountably many…”, and a weak second-order logic with variables and quantifiers for finite sets of individuals. It is well known that first-order logic has a limited ability to express many of the concepts studied by mathematicians, e.g., the concept of a wellordering. However, first-order logic, being among the simplest logics with applications to mathematics, does have an extensively developed and well understood model theory. On the other hand, full second-order logic has all the expressive power needed to do mathematics, but has an unworkable model theory. Indeed, the search for a logic with a semantics complex enough to say something, yet at the same time simple enough to say something about, accounts for the proliferation of logics mentioned above. In this paper, a number of proposed strengthenings of first-order logic are examined with respect to their relative expressive power, i.e., given two logics, what concepts can be expressed in one but not the other?For the most part, the notation is standard. Most of the notation is either explained in the text or can be found in the book [2] of Chang and Keisler. Some notational conventions used throughout the text are listed below: the empty set is denoted by ∅.

Categoricity and the natural numbers

2018 ◽  
Author(s):  
Tim Button ◽  
Sean Walsh
Keyword(s):  
Order Theory ◽  
Second Order ◽  
Order Logic ◽  
Natural Numbers ◽  
Categorical Theory ◽  
First Order Theory ◽  
Standard Models ◽  
Skolem Theorem ◽  
Infinite Model ◽  
Second Order Logic

This chapter focuses on modelists who want to pin down the isomorphism type of the natural numbers. This aim immediately runs into two technical barriers: the Compactness Theorem and the Löwenheim-Skolem Theorem (the latter is proven in the appendix to this chapter). These results show that no first-order theory with an infinite model can be categorical; all such theories have non-standard models. Other logics, such as second-order logic with its full semantics, are not so expressively limited. Indeed, Dedekind's Categoricity Theorem tells us that all full models of the Peano axioms are isomorphic. However, it is a subtle philosophical question, whether one is entitled to invoke the full semantics for second-order logic — there are at least four distinct attitudes which one can adopt to these categoricity result — but moderate modelists are unable to invoke the full semantics, or indeed any other logic with a categorical theory of arithmetic.

Logicism and Logical Consequence

2020 ◽  
pp. 55-95
Author(s):  
Jim Edwards
Keyword(s):  
Proof Theory ◽  
Logical Consequence ◽  
Semantic Relation ◽  
Second Order ◽  
Order Logic ◽  
Hume’S Principle ◽  
Philosophy Of Arithmetic ◽  
Second Order Logic

According to Crispin Wright’s neo-logicist reconstruction of Frege’s philosophy of arithmetic, the truths of arithmetic are logical consequences, in the semantic sense, of second-order logic, augmented with an analytic axiom (Hume’s Principle). Neo-logicism thus views arithmetic truths as analytic, being the logical consequences of an analytic axiom. This chapter argues that the semantic relation of second-order logical consequence that is most naturally suited to the practice of arithmetic is proof-theoretically complete, and that given this, Gödel’s incompleteness result shows that there are arithmetical truths which are not derivable in Wright’s proof theory augmented by Hume’s Principle. The chapter thus challenges Wright’s programme of neo-Fregean logicism.

scholarly journals Second-Order Logic and Foundations of Mathematics

2001 ◽  
Vol 7 (4) ◽  
pp. 504-520 ◽  
Author(s):  
Jouko Väänänen
Keyword(s):  
Set Theory ◽  
Second Order ◽  
Order Logic ◽  
The Other ◽  
Foundations Of Mathematics ◽  
Mathematical Concepts ◽  
Informal Reasoning ◽  
First Order ◽  
Second Order Logic ◽  
Order Set

AbstractWe discuss the differences between first-order set theory and second-order logic as a foundation for mathematics. We analyse these languages in terms of two levels of formalization. The analysis shows that if second-order logic is understood in its full semantics capable of characterizing categorically central mathematical concepts, it relies entirely on informal reasoning. On the other hand, if it is given a weak semantics, it loses its power in expressing concepts categorically. First-order set theory and second-order logic are not radically different: the latter is a major fragment of the former.

scholarly journals Second Order Logic or Set Theory?

2012 ◽  
Vol 18 (1) ◽  
pp. 91-121 ◽  
Author(s):  
Jouko Väänänen
Keyword(s):  
Set Theory ◽  
Order Theory ◽  
Second Order ◽  
Order Logic ◽  
First Order ◽  
First Order Theory ◽  
Standard Models ◽  
The Right ◽  
Second Order Logic ◽  
Proper Context

AbstractWe try to answer the question which is the “right” foundation of mathematics, second order logic or set theory. Since the former is usually thought of as a formal language and the latter as a first order theory, we have to rephrase the question. We formulate what we call the second order view and a competing set theory view, and then discuss the merits of both views. On the surface these two views seem to be in manifest conflict with each other. However, our conclusion is that it is very difficult to see any real difference between the two. We analyze a phenomenonwe call internal categoricity which extends the familiar categoricity results of second order logic to Henkin models and show that set theory enjoys the same kind of internal categoricity. Thus the existence of non-standard models, which is usually taken as a property of first order set theory, and categoricity, which is usually taken as a property of second order axiomatizations, can coherently coexist when put into their proper context. We also take a fresh look at complete second order axiomatizations and give a hierarchy result for second order characterizable structures. Finally we consider the problem of existence in mathematics from both points of view and find that second order logic depends on what we call large domain assumptions, which come quite close to the meaning of the axioms of set theory.

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