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?”

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.

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.

Frege's Theorem and the Peano Postulates

1995 ◽  
Vol 1 (3) ◽  
pp. 317-326 ◽  
Author(s):  
George Boolos
Keyword(s):  
Set Theory ◽  
Cardinal Number ◽  
Formal System ◽  
Second Order ◽  
Order Logic ◽  
Russell’S Paradox ◽  
Russell's Paradox ◽  
Second Order Logic ◽  
Number Of Authors

Two thoughts about the concept of number are incompatible: that any zero or more things have a (cardinal) number, and that any zero or more things have a number (if and) only if they are the members of some one set. It is Russell's paradox that shows the thoughts incompatible: the sets that are not members of themselves cannot be the members of any one set. The thought that any (zero or more) things have a number is Frege's; the thought that things have a number only if they are the members of a set may be Cantor's and is in any case a commonplace of the usual contemporary presentations of the set theory that originated with Cantor and has become ZFC.In recent years a number of authors have examined Frege's accounts of arithmetic with a view to extracting an interesting subtheory from Frege's formal system, whose inconsistency, as is well known, was demonstrated by Russell. These accounts are contained in Frege's formal treatise Grundgesetze der Arithmetik and his earlier exoteric book Die Grundlagen der Arithmetik. We may describe the two central results of the recent re-evaluation of his work in the following way: Let Frege arithmetic be the result of adjoining to full axiomatic second-order logic a suitable formalization of the statement that the Fs and the Gs have the same number if and only if the F sand the Gs are equinumerous.

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.

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).

Categoricity and the sets

2018 ◽  
Author(s):  
Tim Button ◽  
Sean Walsh
Keyword(s):  
Set Theory ◽  
Second Order ◽  
Order Logic ◽  
First Order ◽  
The Status ◽  
Second Order Logic ◽  
Philosophy Of Set Theory ◽  
Order Set

In this chapter, the focus shifts from numbers to sets. Again, no first-order set theory can hope to get anywhere near categoricity, but Zermelo famously proved the quasi-categoricity of second-order set theory. As in the previous chapter, we must ask who is entitled to invoke full second-order logic. That question is as subtle as before, and raises the same problem for moderate modelists. However, the quasi-categorical nature of Zermelo's Theorem gives rise to some specific questions concerning the aims of axiomatic set theories. Given the status of Zermelo's Theorem in the philosophy of set theory, we include a stand-alone proof of this theorem. We also prove a similar quasi-categoricity for Scott-Potter set theory, a theory which axiomatises the idea of an arbitrary stage of the iterative hierarchy.

Algorithmic Logic with Stacks and Its Model-Theoretical Properties

1984 ◽  
Vol 7 (4) ◽  
pp. 391-428
Author(s):  
Wiktor Dańko
Keyword(s):  
Second Order ◽  
Order Logic ◽  
Algorithmic Logic ◽  
Second Order Logic

In this paper we propose to transform the Algorithmic Theory of Stacks (cf. Salwicki [30]) into a logic for expressing and proving properties of programs with stacks. We compare this logic to the Weak Second Order Logic (cf. [11, 15]) and prove theorems concerning axiomatizability without quantifiers (an analogon of Łoś-Tarski theorem) and χ 0 - categoricity (an analogon of Ryll-Nardzewski’s theorem).

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