Frege's Theorem and the Peano Postulates

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.

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Categoricity and the sets

10.1093/oso/9780198790396.003.0008 ◽

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.

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The development of arithmetic in Frege's Grundgesetze der arithmetik

Journal of Symbolic Logic ◽

10.2307/2275220 ◽

1993 ◽

Vol 58 (2) ◽

pp. 579-601 ◽

Cited By ~ 32

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?”

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Krivine's classical realisability from a categorical perspective

Mathematical Structures in Computer Science ◽

10.1017/s0960129512000989 ◽

2013 ◽

Vol 23 (6) ◽

pp. 1234-1256 ◽

Cited By ~ 6

Author(s):

THOMAS STREICHER

Keyword(s):

Recent Work ◽

Set Theory ◽

Axiom Of Choice ◽

Second Order ◽

Order Logic ◽

Current Paper ◽

Categorical Approach ◽

The Axiom Of Choice ◽

Second Order Logic

In a sequence of papers (Krivine 2001; Krivine 2003; Krivine 2009), J.-L. Krivine introduced his notion of classical realisability for classical second-order logic and Zermelo–Fraenkel set theory. Moreover, in more recent work (Krivine 2008), he has considered forcing constructions on top of it with the ultimate aim of providing a realisability interpretation for the axiom of choice.The aim of the current paper is to show how Krivine's classical realisability can be understood as an instance of the categorical approach to realisability as started by Martin Hyland in Hyland (1982) and described in detail in van Oosten (2008). Moreover, we will give an intuitive explanation of the iteration of realisability as described in Krivine (2008).

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Strong Logics of First and Second Order

Bulletin of Symbolic Logic ◽

10.2178/bsl/1264433796 ◽

2010 ◽

Vol 16 (1) ◽

pp. 1-36 ◽

Cited By ~ 10

Author(s):

Peter Koellner

Keyword(s):

Set Theory ◽

Large Cardinal ◽

Second Order ◽

Order Logic ◽

First Order Logic ◽

Close Correspondence ◽

First Order ◽

Second Order Logic

AbstractIn this paper we investigate strong logics of first and second order that have certain absoluteness properties. We begin with an investigation of first order logic and the strong logics ω-logic and β-logic, isolating two facets of absoluteness, namely, generic invariance and faithfulness. It turns out that absoluteness is relative in the sense that stronger background assumptions secure greater degrees of absoluteness. Our aim is to investigate the hierarchies of strong logics of first and second order that are generically invariant and faithful against the backdrop of the strongest large cardinal hypotheses. We show that there is a close correspondence between the two hierarchies and we characterize the strongest logic in each hierarchy. On the first-order side, this leads to a new presentation of Woodin's Ω-logic. On the second-order side, we compare the strongest logic with full second-order logic and argue that the comparison lends support to Quine's claim that second-order logic is really set theory in sheep's clothing.

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Nociones logicistas en filosofía de la matemática

Crítica Revista Hispanoamericana de Filosofía ◽

10.22201/iifs.18704905e.1993.910 ◽

1993 ◽

Vol 25 (75) ◽

pp. 85-103

Author(s):

Ángel Nepomuceno Fernández

Keyword(s):

Mathematical Theory ◽

Philosophy Of Mathematics ◽

Formal System ◽

Second Order ◽

The State ◽

Order Logic ◽

Point Of View ◽

Second Order Logic ◽

Gödel’S Theorem

The point of view according to which logic has priority over mathematics has been maintained by some philosophers and mathematicians in two senses: a strong view and a weak view. Both of them are logicist: The first one regards mathematics as reducible to logic. The second one considers that the essence of mathematics can be known by researching the logical consequences of a certain system of postulates or axioms; thus, the underlying logic (whatever the mathematical theory might be) is crucial for this sort of studies. Some logicist concepts are interesting in philosophy of mathematics. So it is necessary to study the state of being in use of each point of view. In fact, the weak view has prevailed. To show that, we settle down how logicist statements have been influenced by Gödel’s theorem, though that goes against formalist philosophy. Afterwards, we present a formal system for second order logic; treatment of nominalization is enclosed, and every Frege’s law is proved to be a theorem. Finally, a short balance is made.

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Higher‐order Logic Reconsidered

10.1093/oxfordhb/9780195325928.003.0026 ◽

2009 ◽

Author(s):

Ignacio Jané

Keyword(s):

Set Theory ◽

Logical Consequence ◽

Higher Order ◽

Second Order ◽

Order Logic ◽

Higher Order Logic ◽

Consequence Relation ◽

Fourth Section ◽

Second Order Logic

This article discusses canonical (i.e., full, or standard) second-order consequence and argues against it being a case of logical consequence. The discussion is divided into three parts. The first part comprises the first three sections. After stating the problem in Section 1, Sections 2 and 3 examine the role that the consequence relation is expected to play in axiomatic theories. This leads to put forward two requirements on logical consequence, which are called “formality” and “noninterference.” It is this last requirement that canonical second-order consequence violates, as the article sets out to substantiate. The fourth section argues that canonical second-order logic is inadequate for axiomatizing set theory, on the grounds that it codes a significant amount of set-theoretical content.

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Second-Order Logic and Foundations of Mathematics

Bulletin of Symbolic Logic ◽

10.2307/2687796 ◽

2001 ◽

Vol 7 (4) ◽

pp. 504-520 ◽

Cited By ~ 42

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.

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