Neo-Fregeanism and the Burali-Forti Paradox

2018 ◽  
Author(s):  
Ian Rumfitt
Keyword(s):  
Second Order ◽  
Order Logic ◽  
Abstraction Principle ◽  
Ordinal Numbers ◽  
Per Se ◽  
Cardinal Arithmetic ◽  
Second Order Logic ◽  
Predicative Logic

This chapter considers what form a neo-Fregean account of ordinal numbers might take. It begins by discussing how the natural abstraction principle for ordinals yields a contradiction (the Burali-Forti Paradox) when combined with impredicative second-order logic. It continues by arguing that the fault lies in the use of impredicative logic rather than in the abstraction principle per se. As the focus is on a form of predicative logic which reflects a philosophical diagnosis of the source of the paradox, the chapter considers how far Hale and Wright’s neo-logicist programme in cardinal arithmetic can be carried out in that logic.

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).

Monadic second-order logic on finite sequences

2017 ◽  
Vol 52 (1) ◽  
pp. 232-245
Author(s):  
Loris D'Antoni ◽  
Margus Veanes
Keyword(s):  
Second Order ◽  
Order Logic ◽  
Finite Sequences ◽  
Second Order Logic ◽  
Monadic Second Order Logic

An algebraic characterization of indistinguishable cardinals

1970 ◽  
Vol 35 (1) ◽  
pp. 97-104
Author(s):  
A. B. Slomson
Keyword(s):  
Second Order ◽  
Order Logic ◽  
Algebraic Characterization ◽  
Algebraic Criterion ◽  
Characterizable Cardinals ◽  
Second Order Logic ◽  
Interesting Theorem ◽  
Straightforward Proof

Two cardinals are said to beindistinguishableif there is no sentence of second order logic which discriminates between them. This notion, which is defined precisely below, is closely related to that ofcharacterizablecardinals, introduced and studied by Garland in [3]. In this paper we give an algebraic criterion for two cardinals to be indistinguishable. As a consequence we obtain a straightforward proof of an interesting theorem about characterizable cardinals due to Zykov [6].

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