Four Views of Arithmetical Truth

1990 ◽  
Vol 40 (159) ◽  
pp. 155 ◽  
Author(s):  
Charles Sayward
Keyword(s):  
Arithmetical Truth

Supertasks and arithmetical truth

2019 ◽  
Vol 177 (5) ◽  
pp. 1275-1282
Author(s):  
Jared Warren ◽  
Daniel Waxman
Keyword(s):  
Arithmetical Truth

scholarly journals Multi-sorted version of second order arithmetic

2016 ◽  
Vol 13 (5) ◽  
Author(s):  
Farida Kachapova
Keyword(s):  
Reverse Mathematics ◽  
Peano Arithmetic ◽  
Second Order ◽  
Natural Numbers ◽  
Natural Form ◽  
Second Order Arithmetic ◽  
Arithmetical Truth ◽  
Order Arithmetic

This paper describes axiomatic theories SA and SAR, which are versions of second order arithmetic with countably many sorts for sets of natural numbers. The theories are intended to be applied in reverse mathematics because their multi-sorted language allows to express some mathematical statements in more natural form than in the standard second order arithmetic. We study metamathematical properties of the theories SA, SAR and their fragments. We show that SA is mutually interpretable with the theory of arithmetical truth PATr obtained from the Peano arithmetic by adding infinitely many truth predicates. Corresponding fragments of SA and PATr are also mutually interpretable. We compare the proof-theoretical strengths of the fragments; in particular, we show that each fragment SAs with sorts <=s is weaker than next fragment SAs+1.

Logicism and Second-Order Logic

2020 ◽  
pp. 96-115
Author(s):  
George S. Boolos
Keyword(s):  
No Reduction ◽  
Logical Truth ◽  
Second Order ◽  
Order Logic ◽  
First Order Logic ◽  
First Order ◽  
Arithmetical Truth ◽  
Second Order Logic

This chapter outlines an argument to the effect that there is no reduction of arithmetical truth to logical truth, where “logic” is understood to be elementary (first-order) logic, or any system of logic whose theses form an effectively generable set. It suggests, however, that it leaves open the possibility of a significant reduction of arithmetic to something that might be called a system of logic. By investigating metatheoretic differences between first- and second-order logic, it explores the extent to which second-order logic might play a role in facilitating such a reduction.

scholarly journals On the Arithmetical Truth of Self-Referential Sentences

Theoria ◽  
2018 ◽  
Vol 85 (1) ◽  
pp. 8-17
Author(s):  
Kaave Lajevardi ◽  
Saeed Salehi
Keyword(s):  
Arithmetical Truth

THE STRENGTH OF RAMSEY’S THEOREM FOR COLORING RELATIVELY LARGE SETS

2014 ◽  
Vol 79 (01) ◽  
pp. 89-102 ◽  
Author(s):  
LORENZO CARLUCCI ◽  
KONRAD ZDANOWSKI
Keyword(s):  
Current Knowledge ◽  
Type Theorem ◽  
Complete Characterization ◽  
Infinite Subset ◽  
Point Of View ◽  
Ramsey's Theorem ◽  
Ramsey’S Theorem ◽  
Arithmetical Truth ◽  
Image Position

Abstract We characterize the effective content and the proof-theoretic strength of a Ramsey-type theorem for bi-colorings of so-called exactly large sets. An exactly large set is a set $X \subset {\bf{N}}$ such that ${\rm{card}}\left( X \right) = {\rm{min}}\left( X \right) + 1$ . The theorem we analyze is as follows. For every infinite subset M of N, for every coloring C of the exactly large subsets of M in two colors, there exists and infinite subset L of M such that C is constant on all exactly large subsets of L. This theorem is essentially due to Pudlák and Rödl and independently to Farmaki. We prove that—over RCA0 —this theorem is equivalent to closure under the ωth Turing jump (i.e., under arithmetical truth). Natural combinatorial theorems at this level of complexity are rare. In terms of Reverse Mathematics we give the first Ramsey-theoretic characterization of ${\rm{ACA}}_0^ +$ . Our results give a complete characterization of the theorem from the point of view of Computability Theory and of the Proof Theory of Arithmetic. This nicely extends the current knowledge about the strength of Ramsey’s Theorem. We also show that analogous results hold for a related principle based on the Regressive Ramsey’s Theorem. We conjecture that analogous results hold for larger ordinals.

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