Weight of the comprehension axiom in a theory based on logic without contractions

1999 ◽  
Vol 66 (5) ◽  
pp. 533-540
Author(s):  
V. N. Grishin
Keyword(s):  
Comprehension Axiom

scholarly journals Investigations on a comprehension axiom without negation in the defining propositional functions.

1960 ◽  
Vol 1 (1-2) ◽  
pp. 13-22 ◽  
Author(s):  
Thoralf Skolem
Keyword(s):  
Comprehension Axiom

Arithmetical Comprehension

2019 ◽  
pp. 109-129
Author(s):  
John Stillwell
Keyword(s):  
Peano Arithmetic ◽  
Sweet Spot ◽  
Natural Numbers ◽  
Axiom Schema ◽  
Axiom Systems ◽  
Comprehension Axiom

This chapter focuses on arithmetical comprehension. Arithmetical comprehension is the most obvious set existence axiom to use when developing analysis in a system based on Peano arithmetic (PA) with set variables. This axiom asserts the existence of a set X of natural numbers for each property φ‎ definable in the language of PA. More precisely, if φ‎(n) is a property defined in the language of PA plus set variables, but with no set quantifiers, then there is a set X whose members are the natural numbers n such that φ‎(n). Since all such formulas φ‎ are asserted for, the arithmetical comprehension axiom is really an axiom schema. The reason set variables are allowed in φ‎ is to enable sets to be defined in terms of “given” sets. The reason set quantifiers are disallowed in φ‎ is to avoid definitions in which a set is defined in terms of all sets of natural numbers (and hence in terms of itself). The system consisting of PA plus arithmetical comprehension is called ACA0. This system lies at a remarkable “sweet spot” among axiom systems for analysis.

Choice principles, the bar rule and autonomously iterated comprehension schemes in analysis

1983 ◽  
Vol 48 (1) ◽  
pp. 63-70 ◽  
Author(s):  
S. Feferman ◽  
G. Jäger
Keyword(s):  
General Result ◽  
Recursive Relation ◽  
Transfinite Induction ◽  
Conservative Extension ◽  
Main Result Theorem ◽  
Functional Interpretation ◽  
Result Theorem ◽  
Primitive Recursive ◽  
Comprehension Axiom

In [10] Friedman showed that (-AC) is a conservative extension of (-CA)<ε0for-sentences wherei= min(n+ 2, 4), i.e.,i= 2, 3, 4 forn= 0, 1, 2 +m. Feferman [5], [7] and Tait [11], [12] reobtained this result forn= 0, 1 and even with (-DC) instead of (-AC). Feferman and Sieg established in [9] the conservativeness of (-DC) over (-CA)<ε0for-sentences (ias above) for alln. In each paper, different methods of proof have been used. In particular, Feferman and Sieg showed how to apply familiar proof-theoretical techniques by passing through languages with Skolem functionals.In this paper we study the same choice principles in the presence of theBar Rule(BR), which permits one to infer the scheme of transfinite induction on a primitive recursive relation ≺ when it has been proved that ≺ is wellfounded. The main result (Theorem 1 below) characterizes (-DC) + (BR) as a conservative extension of a system of the autonomously iterated-comprehension axiom for-sentences (idepending onnas above). Forn= 0 this has been proved by Feferman in the form that (-DC) + (BR) is a conservative extension of (-CA)<Γ0; this was first done in [8] by use of the Gödel functional interpretation for the stronger systemZω+μ+ (QF-AC) + (BR) and then more recently by the simpler methods of [9]. Jäger showed how the latter methods could also be used to obtain the general result of Theorem 1 below.

scholarly journals A remark on a comprehension axiom without negation

1966 ◽  
Vol 42 (5) ◽  
pp. 425-426
Author(s):  
Takashi Nagashima
Keyword(s):  
Comprehension Axiom

Delineating classes of computational complexity via second order theories with weak set existence principles. I

1995 ◽  
Vol 60 (1) ◽  
pp. 103-121 ◽  
Author(s):  
Aleksandar Ignjatović
Keyword(s):  
Computational Complexity ◽  
Polynomial Time ◽  
Second Order ◽  
Complexity Classes ◽  
Bounded Arithmetic ◽  
Recursive Functions ◽  
Speed Up ◽  
Bounded Complexity ◽  
Polynomial Time Hierarchy ◽  
Comprehension Axiom

AbstractIn this paper we characterize the well-known computational complexity classes of the polynomial time hierarchy as classes of provably recursive functions (with graphs of suitable bounded complexity) of some second order theories with weak comprehension axiom schemas but without any induction schemas (Theorem 6). We also find a natural relationship between our theories and the theories of bounded arithmetic (Lemmas 4 and 5). Our proofs use a technique which enables us to “speed up” induction without increasing the bounded complexity of the induction formulas. This technique is also used to obtain an interpretability result for the theories of bounded arithmetic (Theorem 4).

scholarly journals Bi-Modal Naive Set Theory

2018 ◽  
Vol 15 (2) ◽  
pp. 139
Author(s):  
John Wigglesworth
Keyword(s):  
Set Theory ◽  
Kripke Model ◽  
Axiom Schema ◽  
Modal Operators ◽  
Modal Theory ◽  
Naive Set Theory ◽  
Naïve Comprehension ◽  
Comprehension Axiom ◽  
Forward Looking

This paper describes a modal conception of sets, according to which sets are 'potential' with respect to their members.  A modal theory is developed, which invokes a naive comprehension axiom schema, modified by adding `forward looking' and `backward looking' modal operators.  We show that this `bi-modal' naive set theory can prove modalized interpretations of several ZFC axioms, including the axiom of infinity.  We also show that the theory is consistent by providing an S5 Kripke model.  The paper concludes with some discussion of the nature of the modalities involved, drawing comparisons with noneism, the view that there are some non-existent objects.

Rainbow Ramsey Theorem for Triples is Strictly Weaker than the Arithmetical Comprehension Axiom

2013 ◽  
Vol 78 (3) ◽  
pp. 824-836 ◽  
Author(s):  
Wei Wang
Keyword(s):  
Partial Answer ◽  
Ramsey Theorem ◽  
Mathematical Result ◽  
The Given ◽  
Comprehension Axiom

AbstractWe prove that RCA0 + RRT ⊬ ACA0 where RRT is the Rainbow Ramsey Theorem for 2-bounded colorings of triples. This reverse mathematical result is based on a cone avoidance theorem, that every 2-bounded coloring of pairs admits a cone-avoiding infinite rainbow, regardless of the complexity of the given coloring. We also apply the proof of the cone avoidance theorem to the question whether RCA0 + RRT ⊦ ACA0 and obtain some partial answer.

A consistent combinatory logic with an inverse to equality

1980 ◽  
Vol 45 (3) ◽  
pp. 529-543 ◽  
Author(s):  
Frederic B. Fitch
Keyword(s):  
Mathematical Analysis ◽  
Continuous Functions ◽  
Combinatory Logic ◽  
New Haven ◽  
Yale University ◽  
Free System ◽  
Abstraction Principle ◽  
Unit Class ◽  
Excluded Middle ◽  
Comprehension Axiom

In a previous paper [9] a demonstrably consistent type-free system of combinatory logic CΔ was presented. This system contained a moderately strong exten-sionality principle, the usual equations of combinatory logic, Boolean propositional connectives, unrestricted quantifiers, an unrestricted abstraction principle (“comprehension axiom”), and a limited principle of excluded middle. It also contained elementary arithmetic in its entirety, avoiding arithmetical Gödel incompleteness [14] by being ω-complete. CΔ probably can be shown to contain certain important parts of mathematical analysis, such as the theory of continuous functions, which are contained by the somewhat similar (but nonextensional) system K′ [6], [7]. A stronger system CΓ, having these same properties and more, will be formulated in the present paper. Elsewhere [13] it will be shown that the system Q of my book, Elements of combinatory logic (Yale University Press, New Haven and London, 1974), referred to below as ECL, is essentially a subsystem of CΓ, so that the consistency of CΓ guarantees that of Q.CΓ will be stronger than both CΔ and Q in having an operator ‘ɿ’ (inverted Greek iota) which denotes the inverse of equality in the sense that ‘ɿ(= a) = a’ is a theorem of the system. Thus ‘ɿ’ denotes a function or operator which operates on a unit class to give the only member of that class. (Here ‘=a’ denotes the unit class whose only member is denoted by ‘a’.) If uninverted Greek iota, ‘ι’, is used as an abbreviation for ‘=’, the above equation could be written as ‘ɿ(ιa) = a’. Here ‘ιa’ is analogous to Bertrand Russell's well-known notation ‘ιa’, denoting a unit class. This same operator ‘ɿ’ makes it possible to transform one-many or many-one relations into the corresponding functions or operators, a kind of transformation not usually available in systems of combinatory logic [1], [2]. It also provides a method for restricting functions by restricting the corresponding relations.

scholarly journals PROOF-THEORETIC STRENGTHS OF WEAK THEORIES FOR POSITIVE INDUCTIVE DEFINITIONS

2018 ◽  
Vol 83 (3) ◽  
pp. 1091-1111 ◽  
Author(s):  
TOSHIYASU ARAI
Keyword(s):  
Transfinite Induction ◽  
Boundary Line ◽  
Natural Numbers ◽  
Inductive Definitions ◽  
Image Position ◽  
Weak Theories ◽  
Comprehension Axiom

AbstractIn this article the lightface ${\rm{\Pi }}_1^1$-Comprehension axiom is shown to be proof-theoretically strong even over ${\rm{RCA}}_0^{\rm{*}}$, and we calibrate the proof-theoretic ordinals of weak fragments of the theory ${\rm{I}}{{\rm{D}}_1}$ of positive inductive definitions over natural numbers. Conjunctions of negative and positive formulas in the transfinite induction axiom of ${\rm{I}}{{\rm{D}}_1}$ are shown to be weak, and disjunctions are strong. Thus we draw a boundary line between predicatively reducible and impredicative fragments of ${\rm{I}}{{\rm{D}}_1}$.

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