A NOTE ON PREDICATIVE ORDINAL ANALYSIS I: ITERATED COMPREHENSION AND TRANSFINITE INDUCTION

2019 ◽  
Vol 84 (1) ◽  
pp. 226-265 ◽  
Author(s):  
SATO KENTARO
Keyword(s):  
Second Order ◽  
Transfinite Induction ◽  
Ordinal Analysis ◽  
First Order ◽  
Image Position

AbstractWe determine the proof-theoretic ordinals (i) of ${\cal C} - {\bf{TI}}[\alpha ]$, the transfinite induction along α, for any hyperarithmetical level ${\cal C}$, in the first order setting and (ii) of any combination of iterated arithmetical comprehension and ${\cal C} - {\bf{TI}}[\alpha ]$ for ${\cal C}\, \equiv \,{\rm{\Pi }}_k^i ,{\rm{\Sigma }}_k^i$ ($i\, = \,0,1$) in the second order setting.

A theorem on deducibility for second-order functions

1939 ◽  
Vol 4 (2) ◽  
pp. 77-79 ◽  
Author(s):  
C. H. Langford
Keyword(s):  
Second Order ◽  
Order Function ◽  
Usual Definition ◽  
First Order ◽  
Free Variables ◽  
Usual Convention ◽  
Definition Of ◽  
Image Position

It is known that the usual definition of a dense series without extreme elements is complete with respect to first-order functions, in the sense that any first-order function on the base of a set of postulates defining such a series either is implied by the postulates or is inconsistent with them. It is here understood, in accordance with the usual convention, that when we speak of a function on the base , the function shall be such as to place restrictions only upon elements belonging to the class determined by f; or, more exactly, every variable with a universal prefix shall occur under the hypothesis that its values satisfy f, while every variable with an existential prefix shall have this condition categorically imposed upon it.Consider a set of postulates defining a dense series without extreme elements, and add to this set the condition of Dedekind section, to be formulated as follows. Let the conjunction of the three functions,be written H(ϕ), where the free variables f and g, being parameters throughout, are suppressed. This is the hypothesis of Dedekind's condition, and the conclusion iswhich may be written C(ϕ).

Generalized interpolation theorems

1972 ◽  
Vol 37 (2) ◽  
pp. 343-351
Author(s):  
Stephen J. Garland
Keyword(s):  
Interpolation Theorem ◽  
Second Order ◽  
Cut Elimination ◽  
Real Numbers ◽  
Craig Interpolation ◽  
First Order ◽  
Order Variable ◽  
Analytic Sets ◽  
Image Position ◽  
Special Case

Chang [1], [2] has proved the following generalization of the Craig interpolation theorem [3]: For any first-order formulas φ and ψ with free first- and second-order variables among ν1, …, νn, R and ν1, …, νn, S respectively, and for any sequence Q1, …, Qn of quantifiers such that Q1 is universal whenever ν1 is a second-order variable, ifthen there is a first-order formula θ with free variables among ν1, …, νn such that(Note that the Craig interpolation theorem is the special case of Chang's theorem in which Q1, …, Qn are all universal quantifiers.) Chang also raised the question [2, Remark (k)] as to whether the Lopez-Escobar interpolation theorem [6] for the infinitary language Lω1ω possesses a similar generalization. In this paper, we show that the answer to Chang's question is affirmative and, moreover, that several interpolation theorems for applied second-order languages for number theory also possess such generalizations.Maehara and Takeuti [7] have established independently proof-theoretic interpolation theorems for first-order logic and Lω1ω which have as corollaries both Chang's theorem and its analog for Lω1ω. Our proofs are quite different from theirs and rely on model-theoretic techniques stemming from the analogy between the theory of definability in Lω1ω and the theory of Borel and analytic sets of real numbers, rather than the technique of cut-elimination.

Ordinal bounds for κ-consistency

1974 ◽  
Vol 39 (4) ◽  
pp. 693-699 ◽  
Author(s):  
Warren D. Goldfarb
Keyword(s):  
Nonnegative Integer ◽  
Transfinite Induction ◽  
Axiom Schema ◽  
First Order ◽  
Order Arithmetic ◽  
Definition Of ◽  
Image Position ◽  
Primitive Recursive

In [1] the ω-consistency of arithmetic was proved by a method which yields fine ordinal bounds for κ-consistency, κ ≥ 1. In this paper these bounds are shown to be best possible. The ω-consistency of a number-theoretic system S can be expressed thus: for all sentences ∃xM,where ProvS is the proof predicate for S, if n is a nonnegative integer then n is the formal numeral (of S) for n, and if G is a formula then ˹G˺ is the Gödel number of G. The κ-consistency of S is the restriction of (1) to Σκ0 sentences ∃xM. The proof in [1] establishes the no-counterexample interpretation of (1), that is, the existence of a constructive functional Φ such that, for all sentences ∃xM, all numbers p, and all functions f,(see [1, §2]). A functional Φ is an ω-consistency functional for S if it satisfies (2) for all sentences ∃xM, and a κ-consistency functional for S if it satisfies (2) for all Σκ0 sentences ∃xM.The systems considered in [1] are those obtained from classical first-order arithmetic Z, including the schema for definition of primitive recursive (p.r.) functions, by adjoining, for some p.r. well-founded ordering ≺ of the nonnegative integers, the axiom schemathat is, the least number principle on ≺; it is equivalent to the schema of transfinite induction on ≺.

Two interpolation theorems for a predicate calculus

1971 ◽  
Vol 36 (2) ◽  
pp. 262-270
Author(s):  
Shoji Maehara ◽  
Gaisi Takeuti
Keyword(s):  
Normal Form ◽  
Completeness Theorem ◽  
Interpolation Theorem ◽  
Second Order ◽  
Predicate Calculus ◽  
Cut Elimination ◽  
First Order ◽  
Image Position ◽  
Order Predicate Calculus

A second order formula is called Π1 if, in its prenex normal form, all second order quantifiers are universal. A sequent F1, … Fm → G1 …, Gn is called Π1 if a formulais Π1If we consider only Π1 sequents, then we can easily generalize the completeness theorem for the cut-free first order predicate calculus to a cut-free Π1 predicate calculus.In this paper, we shall prove two interpolation theorems on the Π1 sequent, and show that Chang's theorem in [2] is a corollary of our theorem. This further supports our belief that any form of the interpolation theorem is a corollary of a cut-elimination theorem. We shall also show how to generalize our results for an infinitary language. Our method is proof-theoretic and an extension of a method introduced in Maehara [5]. The latter has been used frequently to prove the several forms of the interpolation theorem.

An Expansion Theorem for a Pair of Singular First Order Equations

1954 ◽  
Vol 6 ◽  
pp. 554-560 ◽  
Author(s):  
S. D. Conte ◽  
W. C. Sangren
Keyword(s):  
Classical Method ◽  
Complex Variables ◽  
Order Equation ◽  
Second Order ◽  
Expansion Theorem ◽  
First Order ◽  
Image Position

Titchmarsh (4) has shown how the classical method of complex variables can be used to obtain expansion theorems for the singular cases of the second order equation(1) .The purpose of this paper is to indicate how these results can be generalized to the singular cases of the pair of first order equations,

scholarly journals Generalised Algebraic Continued Fractions related to Definite Integrals

1958 ◽  
Vol 9 (4) ◽  
pp. 170-182
Author(s):  
L. R. Shenton
Keyword(s):  
Recurrence Relation ◽  
Fourth Order ◽  
Continued Fractions ◽  
Recurrence Formula ◽  
Second Order ◽  
Third Order ◽  
First Order ◽  
Definite Integrals ◽  
Image Position ◽  
Remarkable Relation

The present paper is a continuation of the work initiated in [l]-[5]. In [5] I gave an expansion of the formfor the second order C.F. associated withwhere U8, V8, W8 satisfy a fourth-order recurrence relation, there being a similar expansion for third order C.F.'s. I shall now give simple expressions for U8, V8, W8 (or related forms) in terms of χ2s(Z1), χ2s (Z2), ω2s(Z1), ω2s(Z2), whereand show that there is a remarkable relation between the recurrence formula for the first order C.F. and that satisfied by U3, V3, W3. The generalised form of these results will be stated and proved.

Nonlinear Analysis of Visual Evoked Potentials Elicited by Color Stimulation

1997 ◽  
Vol 36 (04/05) ◽  
pp. 315-318 ◽  
Author(s):  
K. Momose ◽  
K. Komiya ◽  
A. Uchiyama
Keyword(s):  
Visual System ◽  
Evoked Potentials ◽  
Visual Evoked Potentials ◽  
Normal Subjects ◽  
Second Order ◽  
First Order ◽  
The Relationship ◽  
Perceived Color ◽  
Second Order Nonlinearity ◽  
Visual Evoked

Abstract:The relationship between chromatically modulated stimuli and visual evoked potentials (VEPs) was considered. VEPs of normal subjects elicited by chromatically modulated stimuli were measured under several color adaptations, and their binary kernels were estimated. Up to the second-order, binary kernels obtained from VEPs were so characteristic that the VEP-chromatic modulation system showed second-order nonlinearity. First-order binary kernels depended on the color of the stimulus and adaptation, whereas second-order kernels showed almost no difference. This result indicates that the waveforms of first-order binary kernels reflect perceived color (hue). This supports the suggestion that kernels of VEPs include color responses, and could be used as a probe with which to examine the color visual system.

Imaginings

2017 ◽  
Vol 9 (3) ◽  
pp. 17-30
Author(s):  
Kelly James Clark
Keyword(s):  
Religious Tradition ◽  
Second Order ◽  
Fine Tuning ◽  
Intellectual Humility ◽  
First Order ◽  
Natural Processes ◽  
Empirically Supported ◽  
Group Violence ◽  
The World ◽  
The Universe

In Branden Thornhill-Miller and Peter Millican’s challenging and provocative essay, we hear a considerably longer, more scholarly and less melodic rendition of John Lennon’s catchy tune—without religion, or at least without first-order supernaturalisms (the kinds of religion we find in the world), there’d be significantly less intra-group violence. First-order supernaturalist beliefs, as defined by Thornhill-Miller and Peter Millican (hereafter M&M), are “beliefs that claim unique authority for some particular religious tradition in preference to all others” (3). According to M&M, first-order supernaturalist beliefs are exclusivist, dogmatic, empirically unsupported, and irrational. Moreover, again according to M&M, we have perfectly natural explanations of the causes that underlie such beliefs (they seem to conceive of such natural explanations as debunking explanations). They then make a case for second-order supernaturalism, “which maintains that the universe in general, and the religious sensitivities of humanity in particular, have been formed by supernatural powers working through natural processes” (3). Second-order supernaturalism is a kind of theism, more closely akin to deism than, say, Christianity or Buddhism. It is, as such, universal (according to contemporary psychology of religion), empirically supported (according to philosophy in the form of the Fine-Tuning Argument), and beneficial (and so justified pragmatically). With respect to its pragmatic value, second-order supernaturalism, according to M&M, gets the good(s) of religion (cooperation, trust, etc) without its bad(s) (conflict and violence). Second-order supernaturalism is thus rational (and possibly true) and inconducive to violence. In this paper, I will examine just one small but important part of M&M’s argument: the claim that (first-order) religion is a primary motivator of violence and that its elimination would eliminate or curtail a great deal of violence in the world. Imagine, they say, no religion, too.Janusz Salamon offers a friendly extension or clarification of M&M’s second-order theism, one that I think, with emendations, has promise. He argues that the core of first-order religions, the belief that Ultimate Reality is the Ultimate Good (agatheism), is rational (agreeing that their particular claims are not) and, if widely conceded and endorsed by adherents of first-order religions, would reduce conflict in the world.While I favor the virtue of intellectual humility endorsed in both papers, I will argue contra M&M that (a) belief in first-order religion is not a primary motivator of conflict and violence (and so eliminating first-order religion won’t reduce violence). Second, partly contra Salamon, who I think is half right (but not half wrong), I will argue that (b) the religious resources for compassion can and should come from within both the particular (often exclusivist) and the universal (agatheistic) aspects of religious beliefs. Finally, I will argue that (c) both are guilty, as I am, of the philosopher’s obsession with belief. 

Evidence for general base catalysis by protic solvents in a kinetic study of alcoholyses and hydrolyses of 1-(phenoxycarbonyl)pyridinium ions under both solvolytic and non-solvolytic conditions

2009 ◽  
Vol 74 (1) ◽  
pp. 43-55 ◽  
Author(s):  
Dennis N. Kevill ◽  
Byoung-Chun Park ◽  
Jin Burm Kyong
Keyword(s):  
Second Order ◽  
Base Catalysis ◽  
Substitution Reactions ◽  
Third Order ◽  
Methoxy Derivative ◽  
First Order ◽  
General Base ◽  
Protic Solvents ◽  
Kinetics Of ◽  
Pyridinium Ions

The kinetics of nucleophilic substitution reactions of 1-(phenoxycarbonyl)pyridinium ions, prepared with the essentially non-nucleophilic/non-basic fluoroborate as the counterion, have been studied using up to 1.60 M methanol in acetonitrile as solvent and under solvolytic conditions in 2,2,2-trifluoroethan-1-ol (TFE) and its mixtures with water. Under the non- solvolytic conditions, the parent and three pyridine-ring-substituted derivatives were studied. Both second-order (first-order in methanol) and third-order (second-order in methanol) kinetic contributions were observed. In the solvolysis studies, since solvent ionizing power values were almost constant over the range of aqueous TFE studied, a Grunwald–Winstein equation treatment of the specific rates of solvolysis for the parent and the 4-methoxy derivative could be carried out in terms of variations in solvent nucleophilicity, and an appreciable sensitivity to changes in solvent nucleophilicity was found.

Metaontology

2018 ◽  
Author(s):  
Uriah Kriegel
Keyword(s):  
Second Order ◽  
Order Property ◽  
First Order ◽  
First Approximation

Brentano’s theory of judgment serves as a springboard for his conception of reality, indeed for his ontology. It does so, indirectly, by inspiring a very specific metaontology. To a first approximation, ontology is concerned with what exists, metaontology with what it means to say that something exists. So understood, metaontology has been dominated by three views: (i) existence as a substantive first-order property that some things have and some do not, (ii) existence as a formal first-order property that everything has, and (iii) existence as a second-order property of existents’ distinctive properties. Brentano offers a fourth and completely different approach to existence talk, however, one which falls naturally out of his theory of judgment. The purpose of this chapter is to present and motivate Brentano’s approach.

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