The nonconstructive content of sentences of arithmetic

1978 ◽  
Vol 43 (3) ◽  
pp. 497-501
Author(s):  
Nicolas D. Goodman
Keyword(s):  
Natural Number ◽  
Recursive Function ◽  
Function Theory ◽  
First Order ◽  
The Standard Model ◽  
Image Position ◽  
The Relationship ◽  
Excluded Middle ◽  
Degrees Of Unsolvability ◽  
Intuitionistic Arithmetic

This note is concerned with the old topic, initiated by Kleene, of the connections between recursive function theory and provability in intuitionistic arithmetic. More specifically, we are interested in the relationship between the hierarchy of degrees of unsolvability and the interdeducibility of cases of excluded middle. The work described below was motivated by a counterexample, to be given presently, which shows that that relationship is more complicated than one might suppose.Let HA be first-order intuitionistic arithmetic. Let the symbol ⊢ mean derivability in HA. For each natural number n, let n¯ be the corresponding numeral. Let Ω be the standard model of arithmetic. Say that a sentence ϕ is true iff Ω⊨ ϕ. Now suppose ϕ(x) and Ψ(x) are formulas with only the variable x free. SupposeThen it is natural to conjecture that {n∣Ω⊨Ψ(n¯)} is recursive in {n∣Ω⊨ϕ(n¯)}.However, this conjecture is false. Consider the formula is a formalization of Kleene's T-predicate.

Note on an induction axiom

1978 ◽  
Vol 43 (1) ◽  
pp. 113-117
Author(s):  
J. B. Paris
Keyword(s):  
Physical World ◽  
First Order ◽  
Naturally Occurring ◽  
Order Language ◽  
Image Position ◽  
The Relationship

Let θ(ν) be a formula in the first-order language of arithmetic and letIn this note we study the relationship between the schemas I′ and I+.Our interest in I+ lies in the fact that it is ostensibly a more reasonable schema than I′. For, if we believe the hypothesis of I+(θ) then to verify θ(n) only requires at most 2log2(n) steps, whereas assuming the hypothesis of I′(θ) we require n steps to verify θ(n). In the physical world naturally occurring numbers n rarely exceed 10100. For such n applying 2log2(n) steps is quite feasible whereas applying n steps may well not be.Of course this is very much an anthropomorphic argument so we would expect that it would be most likely to be valid when we restrict our attention to relatively simple formulas θ. We shall show that when restricted to open formulas I+ does not imply I′ but that this fails for the classes Σn, Πn, n ≥ 0.We shall work in PA−, where PA− consists of Peano's Axioms less induction together with∀u, w(u + w = w + u ∧ u · w = w · u),∀u, w, t ((u + w) + t = u + (w + t) ∧ (u · w) · t = u · (w · t)),∀u, w, t(u · (w + t) = u · w + u · t),∀u, w(u ≤ w ↔ ∃t(u + t = w)),∀u, w(u ≤ w ∨ w ≤ u),∀u, w, t(u + w = u + t → w = t).The reasons for working with PA− rather than Peano's Axioms less induction is that our additional axioms, whilst intuitively reasonable, will not necessarily follow from some of the weaker forms of I+ which we shall be considering. Of course PA− still contains those Peano Axioms which define + andNotice that, trivially, PA− ⊦ I′(θ) → I+(θ) for any formula θ.

Effective coloration

1976 ◽  
Vol 41 (2) ◽  
pp. 469-480 ◽  
Author(s):  
Dwight R. Bean
Keyword(s):  
Graph Theory ◽  
Planar Graph ◽  
Recursive Function ◽  
Function Theory ◽  
Yield Information ◽  
Countably Infinite ◽  
Degrees Of Unsolvability ◽  
Infinite Planar

AbstractWe are concerned here with recursive function theory analogs of certain problems in chromatic graph theory. The motivating question for our work is: Does there exist a recursive (countably infinite) planar graph with no recursive 4-coloring? We obtain the following results: There is a 3-colorable, recursive planar graph which, for all k, has no recursive k-coloring; every decidable graph of genus p ≥ 0 has a recursive 2(x(p) − 1)-coloring, where x(p) is the least number of colors which will suffice to color any graph of genus p; for every k ≥ 3 there is a k-colorable, decidable graph with no recursive k-coloring, and if k = 3 or if k = 4 and the 4-color conjecture fails the graph is planar; there are degree preserving correspondences between k-colorings of graphs and paths through special types of trees which yield information about the degrees of unsolvability of k-colorings of graphs.

The diversity of quantifier prefixes

1973 ◽  
Vol 38 (1) ◽  
pp. 79-85 ◽  
Author(s):  
H. Jerome Keisler ◽  
Wilbur Walkoe
Keyword(s):  
Positive Integer ◽  
Analogous Result ◽  
Predicate Logic ◽  
Finite Sequence ◽  
Arithmetical Hierarchy ◽  
First Order ◽  
Free Variables ◽  
The Standard Model ◽  
Image Position ◽  
Identity Symbol

The Arithmetical Hierarchy Theorem of Kleene [1] states that in the complete theory of the standard model of arithmetic there is for each positive integer r a Σr0 formula which is not equivalent to any Πr0 formula, and a Πr0 formula which is not equivalent to any Πr0 formula. A Πr0 formula is a formula of the formwhere φ has only bounded quantifiers; Πr0 formulas are defined dually.The Linear Prefix Theorem in [3] is an analogous result for predicate logic. Consider the first order predicate logic L with identity symbol, countably many n-placed relation symbols for each n, and no constant or function symbols. A prefix is a finite sequenceof quantifier symbols ∃ and ∀, for example ∀∃∀∀∀∃. By a Q formula we mean a formula of L of the formwhere v1, …, vr are distinct variables and φ has no quantifiers. A sentence is a formula with no free variables. The Linear Prefix Theorem is as follows.Linear Prefix Theorem. Let Q and q be two different prefixes of the same length r. Then there is a Q sentence which is not logically equivalent to any q sentence.Moreover, for each s there is a Q formula with s free variables which is not logically equivalent to any q formula with s free variables.For example, there is an ∀∃∀∀∀∃ sentence which is not logically equivalent to any ∀∃∃∀∀∃ sentence, and vice versa. Recall that in arithmetic two consecutive ∃'s or ∀'s can be collapsed; for instance all ∀∃∀∀∀∃ and ∀∃∃∀∀∃ formulas are logically equivalent to Π40 formulas. But the Linear Prefix Theorem shows that in predicate logic the number of quantifiers in each block, as well as the number of blocks, counts.

Fragments of Heyting arithmetic

2000 ◽  
Vol 65 (3) ◽  
pp. 1223-1240 ◽  
Author(s):  
Wolfgang Burr
Keyword(s):  
First Order ◽  
Recursive Functions ◽  
Heyting Arithmetic ◽  
Universal Quantification ◽  
Law Of Excluded Middle ◽  
The Law ◽  
Order Arithmetic ◽  
Excluded Middle ◽  
Intuitionistic Arithmetic

AbstractWe define classes Φn of formulae of first-order arithmetic with the following properties:(i) Every φ ϵ Φn is classically equivalent to a Πn-formula (n ≠ 1, Φ1 := Σ1).(ii) (iii) IΠn and iΦn (i.e., Heyting arithmetic with induction schema restricted to Φn-formulae) prove the same Π2-formulae.We further generalize a result by Visser and Wehmeier. namely that prenex induction within intuitionistic arithmetic is rather weak: After closing Φn both under existential and universal quantification (we call these classes Θn) the corresponding theories iΘn still prove the same Π2-formulae. In a second part we consider iΔ0 plus collection-principles. We show that both the provably recursive functions and the provably total functions of are polynomially bounded. Furthermore we show that the contrapositive of the collection-schema gives rise to instances of the law of excluded middle and hence .

scholarly journals RAMSEY’S THEOREM FOR PAIRS ANDKCOLORS AS A SUB-CLASSICAL PRINCIPLE OF ARITHMETIC

2017 ◽  
Vol 82 (2) ◽  
pp. 737-753
Author(s):  
STEFANO BERARDI ◽  
SILVIA STEILA
Keyword(s):  
Natural Number ◽  
Ramsey's Theorem ◽  
Recursively Enumerable ◽  
Ramsey’S Theorem ◽  
Heyting Arithmetic ◽  
Classical Principle ◽  
Image Position ◽  
Intuitionistic Arithmetic

AbstractThe purpose is to study the strength of Ramsey’s Theorem for pairs restricted to recursive assignments ofk-many colors, with respect to Intuitionistic Heyting Arithmetic. We prove that for every natural number$k \ge 2$, Ramsey’s Theorem for pairs and recursive assignments ofkcolors is equivalent to the Limited Lesser Principle of Omniscience for${\rm{\Sigma }}_3^0$formulas over Heyting Arithmetic. Alternatively, the same theorem over intuitionistic arithmetic is equivalent to: for every recursively enumerable infinitek-ary tree there is some$i < k$and some branch with infinitely many children of indexi.

A problem in recursive function theory

1953 ◽  
Vol 18 (3) ◽  
pp. 225-232
Author(s):  
R. L. Goodstein
Keyword(s):  
Classical Theory ◽  
Recursive Function ◽  
Present Note ◽  
Function Theory ◽  
Single Point ◽  
Satisfying Condition ◽  
Mean Value ◽  
Classical Analysis ◽  
Rolle’S Theorem ◽  
Image Position

In a recent paper [4] on mean value theorems in recursive function theory we proved the theorem that(A) iff(n, x) is relatively differentiable with a relative derivative f1(n, x), for a ≤ x ≤ b, and if f(n, a) = f(n, b) = 0 relative to n,then there is a recursive function ck, a < ck < b, and a recursive R(k) such that f1(n, ck) = 0(k) for n ≥ R(k); and we showed further that the added condition(B) f(n, x) is either relatively variable or relatively constantsuffices to ensure that ck is uniformly contained in (a, b), i.e. that there exist α, β such thatA comparison with the conditions under which Rolle's theorem is established in classical analysis suggests that clause (A) itself might suffice to ensure that ck is uniformly contained in (a, b); for in the classical theory there is a single point c, a < c < b, for which lim f1(n, c) = 0, and therefore f1(n, c) = 0(k) for sufficiently great values of n, where of course c is independent of k.The object of the present note is to show that this is not in fact the case, and we shall construct a recursive function f(n, x) satisfying condition (A) in the interval (0, 1) and such that any sequence ck for which f1(n, ck) = 0(k) for large enough values of n is not uniformly contained in (0, 1).

Kleene's Amazing Second Recursion Theorem

2010 ◽  
Vol 16 (2) ◽  
pp. 189-239 ◽  
Author(s):  
Yiannis N. Moschovakis
Keyword(s):  
Natural Number ◽  
Recursive Function ◽  
Short Note ◽  
Partial Function ◽  
Modern Notation ◽  
Recursion Theorem ◽  
Image Position

This little gem is stated unbilled and proved (completely) in the last two lines of §2 of the short note Kleene [1938]. In modern notation, with all the hypotheses stated explicitly and in a strong (uniform) form, it reads as follows:Second Recursion Theorem (SRT). Fix a set V ⊆ ℕ, and suppose that for each natural number n ϵ ℕ = {0, 1, 2, …}, φn: ℕ1+n ⇀ V is a recursive partial function of (1 + n) arguments with values in V so that the standard assumptions (a) and (b) hold with.(a) Every n-ary recursive partial function with values in V is for some e.(b) For all m, n, there is a recursive function : Nm+1 → ℕ such that.Then, for every recursive, partial function f of (1+m+n) arguments with values in V, there is a total recursive function of m arguments such thatProof. Fix e ϵ ℕ such that and let .We will abuse notation and write ž; rather than ž() when m = 0, so that (1) takes the simpler formin this case (and the proof sets ž = S(e, e)).

A note on the representation of general recursive functions and the μ quantifier

1959 ◽  
Vol 55 (2) ◽  
pp. 145-148
Author(s):  
Alan Rose
Keyword(s):  
Natural Number ◽  
Recursive Function ◽  
Natural Numbers ◽  
General Recursive Function ◽  
Recursive Functions ◽  
Image Position ◽  
Primitive Recursive

It has been shown that every general recursive function is definable by application of the five schemata for primitive recursive functions together with the schemasubject to the condition that, for each n–tuple of natural numbers x1,…, xn there exists a natural number xn+1 such that

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.

scholarly journals Simulations of a bubble wall interacting with an electroweak plasma

2021 ◽  
Vol 2021 (2) ◽  
Author(s):  
Zong-Gang Mou ◽  
Paul M. Saffin ◽  
Anders Tranberg
Keyword(s):  
Large Scale ◽  
Baryon Asymmetry ◽  
Winding Number ◽  
Bubble Wall ◽  
Electroweak Phase Transition ◽  
First Order ◽  
The Standard Model ◽  
Technical Details ◽  
Real Time Simulations ◽  
Chern Simons

Abstract We perform large-scale real-time simulations of a bubble wall sweeping through an out-of-equilibrium plasma. The scenario we have in mind is the electroweak phase transition, which may be first order in extensions of the Standard Model, and produce such bubbles. The process may be responsible for baryogenesis and can generate a background of primordial cosmological gravitational waves. We study thermodynamic features of the plasma near the advancing wall, the generation of Chern-Simons number/Higgs winding number and consider the potential for CP-violation at the wall generating a baryon asymmetry. A number of technical details necessary for a proper numerical implementation are developed.

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