An incorrect theorem

1965 ◽  
Vol 30 (1) ◽  
pp. 27-27 ◽  
Author(s):  
Theodore Hailperin
Keyword(s):  
Natural Numbers ◽  
Fourth Edition ◽  
Counter Example ◽  
Image Position ◽  
The Given

In the fourth edition of Hilbert-Ackermann's Grundzüge der Theoretischen Logik (Springer-Verlag: Berlin-Göttingen-Heidelberg 1959) the following assertion is “proved” (p. 131):Ein Ausdruck, der keine freien Individuenvariablen enthält, ist dann und nur dann allgemeingültig, wenn allgemeingültig ist. Die Ausdrücke, die die Identität enthalten, sind dabei eingeschlossen.As a counter example one can cite: ∃x(¬∃xFx ∨ Fx), which is universally valid yet the generalized form: ∀x(¬∃xFx ∨ Fx) is not (e.g. let the domain of individuals be the natural numbers and let ‘Fx’ be ‘x is factorable’).It may be of interest to see where the alleged proof went astray. It is argued (correctly) that any expression can only be derived by the given rules from a formula , or from this formula with any other variable in place of y. Moreover (still correctly) any upper formula of such a formula is of the form (“Upper formula” in the transitive sense of an upper formula of an upper formula being an upper formula). But then it is stated (incorrectly) that if is an upper formula of , then is an upper formula of . To show that this is incorrect we here append a proof of the universally valid formula mentioned above:

Solution of a problem of Tarski

1956 ◽  
Vol 21 (1) ◽  
pp. 49-51 ◽  
Author(s):  
John Myhill
Keyword(s):  
Unary Operation ◽  
Negative Answer ◽  
Operation Symbol ◽  
Counter Example ◽  
Recursively Enumerable ◽  
Consistent Extension ◽  
Axiomatizable Theory ◽  
Definition Of ◽  
Image Position ◽  
The Given

We presuppose the terminology of [1], and we give a negative answer to the following problem ([1], p. 19): Does every essentially undecidable axiomatizable theory have an essentially undecidable finitely axiomatizable subtheory?We use the following theorem of Kleene ([2], p. 311). There exist two recursively enumerable sets α and β such that (1) α and β are disjoint (2) there is no recursive set η for which α ⊂ η, β ⊂ η′. By the definition of recursive enumerability, there are recursive predicates Φ and Ψ for whichWe now specify a theory T which will afford a counter-example to the given problem of Tarski. The only non-logical constants of T are two binary predicates P and Q, one unary operation symbol S, and one individual constant 0. As in ([1], p. 52) we defineThe only non-logical axioms of T are the formulae P(Δm, Δn) for all pairs of integers m, n satisfying Δ(m, n); the formulae Q(Δm, Δn) for all pairs of integers m, n satisfying Ψ(m, n); and the formulaT is consistent, since it has a model. It remains to show that (1) every consistent extension of T is undecidable (2) if T1 is a finitely axiomatizable subtheory of T, there exists a consistent and decidable extension of T1 which has the same constants as T1.

scholarly journals XXV.—The Generating Function of the Reciprocal of a Determinant

1905 ◽  
Vol 40 (3) ◽  
pp. 615-629
Author(s):  
Thomas Muir
Keyword(s):  
Generating Function ◽  
Specific Character ◽  
Partial Fraction ◽  
Homogeneous Functions ◽  
Series Form ◽  
General Proposition ◽  
In Series ◽  
Image Position ◽  
The Given

(1) This is a subject to which very little study has been directed. The first to enunciate any proposition regarding it was Jacobi; but the solitary result which he reached received no attention from mathematicians,—certainly no fruitful attention,—during seventy years following the publication of it.Jacobi was concerned with a problem regarding the partition of a fraction with composite denominator (u1 − t1) (u2 − t2) … into other fractions whose denominators are factors of the original, where u1, u2, … are linear homogeneous functions of one and the same set of variables. The specific character of the partition was only definable by viewing the given fraction (u1−t1)−1 (u2−t2)−1…as expanded in series form, it being required that each partial fraction should be the aggregate of a certain set of terms in this series. Of course the question of the order of the terms in each factor of the original denominator had to be attended to at the outset, since the expansion for (a1x+b1y+c1z−t)−1 is not the same as for (b1y+c1z+a1x−t)−1. Now one general proposition to which Jacobi was led in the course of this investigation was that the coefficient ofx1−1x2−1x3−1…in the expansion ofy1−1u2−1u3−1…, whereis |a1b2c3…|−1, provided that in energy case the first term of uris that containing xr.

scholarly journals A class of harmonically convergent sets

1975 ◽  
Vol 20 (3) ◽  
pp. 301-304
Author(s):  
Torleiv Kløve
Keyword(s):  
Natural Numbers ◽  
Lower Case ◽  
Image Position

Following Craven (1965) we say that a set M of natural numbers is harmonically convergent if converges, and we call μ(M) the harmonic sum of M. (Craven defined these concepts for sequences rather than sets, but we found it convenient to work with sets.) Throughout this paper, lower case italics denote non-negative integers.

scholarly journals Computability and the algebra of fields: Some affine constructions

1980 ◽  
Vol 45 (1) ◽  
pp. 103-120 ◽  
Author(s):  
J. V. Tucker
Keyword(s):  
Algebraic Structure ◽  
Algebraic System ◽  
Finiteness Condition ◽  
Coordinate Systems ◽  
Natural Numbers ◽  
Algebraic Foundation ◽  
Image Position ◽  
Technical Idea ◽  
Natural Way ◽  
General Method

A natural way of studying the computability of an algebraic structure or process is to apply some of the theory of the recursive functions to the algebra under consideration through the manufacture of appropriate coordinate systems from the natural numbers. An algebraic structure A = (A; σ1,…, σk) is computable if it possesses a recursive coordinate system in the following precise sense: associated to A there is a pair (α, Ω) consisting of a recursive set of natural numbers Ω and a surjection α: Ω → A so that (i) the relation defined on Ω by n ≡α m iff α(n) = α(m) in A is recursive, and (ii) each of the operations of A may be effectively followed in Ω, that is, for each (say) r-ary operation σ on A there is an r argument recursive function on Ω which commutes the diagramwherein αr is r-fold α × … × α.This concept of a computable algebraic system is the independent technical idea of M.O.Rabin [18] and A.I.Mal'cev [14]. From these first papers one may learn of the strength and elegance of the general method of coordinatising; note-worthy for us is the fact that computability is a finiteness condition of algebra—an isomorphism invariant possessed of all finite algebraic systems—and that it serves to set upon an algebraic foundation the combinatorial idea that a system can be combinatorially presented and have effectively decidable term or word problem.

On relations as coextensive with classes

1946 ◽  
Vol 11 (3) ◽  
pp. 71-72 ◽  
Author(s):  
W. V. Quine
Keyword(s):  
Natural Numbers ◽  
Previous Note ◽  
Image Position ◽  
Ordered Pairs

In a previous note I showed a new way to define the ordered pair. I made use of the notations ‘Nn’ (for class of natural numbers) and ‘Sv’ (for successor of v), remarking that they are readily defined without appeal to ordered pairs or relations. Adopting the auxiliary abbreviation: I defined the ordered pair thus:

Inequalities for permanents and permanental minors

1965 ◽  
Vol 61 (3) ◽  
pp. 741-746 ◽  
Author(s):  
R. A. Brualdi ◽  
M. Newman
Keyword(s):  
Convex Function ◽  
Convex Set ◽  
Identity Matrix ◽  
Stochastic Matrices ◽  
Doubly Stochastic ◽  
Counter Example ◽  
Doubly Stochastic Matrices ◽  
Permanent Function ◽  
Image Position

Let Ωndenote the convex set of alln×ndoubly stochastic matrices: chat is, the set of alln×nmatrices with non-negative entries and row and column sums 1. IfA= (aij) is an arbitraryn×nmatrix, then thepermanentofAis the scalar valued function ofAdefined bywhere the subscriptsi1,i2, …,inrun over all permutations of 1, 2, …,n. The permanent function has been studied extensively of late (see, for example, (1), (2), (3), (4), (6)) and it is known that ifA∈ Ωnthen 0 <cn≤ per (A) ≤ 1, where the constantcndepends only onn. It is natural to inquire if per (A) is a convex function ofAforA∈ Ωn. That this is not the case was shown by a counter-example given by Marcus and quoted by Perfect in her paper ((5)). In this paper, however, she shows that per (½I+ ½A) ≤ ½ + ½ per (A) for allA∈ Ωn. HereI=Inis the identity matrix of ordern.

k-Degenerate Graphs

1970 ◽  
Vol 22 (5) ◽  
pp. 1082-1096 ◽  
Author(s):  
Don R. Lick ◽  
Arthur T. White
Keyword(s):  
Bipartite Graph ◽  
Chromatic Number ◽  
Complete Graph ◽  
Planar Graphs ◽  
Complete Bipartite Graph ◽  
Natural Numbers ◽  
Image Position ◽  
Disconnected Graphs ◽  
Totally Disconnected

Graphs possessing a certain property are often characterized in terms of a type of configuration or subgraph which they cannot possess. For example, a graph is totally disconnected (or, has chromatic number one) if and only if it contains no lines; a graph is a forest (or, has point-arboricity one) if and only if it contains no cycles. Chartrand, Geller, and Hedetniemi [2] defined a graph to have property Pn if it contains no subgraph homeomorphic from the complete graph Kn+1 or the complete bipartite graphFor the first four natural numbers n, the graphs with property Pn are exactly the totally disconnected graphs, forests, outerplanar and planar graphs, respectively. This unification suggested the extension of many results known to hold for one of the above four classes of graphs to one or more of the remaining classes.

scholarly journals Note on the Application of Compound Poisson Processes to Sickness and Accident Statistics

1960 ◽  
Vol 1 (4) ◽  
pp. 224-237 ◽  
Author(s):  
Carl Philipson
Keyword(s):  
Random Process ◽  
Conditional Probability ◽  
Compound Poisson Process ◽  
Compound Poisson ◽  
Continuous Parameter ◽  
Absolute Probability ◽  
Image Position ◽  
The Given ◽  
Accident Statistics ◽  
Uniformly Bounded

In order to fix our ideas an illustration of the theory for (a) a general elementary random process, (b) a compound Poisson process and (c) a Polya process shall be given here below following Ove Lundberg (On Random Processes and Their Application to Accident and Sickness Statistics, Inaug. Diss., Uppsala 1940).Let the continuous parameter t* be measured on an absolute scale from a given point of zero and consider the random function N* (t*) which takes only non-negative and integer values with N* (o) = o. This function constitutes a general elementary random process for which the conditional probability that N* (t*) = n relative to the hypothesis that shall be denoted , while the absolute probability that N* (t) = n i.e. shall be written If quantities of lower order than dt* are neglected, we may write for the conditional probability that N* (t* + dt*) = n + 1 relative to thehyp othesis that N* (t*) = n, i.e. is the intensity function of the process which is assumed to be a continuous function of t* (the condition of existence for the integral over the given interval of t* for every n > m may be substituted for the condition of continuity). The expectations for an arbitrary but fix value of t* of N* (t*) and p* (t*) will be denoted by the corresponding symbol with a bar so thatIf is uniformly bounded for all n in the interval o ≤ t* < T*, where T* is an arbitrary but fix value of t*, we have i.a. that

scholarly journals Some Problems Connected with the Calculation of Stop Loss Premiums for Large Portfolios

1967 ◽  
Vol 4 (2) ◽  
pp. 170-174 ◽  
Author(s):  
Fredrik Esscher
Keyword(s):  
Distribution Function ◽  
Expected Number ◽  
Limit Values ◽  
Empirical Determination ◽  
The Mean ◽  
Premium Rates ◽  
Image Position ◽  
Stop Loss ◽  
The Given

When experience is insufficient to permit a direct empirical determination of the premium rates of a Stop Loss Cover, we have to fall back upon mathematical models from the theory of probability—especially the collective theory of risk—and upon such assumptions as may be considered reasonable.The paper deals with some problems connected with such calculations of Stop Loss premiums for a portfolio consisting of non-life insurances. The portfolio was so large that the values of the premium rates and other quantities required could be approximated by their limit values, obtained according to theory when the expected number of claims tends to infinity.The calculations were based on the following assumptions.Let F(x, t) denote the probability that the total amount of claims paid during a given period of time is ≤ x when the expected number of claims during the same period increases from o to t. The net premium II (x, t) for a Stop Loss reinsurance covering the amount by which the total amount of claims paid during this period may exceed x, is defined by the formula and the variance of the amount (z—x) to be paid on account of the Stop Loss Cover, by the formula As to the distribution function F(x, t) it is assumed that wherePn(t) is the probability that n claims have occurred during the given period, when the expected number of claims increases from o to t,V(x) is the distribution function of the claims, giving the conditioned probability that the amount of a claim is ≤ x when it is known that a claim has occurred, andVn*(x) is the nth convolution of the function V(x) with itself.V(x) is supposed to be normalized so that the mean = I.

Differences, Derivatives, and Decreasing Rearrangements

1967 ◽  
Vol 19 ◽  
pp. 1153-1178 ◽  
Author(s):  
G. F. D. Duff
Keyword(s):  
Finite Sequence ◽  
Real Numbers ◽  
Decreasing Rearrangement ◽  
Image Position ◽  
Decreasing Rearrangements ◽  
The Given

The decreasing rearrangement of a finite sequence a1, a2, … , an of real numbers is a second sequence aπ(1), aπ(2), … , aπ(n), where π(l), π(2), … , π(n) is a permutation of 1, 2, … , n and(1, p. 260). The kth term of the rearranged sequence will be denoted by . Thus the terms of the rearranged sequence correspond to and are equal to those of the given sequence ak, but are arranged in descending (non-increasing) order.

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