A system of abstract constructive ordinals

Infinitely Long Terms of Transfinite Type

which is intended to show that the concept of continuity is more subtle than one expects, because f is continuous exactly the irrational points (see [1, p. 86] or [2, p. 94]). And for the average student, this is the only property of the function f, that is, it is not interesting otherwise because there seems to be no reasonable context in which this function arises naturally. Its strange continuity behavior is only just punishment for having created a function by an artificial f-definition! But there is an interesting interpretation of this function: We consider only positive arguments, and we interpret t as the slope of a ray in the first quadrant of the x-y plane, starting the origin. Let us intercept the ray x = n, where n is any natural number chosen at random. Then f(t) is the that the y-coordinate of the interception point is also an integer, because this is the case if and only if n is divisible by q, and thus in making a great number of random guesses of the integer x-coordinate n we shall have also an integer y-coordinate in one of q cases. (Strictly speaking, there is no probability measure on N that makes the choice of each natural number equally probable! But if you choose your number n among the natural numbers 1, 2,. . ., N, randomly with respect to the uniform probability measure, your chance to get one which is divisible by q is [N/q]/N, which tends to l/q for N -> oo. So f(t) = l/q is the asymptotic density that n chosen in N is divisible by q.) Of course, this interpretation of the function f may be reformulated for a torus instead of the x-y plane. A slight modification of the function f doesn't change its continuity behavior but admits another nice interpretation: Place a square billiard table in the first quadrant of the x-y plane, with its lower-left corner the origin, and let a ball start this corner and proceed along the line with slope t > 0. Then

Predicative functionals and an interpretation of [formula omitted

We consider some very robust semi-constructive theories related to Kripke–Platek set theory, with and without the powerset operation. These theories include the law of excluded middle for bounded formulas, a form of Markov’s principle, the unrestricted collection scheme and, also, the classical contrapositive of the bounded collection scheme. We analyse these theories using forms of a functional interpretation which work in tandem with the constructible hierarchy (or the cumulative hierarchy, if the powerset operation is present). The main feature of these functional interpretations is to treat bounded quantifications as “computationally empty.” Our analysis is extended to a second-order setting enjoying some forms of class comprehension, including strict-\(\Pi ^1_1\) reflection. The key idea of the extended analysis is to treat second-order (class) quantifiers as bounded quantifiers and strict-\(\Pi ^1_1\) reflection as a form of collection. We will be able to extract some effective bounds from proofs in these systems in terms of the constructive tree ordinals up to the Bachmann–Howard ordinal.

The essence of the natural number concept is … closure under the successor operation. Richard Dedekind (1888) In this chapter we discuss a special proof technique which is particularly useful when proving statements about the positive integers. Proof by induction Suppose that we wish to prove that some property holds for all the positive integers 1, 2, 3, 4, …. It is often difficult, or even impossible, to prove such statements simply from basic rules of arithmetic. What these rules fail to capture is the fact that the positive integers come in a sequence with any number obtainable by starting from the number 1 and adding 1 to it enough times. The integer n + 1 is called the successor of the integer n . Thus, if we start with the integer 1 and form its successor, and then its successor, and so on, then given any positive integer eventually it will be reached. This idea can be formulated more precisely as follows. Axiom 5.1.1 (The induction principle) Suppose, that P(n) is a statement involving a general positive integer n. Then P(n) is true for all positive integers 1, 2, 3, … if (i) P (1) is true, and (ii) P(k) ⇒ P ( k + 1) for all positive integers k . Before discussing the induction principle further, here is an example of how it is used. Proposition 5.1.2 For all positive integers n we have the inequality n ≤ 2 n .

The aim of this article is to give an introduction to functional interpretations of set theory given by the authorin Burr (2000a). The first part starts with some general remarks on Godel’s functional interpretation with a focus on aspects related to problems that arise in the context of set theory. The second part gives an insight in the techniques needed to perform a functional interpretation of systems of set theory. However, the first part of this article is not intended to be a complete survey of functional interpretations and here we recommend, for example, Avigad and Feferman (1998),Troelstra (1990) and Troelstra (1973).

In 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 ${\rm{I}}{{\rm{D}}_1}$.

We prove in the metalanguage, by induction on natural numbers n, that derivations can be given in free Core Logic of all instances of Schema N (#xΦ‎x = n ⊣⊢ ∇nxΦ‎x). We also prove all of the Dedekind‒Peano postulates for successor arithmetic, including the Principle of Mathematical Induction. To this end one needs only the rules of free Core Logic itself, the rules for 0, s and # set out in Chapter 10, and certain logical rules about one-one mappings that are clearly stated. Among the lemmas proved along the way to the Dedekind‒Peano postulates is the result known as ‘Frege’s trick’: any natural number is the number of naturals preceding it. All derivations are given with complete formal rigor, but with additional commentary in ‘logician’s English’ to convey the gist of the formal work.

Channel coding linear programming decoding (CC- LPD) and compressed sensing linear programming decoding (CS- LPD) are two setups that are formally tightly related. Recently, a connection between CC-LPD and CS-LPD was exhibited that goes beyond this formal relationship. The main ingredient was a lemma that allowed one to map vectors in the nullspace of some zero-one measurement matrix into vectors of the fundamental cone defined by that matrix. The aim of the present paper is to extend this connection along several directions. In particular, the above-mentioned lemma is extended from real measurement matrices where every entry is equal to either zero or one to complex measurement matrices where the absolute value of every entry is a non-negative integer. Moreover, this lemma and its generalizations are used to translate performance guarantees from CC-LPD to CS-LPD. In addition, the present paper extends the formal relationship between CC-LPD and CS-LPD with the help of graph covers. First, this graph-cover viewpoint is used to obtain new connec- tions between, on the one hand, CC-LPD for binary parity- check matrices, and, on the other hand, CS-LPD for complex measurement matrices. Secondly, this graph-cover viewpoint is used to see CS-LPD not only as a well-known relaxation of some zero-norm minimization problem but (at least in the case of real measurement matrices with only zeros, ones, and minus ones) also as a relaxation of a problem we call the zero-infinity operator minimization problem.

In the present paper we investigate commutativity of rings with unity satisfying any one of the properties (Error rendering LaTeX formula) , (Error rendering LaTeX formula) , and for some in where are non-negative integers. Finally, under different appopriate constraints on commutators, commutativity of R has been established.

Godel's functional interpretation [1] can be used to extract programs from non-constructive proofs. Though correct by construction, the obtained terms can be computationally inefficient. One reason for slow execution is the re-evaluation of equal subterms due to the use of substitution during the extraction process. In the present paper we define a variant of the interpretation, which avoids subterm repetition and achieves an almost linear bound on the size of extracted programs.

In a recent paper [l] K . CORRADI and A. HAJNAL proved that if a finite graph without multiple edges contains at least 3k vertices and the valency of every vertex is at least 2k, where k is a positive integer, then the graph contains k independent circuits, i . e . the graph contains as a subgraph a set of k circuits no two of which have a vertex in common . The present paper contains extensions of this theorem. In a recent paper [2] P . ERDŐS and L. POSA proved, among other things, that if a finite graph with or without loops and multiple edges contains n vertices and at least n + 4 edges, then the graph contains two circuits without an edge in common . The present paper contains analogous results for planar graphs . We adopt the following notation : O k denotes a graph consisting of k independent circuits, kO denotes a graph consisting of k or more circuits no two of which have an edge in common . If q is a graph then 'V (q) denotes the set of vertices of q, `V i(q ) denotes the set of vertices of q having valency i in q (i being a non-negative integer), T 5 i (q), Ty i (q) denote the set of vertices of 4 having valency i and ' i, respectively, and &(q) denotes the set of edges of c~ . The valency of the vertex x in the graph will be denoted by v (x, C~) . I ? (() I will be denoted by V (q), J & (q) I by E(q) etc. In this notation the theorem Of CORRADI and HAJNAL quoted above states that if q is a finite graph without multiple edges and if V(q) 3k and T -2k_ 1 (4) =0, then q Ok ; and the theorem of ERDOS and POSA quoted above states that if is a finite graph and E(C) _V(q) + 4, then qD 2 O .

I - Elementary Principles

Let A be a semifinite and properly infinite von Neumann algebra of operators of a complex Hilbert space H. Call C e A compact (relative to A) if it is the limit in the norm of elements with finite support. The investigation of analytic properties of compact elements of A, generalizing welt-known classical theorems on compact operators, was started in [2] and is continued in the present paper. Let N i (resp. Ri) be the null (resp. range) projection of(1 C ) ~ where i is a natural number. Contrary to the classical case, the sequences N1 =R2 > R3 ~ "" need not be stationary, even if A is of type I. The question of the finiteness (resp. cofiniteness) of the projection Noo = sup Ni (resp. R~ = infRi) was left open in [2]. In the present paper it is shown that No~ is in fact always finite (and R~ is cofinite). It was shown already in [2] that if N® is finite, then inf(N®, Ro0) = 0 and sup(No~,R~)= 1. This is a decomposition of H in the sense of lattice theory or perhaps continuous geometry. In the present paper this result is refined by introducing the concepts of essentially closed subspaces and essentially topological direct sums. It is shown that the algebraic sum No~(H)+R®(H) is essentially topological direct and essentially closed. The operator 1 C decomposes into the sum of two closed operators affiliated with A, one summand being essentially nilpotent on Noo(H), the other one essentially regular on Roo(H). Throughout this paper H denotes a complex Hilbert space and A a properly infinite semifinite von Neumann algebra of linear operators of H. § 1. Preliminaries

Systems of explicit mathematics with non-constructive μ-operator. Part I