This paper is the first of a series of two. It contains proof–theoretic investigations on subtheories of second order arithmetic and set theory. Among the principles on which these theories are based one finds autonomously iterated positive and monotone inductive definitions, Π1 transfinite recursion, ∆ 1 2 transfinite recursion, transfinitetely iterated Π 1 1 dependent choices, extended Bar ru...

Among various barycentric coordinates and their extensions, the linear and cubic (Hermite) Gordon-Wixom transfinite interpolation schemes deliver the most accurate approximations of the harmonic and biharmonic functions, respectively. However interpolation properties of the original Gordon-Wixom interpolations are studied for convex domains only and, therefore, their current practical importanc...

The galaxies of the nonstandard enlargements of conventionally infinite graphs as well as of transfinite graphs are defined, analyzed, and illustrated by some examples. It is then shown that any such enlargement either has exactly one galaxy, its principal one, or it has infinitely many galaxies. In the latter case, the galaxies are partially ordered by there “closeness” to the principal galaxy...

We provide a methodology for efficiently checking whether a tetrahedral transfinite interpolation is regular. From given four triangular surfaces in form of B-patches fulfilling some compatibility conditions, we generate a transfinite interpolation defined on the unit 3D simplex. An efficient subdivision scheme is provided for a B-patch in order to obtain criteria which are verifiable in a disc...

Programming languages theory is full of problems that reduce to proving the consistency of a logic, such as the normalization of typed lambda-calculi, the decidability of equality in type theory, equivalence testing of traces in security, etc. Although the principle of transfinite induction is routinely employed by logicians in proving such theorems, it is rarely used by programming languages r...

This article reports that some robustness of the notions of predicativity and of autonomous progression is broken down if as the given infinite total entity we choose some mathematical entities other than the traditional ω. Namely, the equivalence between normal transfinite recursion scheme and new dependent transfinite recursion scheme, which does hold in the context of subsystems of second or...

This report is an improvement of a prior report [4, Report 814]. It sharpens the principal theorem (Theorem 5.1 of Report 814) and also simplifies its proof. There are also several minor changes involving clarifications and corrections of misprints. The prior abstract remains the same as follows: In a previous work, the galaxies of the nonstandard enlargements of connected, conventionally infin...

We study the problem of finding nonconstant monic integer polynomials, normalized by their degree, with small supremum on an interval I. The monic integer transfinite diameter tM(I) is defined as the infimum of all such supremums. We show that if I has length 1, then tM(I) = 1 2 . We make three general conjectures relating to the value of tM(I) for intervals I of length less than 4. We also con...

An ordinal-valued metric, taking its values in the set א1 of all countable ordinals, can be assigned to a metrizable set M of nodes in any transfinite graph. M contains all the nonsingleton nodes, as well as certain singleton nodes. Moreover, this yields a graphical realization of Cantor’s countable ordinals, as well as of the Aristotelian ideas of “potential” and “actual” infinities, the forme...

In this note, we give some applications of projective logarithmic potentials. First introduce the notions energy and capacity associated to kernel. We compare quantitatively with complex Monge-Ampère on space deduce that set zero is zero. Further, define transfinite diameter a compact show it coincides capacity. Finally there an analogous classical Evans's theorem.