The relative strengths of first-order theories axiomatized by transfinite induction, for ordinals less-than ~0, and formulas restricted in quantifier complexity, is determined. This is done, in part, by describing the provably recursive functions of such theories. Upper bounds for the provably recursive functions are obtained using model-theoretic techniques. A variety of additional results tha...

We investigate a hierarchy of arithmetical structures obtained by transfinite addition of a canonic universal predicate, where the canonic universal predicate for M is defined as a minimum universal predicate for M in terms of definability. We determine the upper bound of the hierarchy and give a characterisation for the sets definable in the hierarchy.

Let M = (Mr)reW be a system of matroids on a set S. For every transfinite sequence / of distinct elements of 5, we define a number TJ(/). In [12] we proved that the condition that r](f) 3=0 for every possible choice of/is necessary for M to have a system of mutually disjoint bases. Further, we showed that this condition is sufficient if R is countable and Mr is a rank-finite transversal matroid...

An axiom system for machine spaces is introduced. Time is modeled as a totally ordered monoid, thus allowing discrete, continuous, and transfinite time structures. A generalized metric on these machine spaces is defined, having values in a directedly ordered monoid. This enables the notion of ε-balls in machine space, which, for example, can be used to explore model risk and robustness in stati...

Infinite time Turing machines extend the classical Turing machine concept to transfinite ordinal time, thereby providing a natural model of infinitary computability that sheds light on the power and limitations of supertask algorithms.

Pada artikel ini, diteliti sifat-sifat fungsi terfragmentasi dan terhitung fungsional. Dengan menggunakan barisan \textit{transfinite} regular, dibuktikan bahwa himpunan semua bernilai real fungsional terbatas yang merupakan ring. Selain itu, pula sifat analog dengan Teroema Weierstrass M-Test. Lebih lanjut, diberikan syarat tambahan supaya juga kontinu.

A straightforward explanation of the Young’s two-slit experiment of a quantum particle is obtained within the framework of the Noncommutative Geometric associated with El Naschie’s Cantorian-Fractal transfinite Spacetime continuum.