Transfinite Ordinals Research Articles

Positive Almost-Sure Termination: Complexity and Proof Rules

We study the recursion-theoretic complexity of Positive Almost-Sure Termination ( PAST ) in an imperative programming language with rational variables, bounded nondeterministic choice, and discrete probabilistic choice. A program terminates positive almost-surely if, for every scheduler, the program terminates almost-surely and the expected runtime to termination is finite. We show that PAST for our language is complete for the (lightface) co-analytic sets (Π 1 1 -complete). This is in contrast to the related notions of Almost-Sure Termination ( AST ) and Bounded Termination ( BAST ), both of which are arithmetical (Π 2 0 - and Σ 2 0 -complete respectively). Our upper bound implies an effective procedure to reduce reasoning about probabilistic termination to non-probabilistic fair termination in a model with bounded nondeterminism, and to simple program termination in models with unbounded nondeterminism. Our lower bound shows the opposite: for every program with unbounded nondeterministic choice, there is an effectively computable probabilistic program with bounded choice such that the original program is terminating if, and only if, the transformed program is PAST . We show that every program has an effectively computable normal form, in which each probabilistic choice either continues or terminates execution immediately, each with probability 1/2. For normal form programs, we provide a sound and complete proof rule for PAST . Our proof rule uses transfinite ordinals. We show that reasoning about PAST requires transfinite ordinals up to ω 1 CK ; thus, existing techniques for probabilistic termination based on ranking supermartingales that map program states to reals do not suffice to reason about PAST .

Construction of Ordinal Numbers and Arithmetic of Ordinal Numbers

The purpose of this paper is to introduce the idea of how to construct transfinite numbers and study transfinite arithmetic. The research method used is a literature review, which involves collecting various sources such as scientific papers and books related to Cantorian set theory, infinity, ordinal or transfinite arithmetic, as well as the connection between infinity and theology. The study also involves constructing the objects of study, namely ordinal numbers such as finite ordinals and transfinite ordinals, and examining their arithmetic properties. The results of this research include the methods of constructing both finite and transfinite ordinal numbers using two generation principles. Both finite and transfinite ordinal numbers are defined as well-orderings that are also transitive sets. Arithmetic of finite ordinal numbers is well-known, but the arithmetic of transfinite ordinal numbers will be introduced in this paper, including addition, multiplication, and exponentiation.

Main Concepts in Philosophy of Quantum Information

Quantum mechanics involves a generalized form of information, that of quantum information. It is the transfinite generalization of information and re-presentable by transfinite ordinals. The physical world being in the current of time shares the quality of “choice”. Thus quantum information can be seen as the universal substance of the world serving to describe uniformly future, past, and thus the present as the frontier of time. Future is represented as a coherent whole, present as a choice among infinitely many alternatives, and past as a well-ordering obtained as a result of a series of choices. The concept of quantum information describes the frontier of time, that “now”, which transforms future into past. Quantum information generalizes information from finite to infinite series or collections. The concept of quantum information allows of any physical entity to be interpreted as some nonzero quantity of quantum information. The fundament of quantum information is the concept of ‘quantum bit’, “qubit”. A qubit is a choice among an infinite set of alternatives. It generalizes the unit of classical information, a bit, which refer to a finite set of alternatives. The qubit is also isomorphic to a ball in Euclidean space, in which two points are chosen.

Fixed-point solutions to the regress problem in normative uncertainty

When we are faced with a choice among acts, but are uncertain about the true state of the world, we may be uncertain about the acts’ “choiceworthiness”. Decision theories guide our choice by making normative claims about how we should respond to this uncertainty. If we are unsure which decision theory is correct, however, we may remain unsure of what we ought to do. Given this decision-theoretic uncertainty, meta-theories attempt to resolve the conflicts between our decision theories...but we may be unsure which meta-theory is correct as well. This reasoning can launch a regress of ever-higher-order uncertainty, which may leave one forever uncertain about what one ought to do. There is, fortunately, a class of circumstances under which this regress is not a problem. If one holds a cardinal understanding of subjective choiceworthiness, and accepts certain other criteria (which are too weak to specify any particular decision theory), one’s hierarchy of metanormative uncertainty ultimately converges to precise definitions of “subjective choiceworthiness” for any finite set of acts. If one allows the metanormative regress to extend to the transfinite ordinals, the convergence criteria can be weakened further. Finally, the structure of these results applies straightforwardly not just to decision-theoretic uncertainty, but also to other varieties of normative uncertainty, such as moral uncertainty.

NOMINALISTIC ORDINALS, RECURSION ON HIGHER TYPES, AND FINITISM

AbstractIn 1936, Gerhard Gentzen published a proof of consistency for Peano Arithmetic using transfinite induction up to ε0, which was considered a finitistically acceptable procedure by both Gentzen and Paul Bernays. Gentzen’s method of arithmetising ordinals and thus avoiding the Platonistic metaphysics of set theory traces back to the 1920s, when Bernays and David Hilbert used the method for an attempted proof of the Continuum Hypothesis. The idea that recursion on higher types could be used to simulate the limit-building in transfinite recursion seems to originate from Bernays. The main difficulty, which was already discovered in Gabriel Sudan’s nearly forgotten paper of 1927, was that measuring transfinite ordinals requires stronger methods than representing them. This paper presents a historical account of the idea of nominalistic ordinals in the context of the Hilbert Programme as well as Gentzen and Bernays’ finitary interpretation of transfinite induction.

Alternating Turing machines for inductive languages

We show that alternating Turing machines, with a novel and natural definition of acceptance, accept precisely the inductive (Pi-1-1) languages. Total alternating machines, that either accept or reject each input, accept precisely the hyper-elementary (Delta-1-1) languages. Moreover, bounding the permissible number of alternations yields a characterization of the levels of the arithmetical hierarchy. Notably, these results use simple finite computing devices, with finitary and discrete operational semantics, and neither the results nor their proofs make any use of transfinite ordinals. Our characterizations elucidate the analogy between the polynomial-time hierarchy and the arithmetical hierarchy, as well as between their respective limits, namely polynomial-space and Pi-1-1.

カントルの集合論の哲学的側面―連続・時間・空間の問題―

This is a complement of 'Cantor's set theory and his philosophy', compiled in my collected papers, Borderland between Mathematics and Philosophy (1998, in Japanese). Cantor (1845-1918) was a mathematician with a strong tendency to philosophy, which tendency appeared as early as 1867 (Dissertation), and finally exerted latent influences upon the formation of transfinite set theory. For a purely mathematical purpose, this theory as well as its formation process can be explained without any reflection on his philosophical writings. However, in this case, we shall miss the complete figure of his conception. In fact, he expressed an opinion on Time and Space, opposed to Kant, on the basis of his Continuum, and gave an interpretation of-Spinoza's 'Deus' (in Ethica) through his transfinite ordinals (1883). He attempted, too, to establish a sort of Pythagorean Natural Philosophy, but in vain, in which was applied the Continuum Hypothesis (1885). This Natural Philosophy, fantastic as it was, can be traced back via several marks in his papers up to 1867. On the other hand, to speak frankly, Cantor's philosophical reasonings are far from systematic, and sometimes remain casual remarks.

Ackermann functions and transfinite ordinals

A set of binary operators are defined and shown to be equivalent to Ackermann functions. The same set of operators are used to develop a notation for writing the sequence of transfinite ordinals.

Logical organization of knowledge with inconsistent and undecidable algorithms using imaginary and transfinite exponential number forms in a non-Boolean field-I. Basic principles

In order to consider the organization of knowledge using inconsistent algorithms, a mathematical set-theoretic definition of axioms and undecidability is discussed. Ways in which imaginary numbers, exponentials, and transfinite ordinals can be given logical meanings that result in a new way to definite axioms are presented. This presentation is based on a proposed logical definition for axioms that includes an axiom and its negation as parts of an undecidable statement which is forced to the tautological truth value: true. The logical algebraic expression for this is shown to be isomorphic to the algebraic expression defining the imaginary numbers. This supports a progressive and Hegelian view of theory development, which means that thesis and antithesis axioms that exist in quantum mechanics (QM) and the special theory of relativity (STR) can be carried along at present and might be replaced by a synthesis of a deeper theory prompted by subsequently discovered experimental concept.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

A dialogue on Zeno's paradox of Achilles and the tortoise

The five participants in this dialogue critically discuss Zeno of Elea's paradox of Achilles and the tortoise. They consider a number of solutions to and restatements of the paradox, together with their philosophical implications. Among the issues investigated include the appearance-reality distinction, Aristotle's distinction between actual and potential infinity, the concept of a continuum, Cantor's continuum hypothesis and theory of transfinite ordinals, and, as a solution to Zeno's puzzle, the distinction between infinite and indeterminate or inexhaustible divisibility.

Transfinite cascades

Just as the natural numbers can be extended to the transfinite ordinals, so too can ladder networks and cascaded three-terminal networks be extended beyond infinity-so long as appropriate assumptions are imposed on their parameters. The author establishes the existence of such transfinite, linear or nonlinear, resistive, electrical networks. In particular, there are cascades of three-terminal networks whose nodes are sequentially numbered form the input to the output first by the natural numbers and then by transfinite ordinals all the way out to omega /sup p/, where rho is any natural number and omega is the first transfinite ordinal. Upon specifying an appropriate source or load at the input (node 0) and also at the output (node omega /sup p/), one obtains as a result a unique set of voltages and currents along the nodes of the cascade. This implies, in turn, that an output load far beyond infinity can be perceived by an observer at the input of the cascade. In fact, these ideas can be extended to cascades that reach still further, for example, out to omega /sup w/ and beyond. All this is a natural extension of a certain kind of infinite ladder network whose load at infinity is perceptible to an observer at the input to the ladder. Such ladder networks arise in practical applications as models of various physical phenomena taking place in infinite domains.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Generalizations of the Kruskal-Friedman theorems

AbstractKruskal proved that finite trees are well-quasi-ordered by hom(e)omorphic embeddability. Friedman observed that this statement is not provable in predicative analysis. Friedman also proposed (see in [Simpson]) some stronger variants of the Kruskal theorem dealing with finite labeled trees under hom(e)omorphic embeddability with a certain gap-condition, where labels are arbitrary finite ordinals from a fixed initial segment of ω. The corresponding limit statement, expressing that for all initial segments of ω these labeled trees are well-quasi-ordered, is provable in -CA, but not in the analogous theory -CA0 with induction restricted to sets. Schütte and Simpson proved that the one-dimensional case of Friedman's limit statement dealing with finite labeled intervals is not provable in Peano arithmetic. However, Friedman's gap-condition fails for finite trees labeled with transfinite ordinals. In [Gordeev 1] I proposed another gap-condition and proved the resulting one-dimensional modified statements for all (countable) transfinite ordinal-labels. The corresponding universal modified one-dimensional statement UM1 is provable in (in fact, is equivalent to) the familiar theory ATR0 whose proof-theoretic ordinal is Γ0. In [Gordeev 1] I also announced that, in the general case of arbitrarily-branching finite trees labeled with transfinite ordinals, in the proof-theoretic sense the hierarchy of the limit modified statements M&lt;λ (which are denoted by LMλ in the present note) is as strong as the hierarchy of the familiar theories of iterated inductive definitions (more precisely, see [Gordeev 1, Concluding Remark 3]). In this note I present a “positive” proof of the full universal modified statement UM, together with a short proof of the crucial “reverse” results which is based on Okada's interpretation of the well-established ordinal notations of Buchholz corresponding to the theories of iterated inductive definitions. Formally the results are summarized in §5 below.

Artificial Intelligence as a Possible Tool for Discovering Laws of Logic

The present essay has its origins in some technical work I did several years ago in mathematical logic. At that time I was busy in an obscure area on the borders of set theory and proof theory trying to describe notations for transfinite ordinals. (The best known example of this sort of result is Cantor's of a normal form for certain countable ordinals. One builds notations for these out of some basic functions and the resulting is similar to the ordinary Indian notation for finite numbers.) As time passed, I found myself increasingly puzzled by the circumstance that although I could manipulate my ideas with some facility, I really had no clear notion of what I was doing. From one point of view, my unease was not justified: if one made use of a few notions from standard set theory--some material about ordinal functions, inaccessible ordinals of various sorts, e tc . -one could obtain a neat, classical of the situation (Buchholz, 1975). But this description seemed to miss the mark for it did not provide an elucidation of what I felt was the basically computational and procedural nature of the experience that I underwent. I felt, in some vague way, that I was just doing a construction and that an adequate explication should be a of the essentials of this construction. The set-theoretic explanation lacked any dynamic quality. Confused by this circumstance, I began to pay more attention to the introductory chapters of the various texts on logic and mathematical logic that I was using in my teaching, for it was there that was usually concentrated a discussion of what was called foundations of mathematics, a discussion which, I naively thought, justified the long and intricate technical chapters which followed. In fact, most of these chapters (if

Transfinite ordinals as axiom number symbols for unification of quantum and electromagnetic wave functions

The mapping of axioms into transfinite number fields provides a method whereby axioms and the magnitudes of experimental values can be distinguished in a clear manner. This procedure is shown also to result in a logical interpretation for the presence of exponential forms and for their imaginary arguments.

On some equation in transfinite ordinals

On some equation in transfinite ordinals

Transfinite ordinals in recursive number theory

The possibility of constructing a numerical equivalent of a system of trans-finite ordinals, in recursive number theory, was briefly indicated in a previous paper, where consideration was confined to ordinals less than ε (the first to satisfy ε = ω). In the present paper we construct a representation, by functions of number-theoretic variables, for ordinals of any type.In addition to definite numerals, and numeral variables, we introduce majorant variables σ, ω, ωr, r ≧ 1. A relation containing a single majorant variable σ is required to hold, not necessarily for all non-negative integral values of σ, but for all values greater than some assigned constant.

The Transfinite Ordinals of the Second Class

THERE is a theorem in the transfinite calculus that any ascending sequence of ordinal numbers of the second class has a limit which is also of the second class. This theorem is important, being wanted to prove that the aggregate of these ordinals is unenumerable.

Some Arithmetic Operations with Transfinite Ordinals

Some Arithmetic Operations with Transfinite Ordinals

Some Arithmetic Operations with Transfinite Ordinals

Some Arithmetic Operations with Transfinite Ordinals

Continuous Increasing Functions of Finite and Transfinite Ordinals

Continuous Increasing Functions of Finite and Transfinite Ordinals