Ackermann functions and transfinite ordinals

Nambiar, K.K.

doi:10.1016/0893-9659(95)00084-4

Cited by 8 publications

(4 citation statements)

References 2 publications

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“…Interestingly, this exclusion leads to a contradiction. This can be shown by the simple observation that hyperexponential operations exist (Nambiar 1995). Addition, multiplication and exponentiation, which can be called hyper 1, hyper 2 and hyper 3, can be extended easily to hyper 4, hyper 5, etc.…”

Section: Occam's Razor Rejectedmentioning

confidence: 99%

An Evolutionary Argument for a Self-Explanatory, Benevolent Metaphysics

Blondé¹

2015

Symposion

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In this paper, a metaphysics is proposed that includes everything that can be represented by a well-founded multiset. It is shown that this metaphysics, apart from being self-explanatory, is also benevolent. Paradoxically, it turns out that the probability that we were born in another life than our own is zero. More insights are gained by inducing properties from a metaphysics that is not self-explanatory. In particular, digital metaphysics is analyzed, which claims that only computable things exist. First of all, it is shown that digital metaphysics contradicts itself by leading to the conclusion that the shortest computer program that computes the world is infinitely long. This means that the Church-Turing conjecture must be false. Secondly, the applicability of Occam's razor is explained by evolution: in an evolving physics it can appear at each moment as if the world is caused by only finitely many things. Thirdly and most importantly, this metaphysics is benevolent in the sense that it organizes itself to fulfill the deepest wishes of its observers. Fourthly, universal computers with an infinite memory capacity cannot be built in the world. And finally, all the properties of the world, both good and bad, can be explained by evolutionary conservation.

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Section: Occam's Razor Rejectedmentioning

confidence: 99%

An Evolutionary Argument for a Self-Explanatory, Benevolent Metaphysics

Blondé¹

2015

Symposion

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“…One of the most known methodologies is the so-called Knuth up-arrow notation introduced by D.E. Knuth in 1976 (see 56 ANTONINO LEONARDIS, GIANFRANCO D'ATRI AND FABIO CALDAROLA [19]) and strictly linked to the concept of hyper-operation and Ackermann function (see [1,23]). The idea of hyper-operation dates back to the early 1900s by A.A. Bennet (see [5]), and subsequently we re-find it in a group of Hilbert's students as W. Ackermann and G. Sudan.…”

Section: Introductionmentioning

confidence: 99%

Beyond Knuth's notation for unimaginable numbers within computational number theory

Leonardis

d’Atri

Caldarola

2022

International Electronic Journal of Algebra

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Literature considers under the name "unimaginable numbers" any positive integer going beyond any physical application. One of the most known methodologies to conceive such numbers is using hyper-operations, that is a sequence of binary functions defined recursively starting from the usual chain: addition -multiplication -exponentiation. The most important notations to represent such hyper-operations have been considered by Knuth, Goodstein, Ackermann and Conway as described in this work's introduction. Within this work we will give an axiomatic setup for this topic, and then try to find on one hand other ways to represent unimaginable numbers, as well as on the other hand applications to computer science, where the algorithmic nature of representations and the increased computation capabilities of computers give the perfect field to develop further the topic, exploring some possibilities to effectively operate with such big numbers. In particular, we will give some axioms and generalizations for the up-arrow notation and, considering a representation via rooted trees of the hereditary base-n notation, we will determine in some cases an effective bound related to "Goodstein sequences" using Knuth's notation. Finally, we will also analyze some methods to compare big numbers, proving specifically a theorem about approximation using scientific notation and a theorem on hyperoperation bounds for Steinhaus-Moser notation.

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“…Knuth himself had instead introduced the notation (2) a few years earlier in [16] (1976), but these ideas actually date from the beginning of the century (see [1,2,13,19]). More recent works that start from "extremely large" or "infinite" numbers are [7,8,12,14,15,17,18,20,21,23]. There are also the online resources [5,6,22].…”

Section: Introduction: the Unimaginable Numbersmentioning

confidence: 99%

On the Arithmetic of Knuth’s Powers and Some Computational Results About Their Density

Caldarola

d’Atri

Mercuri

et al. 2020

Lecture Notes in Computer Science

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The object of the paper are the so-called "unimaginable numbers". In particular, we deal with some arithmetic and computational aspects of the Knuth's powers notation and move some first steps into the investigation of their density. Many authors adopt the convention that unimaginable numbers start immediately after 1 googol which is equal to 10 100 , and G.R. Blakley and I. Borosh have calculated that there are exactly 58 integers between 1 and 1 googol having a nontrivial "kratic representation", i.e., are expressible nontrivially as Knuth's powers. In this paper we extend their computations obtaining, for example, that there are exactly 2 893 numbers smaller than 10 10 000 with a nontrivial kratic representation, and we, moreover, investigate the behavior of some functions, called krata, obtained by fixing at most two arguments in the Knuth's power a ↑ b c.

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Ackermann functions and transfinite ordinals

Abstract: 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.

Cited by 8 publications

References 2 publications

An Evolutionary Argument for a Self-Explanatory, Benevolent Metaphysics

An Evolutionary Argument for a Self-Explanatory, Benevolent Metaphysics

Beyond Knuth's notation for unimaginable numbers within computational number theory

On the Arithmetic of Knuth’s Powers and Some Computational Results About Their Density

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