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Journal articles on the topic 'Transfinite numbers'

Author: Grafiati

Published: 4 June 2021

Last updated: 1 August 2024

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1

Agustito, Denik, Krida Singgih Kuncoro, Istiqomah Istiqomah, and Agus Hendriyanto. "Construction of Ordinal Numbers and Arithmetic of Ordinal Numbers." JTAM (Jurnal Teori dan Aplikasi Matematika) 7, no. 3 (2023): 781. http://dx.doi.org/10.31764/jtam.v7i3.15039.

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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 t

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2

Srivastava, S. M. "Transfinite numbers." Resonance 2, no. 3 (1997): 58–68. http://dx.doi.org/10.1007/bf02838969.

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3

Weinert, Thilo. "Transfinite Ramsey Numbers." Electronic Notes in Discrete Mathematics 43 (September 2013): 231–34. http://dx.doi.org/10.1016/j.endm.2013.07.038.

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4

Álvarez J., Carlos. "De la determinación del infinito a la inaccesibilidad en los cardinales transfinitos." Crítica (México D. F. En línea) 26, no. 78 (1994): 27–71. http://dx.doi.org/10.22201/iifs.18704905e.1994.956.

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In this paper I deal with two problems in mathematical philosophy: the (very old) question about the nature of infinity, and the possible answer to this question after Cantor’s theory of transfinite numbers. Cantor was the first to consider that his transfinite numbers theory allows to speak, within mathematics, of an actual infinite and allows to leave behind the Aristotelian statement that infinity exists only as potential infinity. In the first part of this paper I discuss Cantorian theory of transfinite numbers and his particular point of view about this matter. But the development of the

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5

Velev, Milen V. "Infinite multisets: Basic properties and cardinality." Notes on Number Theory and Discrete Mathematics 30, no. 2 (2024): 335–56. http://dx.doi.org/10.7546/nntdm.2024.30.2.335-356.

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This research work presents the topic of infinite multisets, their basic properties and cardinality from a somewhat different perspective. In this work, a new property of multisets, ‘m-cardinality’, is defined using multiset functions. M-cardinality unifies and generalizes the definitions of cardinality, injection, bijection, and surjection to apply to multisets. M-cardinality takes into account both the number of distinct elements in a multiset and the number of copies of each element (i.e., the multiplicity of the elements). Based on m-cardinality, ‘m-cardinal numbers’ are defined as a gener

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6

WEBER, ZACH. "TRANSFINITE NUMBERS IN PARACONSISTENT SET THEORY." Review of Symbolic Logic 3, no. 1 (2010): 71–92. http://dx.doi.org/10.1017/s1755020309990281.

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This paper begins an axiomatic development of naive set theory—the consequences of a full comprehension principle—in a paraconsistent logic. Results divide into two sorts. There is classical recapture, where the main theorems of ordinal and Peano arithmetic are proved, showing that naive set theory can provide a foundation for standard mathematics. Then there are major extensions, including proofs of the famous paradoxes and the axiom of choice (in the form of the well-ordering principle). At the end I indicate how later developments of cardinal numbers will lead to Cantor’s theorem, the exist

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7

Beyer, W. A., and J. D. Louck. "Transfinite Function Iteration and Surreal Numbers." Advances in Applied Mathematics 18, no. 3 (1997): 333–50. http://dx.doi.org/10.1006/aama.1996.0513.

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8

WEBER, ZACH. "TRANSFINITE CARDINALS IN PARACONSISTENT SET THEORY." Review of Symbolic Logic 5, no. 2 (2012): 269–93. http://dx.doi.org/10.1017/s1755020312000019.

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This paper develops a (nontrivial) theory of cardinal numbers from a naive set comprehension principle, in a suitable paraconsistent logic. To underwrite cardinal arithmetic, the axiom of choice is proved. A new proof of Cantor’s theorem is provided, as well as a method for demonstrating the existence of large cardinals by way of a reflection theorem.

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9

Rioux, Jean W. "Cantor’s Transfinite Numbers and Traditional Objections to Actual Infinity." Thomist: A Speculative Quarterly Review 64, no. 1 (2000): 101–25. http://dx.doi.org/10.1353/tho.2000.0003.

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10

Pąk, Karol. "The Ring of Conway Numbers in Mizar." Formalized Mathematics 31, no. 1 (2023): 215–28. http://dx.doi.org/10.2478/forma-2023-0020.

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Summary Conway’s introduction to algebraic operations on surreal numbers with a rather simple definition. However, he combines recursion with Conway’s induction on surreal numbers, more formally he combines transfinite induction-recursion with the properties of proper classes, which is diffcult to introduce formally. This article represents a further step in our ongoing e orts to investigate the possibilities offered by Mizar with Tarski-Grothendieck set theory [4] to introduce the algebraic structure of Conway numbers and to prove their ring character.

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11

Pringe, Hernán. "Dimitry Gawronsky: Reality and Actual Infinitesimals." Kant-Studien 114, no. 1 (2023): 68–97. http://dx.doi.org/10.1515/kant-2023-2012.

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Abstract The aim of this paper is to analyze Dimitry Gawronsky’s doctrine of actual infinitesimals. I examine the peculiar connection that his critical idealism establishes between transcendental philosophy and mathematics. In particular, I reconstruct the relationship between Gawronsky’s differentials, Cantor’s transfinite numbers, Veronese’s trans-Archimedean numbers and Robinson’s hyperreal numbers. I argue that by means of his doctrine of actual infinitesimals, Gawronsky aims to provide an interpretation of calculus that eliminates any alleged given element in knowledge.

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12

Rabi, Lior. "Ortega y Gasset on Georg Cantor’s Theory of Transfinite Numbers." Kairos. Journal of Philosophy & Science 15, no. 1 (2016): 46–70. http://dx.doi.org/10.1515/kjps-2016-0003.

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Abstract Ortega y Gasset is known for his philosophy of life and his effort to propose an alternative to both realism and idealism. The goal of this article is to focus on an unfamiliar aspect of his thought. The focus will be given to Ortega’s interpretation of the advancements in modern mathematics in general and Cantor’s theory of transfinite numbers in particular. The main argument is that Ortega acknowledged the historical importance of the Cantor’s Set Theory, analyzed it and articulated a response to it. In his writings he referred many times to the advancements in modern mathematics an

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13

Rauff, James V. "TRANSFINITE NUMBERS AND NATIVE AMERICAN COSMOLOGY: A COURSE ABOUT INFINITY." PRIMUS 12, no. 1 (2002): 1–10. http://dx.doi.org/10.1080/10511970208984013.

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14

Burgin, Mark. "Introduction to Hyperspaces." International Journal of Pure Mathematics 7 (February 8, 2021): 36–42. http://dx.doi.org/10.46300/91019.2020.7.5.

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The development of mathematics brought mathematicians to infinite structures. This process started with transcendent real numbers and infinite sequences going through infinite series to transfinite numbers to nonstandard numbers to hypernumbers. From mathematics, infinity came to physics where physicists have been trying to get rid of infinity inventing a variety of techniques for doing this. In contrast to this, mathematicians as well as some physicists suggested ways to work with infinity introducing new mathematical structures such distributions and extrafunctions. The goal of this paper is

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15

Ferreirós, José. "“What fermented in me for years”: Cantor's discovery of transfinite numbers." Historia Mathematica 22, no. 1 (1995): 33–42. http://dx.doi.org/10.1006/hmat.1995.1003.

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16

ARAI, TOSHIYASU. "PROOF-THEORETIC STRENGTHS OF WEAK THEORIES FOR POSITIVE INDUCTIVE DEFINITIONS." Journal of Symbolic Logic 83, no. 3 (2018): 1091–111. http://dx.doi.org/10.1017/jsl.2018.36.

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

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17

Chatyrko, Vitalij, Sang-Eon Han, and Yasunao Hattori. "The small inductive dimension of subsets of Alexandroff spaces." Filomat 30, no. 11 (2016): 3007–14. http://dx.doi.org/10.2298/fil1611007c.

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We describe the small inductive dimension ind in the class of Alexandroff spaces by the use of some standard spaces. Then for ind we suggest decomposition, sum and product theorems in the class. The sum and product theorems there we prove even for the small transfinite inductive dimension trind. As an application of that, for each positive integers k,n such that k ? n we get a simple description in terms of even and odd numbers of the family S(k,n) = {S ? Kn : |S|=k+1 and indS=k}, where K is the Khalimsky line.

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18

Radul, D. N. "FLORENSKY’S PHILOSOPHICAL IDEAS AND THE THEORY OF SETS OF CANTOR." Metaphysics, no. 2 (December 15, 2021): 125–32. http://dx.doi.org/10.22363/2224-7580-2021-2-125-132.

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The article considers the role of the idea of actual infinity in the works of Florensky. The introduction briefly traces the history of ideas about the actual infinity in European culture to the works of George Cantor. The reaction of European scientists and religious figures to the emergence of the “naïve” theory of Cantor sets is characterized. A detailed analysis of the connection between Florensky and George Cantor’s ideas is given. Many quotations from the 1904 work on the symbols of Infinity are given, which illustrate the influence of Cantor’s works on Florensky. The presentation of Flo

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19

Escardó, Martín H. "Infinite sets that Satisfy the Principle of Omniscience in any Variety of Constructive Mathematics." Journal of Symbolic Logic 78, no. 3 (2013): 764–84. http://dx.doi.org/10.2178/jsl.7803040.

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AbstractWe show that there are plenty of infinite sets that satisfy the omniscience principle, in a minimalistic setting for constructive mathematics that is compatible with classical mathematics. A first example of an omniscient set is the one-point compactification of the natural numbers, also known as the generic convergent sequence. We relate this to Grilliot's and Ishihara's Tricks. We generalize this example to many infinite subsets of the Cantor space. These subsets turn out to be ordinals in a constructive sense, with respect to the lexicographic order, satisfying both a well-foundedne

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20

Macdonald, Ranald R. "The limits of probability modelling: A serendipitous tale of goldfish, transfinite numbers, and pieces of string." Mind & Society 1, no. 2 (2000): 17–38. http://dx.doi.org/10.1007/bf02512312.

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21

Gomułka, Jakub. "Cantor’s paradise from the perspective of non‐revisionist Wittgensteinianism." Argument: Biannual Philosophical Journal 10, no. 2 (2021): 351–72. http://dx.doi.org/10.24917/20841043.10.2.5.

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   Cantor’s paradise from the perspective of non‐revisionist Wittgensteinianism: Ludwig Wittgenstein is known for his criticism of transfinite set theory. He forwards the claim that we tend to conceptualise infinity as an object due to the systematic confusion of extension with in‐ tension. There can be no mathematical symbol that directly refers to infinity: a rule is the only form by which the latter can appear in our symbolic operations. In consequence, Wittgenstein rejects such ideas as infinite cardinals, the Cantorian understanding of non‐denumerability, and the view of

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22

Gomułka, Jakub. "Cantor’s paradise from the perspective of non‐revisionist Wittgensteinianism." Argument: Biannual Philosophical Journal 10, no. 2 (2021): 351–72. http://dx.doi.org/10.24917/20841043.10.2.5.

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   Cantor’s paradise from the perspective of non‐revisionist Wittgensteinianism: Ludwig Wittgenstein is known for his criticism of transfinite set theory. He forwards the claim that we tend to conceptualise infinity as an object due to the systematic confusion of extension with in‐ tension. There can be no mathematical symbol that directly refers to infinity: a rule is the only form by which the latter can appear in our symbolic operations. In consequence, Wittgenstein rejects such ideas as infinite cardinals, the Cantorian understanding of non‐denumerability, and the view of

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23

Hoare, Graham. "R. L. Goodstein and mathematical logic." Mathematical Gazette 97, no. 540 (2013): 409–12. http://dx.doi.org/10.1017/s0025557200000139.

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Born in London, Reuben Louis Goodstein (1912-1985) completed his secondary education at St Paul's School and in 1931 proceeded to Magdalene College, Cambridge, with a Major Open Scholarship to read mathematics. He graduated in 1933 having taken firsts in Parts I and II of the Mathematical Tripos. From 1933 to 1935 his research on transfinite numbers was supervised by Professor J. E. Littlewood. He took a MSc and left Cambridge in 1935 to take up an appointment as lecturer in pure and applied mathematics at Reading University, a position he held until late 1947. While undertaking a strenuous te

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24

Peters, John, and Hugo Burgos. "Semblanza absolutamente exacta: mapas y medios en Borges y Royce." post(s) 7 (December 13, 2021): 134–53. http://dx.doi.org/10.18272/post(s).v7i7.2528.

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Josiah Royce, the American idealist philosopher (1855-1916), is best known to readers of Borges in connection with a recursive map-within-a-map drawn upon the soil of England. Indeed, Borges ranks "el mapa de Royce" side-by-side with his beloved Zeno´´´ s paradox in “Otro poema de los dones” (336), a Whitmanesque catalog of a few of his favorite things. Borges appreciated Royce as a fellow-wanderer through the late nineteenth-century thickets of both Anglo-American idealism and the new mathematics of transfinite numbers. Royce was not so much an influence on Borges as a fellow traveler who h

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25

Paleva-Kadiyska, B. I., R. А. Roussev, and V. В. Galabov. "Rational Possibility of Generating Power Laws in the Synthesis of Cam Mechanisms." Advanced Engineering Research 21, no. 2 (2021): 184–90. http://dx.doi.org/10.23947/2687-1653-2021-21-2-184-190.

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Introduction.The generation of polynomial power laws of motion for the synthesis of cam mechanisms is complicated by the need to determine the coefficients of power polynomials. The study objective is to discover a rational capability of generating рower law swith arbitrary terms number under s with an rbitrary number of terms under the synthesis of cam mechanisms.Materials and Methods.A unified formula for determining the values of coefficients of power polynomials with any number of integers and/or non-integer exponents is derived through the so-called transfinite mathematical induction. Res

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26

Goodman, Nicolas D. "Replacement and collection in intuitionistic set theory." Journal of Symbolic Logic 50, no. 2 (1985): 344–48. http://dx.doi.org/10.2307/2274220.

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Intuitionistic Zermelo-Fraenkel set theory, which we call ZFI, was introduced by Friedman and Myhill in [3] in 1970. The idea was to have a theory with the same axioms as ordinary classical ZF, but with Heyting's predicate calculus HPC as the underlying logic. Since some classically equivalent statements are intuitionistically inequivalent, however, it was not always obvious which form of a classical axiom to take. For example, the usual formulation of the axiom of foundation had to be replaced with a principle of transfinite induction on the membership relation in order to avoid having exclud

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27

Gordeev, L. "Generalizations of the one-dimensional version of the Kruskal-Friedman theorems." Journal of Symbolic Logic 54, no. 1 (1989): 100–121. http://dx.doi.org/10.2307/2275019.

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The paper [Schütte + Simpson] deals with the following one-dimensional case of Friedman's extension (see in [Simpson 1]) of Kruskal's theorem ([Kruskal]). Given a natural number n, let Sn+1 be the set of all finite sequences of natural numbers <n + 1. If s1 = (a0,…,ak) ∈Sn+1 and s2 = (b0,…,bm) ∈Sn + 1, then a strictly monotone function f: {0,…, k} → {0,…, m} is called an embedding of s1 into s2 if the following two assertions are satisfied:1) ai, = bf(i), for all i < k;2) if f(i) < j < f(i + 1) then bj > bf(i+1), for all i < k, j < m.Then for every infinite sequence s1, s2

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Amoroso, Francesco. "$f$- transfinite diameter and number theoretic applications." Annales de l’institut Fourier 43, no. 4 (1993): 1179–98. http://dx.doi.org/10.5802/aif.1368.

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29

Wojciechowski, Jerzy. "Finite Automata on Transfinite Sequences and Regular Expressions." Fundamenta Informaticae 8, no. 3-4 (1985): 379–96. http://dx.doi.org/10.3233/fi-1985-83-407.

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In this paper the notion of regular expression for finite automata on transfinite sequences /TF-automata/ is introduced. The characterization theorem for TF-automata is proved. From this theorem we conclude the decidability of the emptiness problem for TF-automata and the characterization theorem for finite automata on transfinite sequences of bounded lenght.

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30

Hare, K. G., and C. J. Smyth. "The monic integer transfinite diameter." Mathematics of Computation 75, no. 256 (2006): 1997–2019. http://dx.doi.org/10.1090/s0025-5718-06-01843-6.

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31

Flammang, V. "The absolute trace of totally positive algebraic integers." International Journal of Number Theory 15, no. 01 (2019): 173–81. http://dx.doi.org/10.1142/s1793042119500064.

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Thanks to our recursive algorithm developed in [Trace of totally positive algebraic integers and integer transfinite diameter, Math. Comp. 78(266) (2009) 1119–1125], we prove that, if [Formula: see text] is a totally positive algebraic integer of degree [Formula: see text] with minimum conjugate [Formula: see text] then, with a finite number of explicit exceptions, [Formula: see text]

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32

Mátrai, Tamás. "On the closure of Baire classes under transfinite convergences." Fundamenta Mathematicae 183, no. 2 (2004): 157–68. http://dx.doi.org/10.4064/fm183-2-6.

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33

Hare, K. G., and C. J. Smyth. "Corrigendum to ``The monic integer transfinite diameter''." Mathematics of Computation 77, no. 263 (2008): 1869. http://dx.doi.org/10.1090/s0025-5718-07-02077-7.

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34

Trlifaj, Jan. "Categoricity for transfinite extensions of modules." Proceedings of the American Mathematical Society, Series B 10, no. 32 (2023): 369–81. http://dx.doi.org/10.1090/bproc/194.

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For each deconstructible class of modules D \mathcal D , we prove that the categoricity of D \mathcal D in a big cardinal is equivalent to its categoricity in a tail of cardinals. We also prove Shelah’s Categoricity Conjecture for ( D , ⪯ ) (\mathcal D, \preceq ) , where ( D , ⪯ ) (\mathcal D, \preceq ) is any abstract elementary class of roots of Ext.

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35

Charalambous, M. G. "A factorization theorem for the transfinite kernel dimension of metrizable spaces." Fundamenta Mathematicae 157, no. 1 (1998): 79–84. http://dx.doi.org/10.4064/fm-157-1-79-84.

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36

Trlifaj, Jan. "Weak diamond, weak projectivity, and transfinite extensions of simple artinian rings." Journal of Algebra 601 (July 2022): 87–100. http://dx.doi.org/10.1016/j.jalgebra.2022.03.009.

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37

Flammang, V., G. Rhin, and C. J. Smyth. "The integer transfinite diameter of intervals and totally real algebraic integers." Journal de Théorie des Nombres de Bordeaux 9, no. 1 (1997): 137–68. http://dx.doi.org/10.5802/jtnb.193.

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38

Wu, Qiang. "A new exceptional polynomial for the integer transfinite diameter of $[0,1]$." Journal de Théorie des Nombres de Bordeaux 15, no. 3 (2003): 847–61. http://dx.doi.org/10.5802/jtnb.430.

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39

Flammang, Valérie, Georges Rhin, and Jean-Marc Sac-Épée. "Integer transfinite diameter and polynomials with small Mahler measure." Mathematics of Computation 75, no. 255 (2006): 1527–41. http://dx.doi.org/10.1090/s0025-5718-06-01791-1.

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40

Flammang, V. "Trace of totally positive algebraic integers and integer transfinite diameter." Mathematics of Computation 78, no. 266 (2008): 1119–25. http://dx.doi.org/10.1090/s0025-5718-08-02120-0.

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41

Honig, William M. "The Correspondence Between the Axioms of Quantum Mechanics and Imaginary and Transfinite Number Forms." Physics Essays 1, no. 4 (1988): 247–58. http://dx.doi.org/10.4006/1.3033419.

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42

Conidis, Chris J., Pace P. Nielsen, and Vandy Tombs. "Transfinitely valued Euclidean domains have arbitrary indecomposable order type." Communications in Algebra 47, no. 3 (2019): 1105–13. http://dx.doi.org/10.1080/00927872.2018.1501569.

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43

Gifford. "Mathematical Transfinites and Modernism: Literary Infinities: Number and Narrative in Modern Fiction." Journal of Modern Literature 43, no. 2 (2020): 190. http://dx.doi.org/10.2979/jmodelite.43.2.13.

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44

Mancosu, Paolo. "Between Russell and Hilbert: Behmann on the Foundations of Mathematics." Bulletin of Symbolic Logic 5, no. 3 (1999): 303–30. http://dx.doi.org/10.2307/421183.

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AbstractAfter giving a brief overview of the renewal of interest in logic and the foundations of mathematics in Göttingen in the period 1914-1921, I give a detailed presentation of the approach to the foundations of mathematics found in Behmann's doctoral dissertation of 1918, Die Antinomie der transfiniten Zahl und ihre Auflösung durch die Theorie von Russell und Whitehead. The dissertation was written under the guidance of David Hilbert and was primarily intended to give a clear exposition of the solution to the antinomies as found in Principia Mathematica. In the process of explaining the t

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45

Kanovei, Vladimir, та Vassily Lyubetsky. "Jensen Δn1 Reals by Means of ZFC and Second-Order Peano Arithmetic". Axioms 13, № 2 (2024): 96. http://dx.doi.org/10.3390/axioms13020096.

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It was established by Jensen in 1970 that there is a generic extension L[a] of the constructible universe L by a non-constructible real a∉L, minimal over L, such that a is Δ31 in L[a]. Our first main theorem generalizes Jensen’s result by constructing, for each n≥2, a generic extension L[a] by a non-constructible real a∉L, still minimal over L, such that a is Δn+11 in L[a] but all Σn1 reals are constructible in L[a]. Jensen’s forcing construction has found a number of applications in modern set theory. A problem was recently discussed as to whether Jensen’s construction can be reproduced entir

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46

Honig, W. M. "Logical organization of knowledge with inconsistent and undecidable algorithms using imaginary and transfinite exponential number forms in a non-Boolean field-I. Basic principles." IEEE Transactions on Knowledge and Data Engineering 5, no. 2 (1993): 190–203. http://dx.doi.org/10.1109/69.219730.

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47

Agustito, Denik, Sukiyanto Sukiyanto, and Krida Singgih Kuncoro. "Mathematical induction, transfinite induction, and induction over the continuum." International Journal of Mathematics and Mathematics Education, June 14, 2023, 180–91. http://dx.doi.org/10.56855/ijmme.v1i02.385.

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This article examines three types of induction methods in mathematics: mathematical induction, transfinite induction, and induction over the continuum. If a statement holds true for all natural numbers, it is proven using mathematical induction. If a statement holds true for all ordinal numbers, it is proven using transfinite induction. Since induction over the continuum cannot be applied to a statement, when something is said to be proven true for every point in [a, b), the proof is done using induction over the continuum.

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48

Connelly, James R. "Transfinite Number in Wittgenstein's Tractatus." Journal for the History of Analytical Philosophy 9, no. 2 (2021). http://dx.doi.org/10.15173/jhap.v9i2.4029.

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In his highly perceptive, if underappreciated introduction to Wittgenstein’s Tractatus, Russell identifies a “lacuna” within Wittgenstein’s theory of number, relating specifically to the topic of transfinite number. The goal of this paper is two-fold. The first is to show that Russell’s concerns cannot be dismissed on the grounds that they are external to the Tractarian project, deriving, perhaps, from logicist ambitions harbored by Russell but not shared by Wittgenstein. The extensibility of Wittgenstein’s theory of number to the case of transfinite cardinalities is, I shall argue, a desidera

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Ganguli, Nirendra Mohan. "Counting of Numbers." Indian Science Cruiser, August 9, 2023, 33–38. http://dx.doi.org/10.24906/isc/2023/v37/i1/222818.

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‘χ0’ and ‘c’, termed as transfinite numbers, signify the number of elements in the infinite set of positive integers and that of real numbers respectively. Cantor proved that c > χ0 using the diagonal argument; but the infinitude of ‘c’ is on two counts i.e. magnitude and type of the element while that of χ0 is on single count i.e. magnitude. ‘c’ is not numerically greater than ‘χ0’, but denotes a higher degree of infinity. Hence the continuum hypothesis is rather irrelevant.

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Friedman, Harvey M., and Andreas Weiermann. "Some independence results related to finite trees." Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 381, no. 2248 (2023). http://dx.doi.org/10.1098/rsta.2022.0017.

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Abstract:

We investigate some concrete independence results for systems of reverse mathematics which emerge from monotonicity properties of number-theoretic functions. Natural properties of the less than or equal to relation with respect to sums of natural numbers lead to independence results for first-order Peano arithmetic. Natural properties of the less than or equal to relation with respect to sums and products of natural numbers lead to independence results for arithmetical transfinite recursion. By considering number-theoretic functions of arbitrary arities, we obtain independence results for syst

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