DOCSLIB.ORG

The Infinity Theorem Is Presented Stating That There Is at Least One Multivalued Series That Diverge to Infinity and Converge to Infinite Finite Values

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
The Infinity Theorem Is Presented Stating That There Is at Least One Multivalued Series That Diverge to Infinity and Converge to Infinite Finite Values Open Journal of Mathematics and Physics | Volume 2, Article 75, 2020 | ISSN: 2674-5747 https://doi.org/10.31219/osf.io/9zm6b | published: 4 Feb 2020 | https://ojmp.wordpress.com CX [microresearch] Diamond Open Access The infinity theorem Open Mathematics Collaboration∗† March 19, 2020 Abstract The infinity theorem is presented stating that there is at least one multivalued series that diverge to infinity and converge to infinite finite values. keywords: multivalued series, infinity theorem, infinite Introduction 1. 1, 2, 3, ..., ∞ 2. N =x{ N 1, 2,∞3,}... ∞ > ∈ = { } The infinity theorem 3. Theorem: There exists at least one divergent series that diverge to infinity and converge to infinite finite values. ∗All authors with their affiliations appear at the end of this paper. †Corresponding author: [email protected] | Join the Open Mathematics Collaboration 1 Proof 1 4. S 1 1 1 1 1 1 ... 2 [1] = − + − + − + = (a) 5. Sn 1 1 1 ... 1 has n terms. 6. S = lim+n + S+n + 7. A+ft=er app→ly∞ing the limit in (6), we have S 1 1 1 ... + 8. From (4) and (7), S S 2 = 0 + 2 + 0 + 2 ... 1 + 9. S 2 2 1 1 1 .+.. = + + + + + + 1 10. Fro+m (=7) (and+ (9+), S+ )2 2S . + +1 11. Using (6) in (10), limn+ =Sn 2 2 limn Sn. 1 →∞ →∞ 12. limn Sn 2 + = →∞ 1 13. From (6) a=nd (12), S 2. + 1 14. From (7) and (13), S = 1 1 1 ... 2. + = + + + = (b) 15. S 1 1 1 1 1 ... + 1 16. S =0 +1 +1 +1 +1 +1 1 ... 2 17. S = S+ 1− 2+ 2− 2+ .−.. + = + 1 18. S + 2 =1 +2S+ + + + 1 + 19. S + 2= + + = − 2 (c) 20. S 1 1 1 1 1 ... + 1 21. S =0 +0 +1 +1 +1 +1 1 1 ... 2 22. S = S+ 1+ 1− 2+ 2− 2+ .−.. + = + 1 23. S + 2 =2 +2S+ + + + + 3 + 24. S + 2= + + = − (n) 2n 1 25. S 2 for n N 1, 2, 3, ... 0 . ( − ) + = − ∈ = { } ∪ { } (n ) 26. Now, we will take n . → ∞ 1 27. (25) can be written a→s S∞ n 2. 28. From (26) and (27), + = − + 1 S lim n . n 2 + = →∞ (− + ) = −∞ 29. From (7) and (26), n S 1 lim 1 lim n . ∞ n n i 1 i 1 + →∞ →∞ = ∑= = ∑= = = ∞ Final Remarks 30. S converges. 31. S+ diverges. + 3 32. In summary, the series S diverges to infinity and converges to infinite finite values. + 33. The infinite is the quantum superposition of mathematics [2–5]. 34. S is a multivalued series. + 2n 1 35. S 2 , n N 0 ( − ) + = {− ±∞ ∶ ∈ ∪ { }} A complex multivalued integral 1 x 36. Let S1 0 1 dx. iθ 37. e co=s θ∫ (i−sin)θ 38. eiπ = 1 + 39. For=n− Z, ei π 2nπ 1. ( + ) 40. Substi∈tuting (39) in=(−36), 1 iπ 2n 1 x S1 e dx. 0 ( + ) 41. = ∫ 1 eiπ 2n 1 x S1 ( + ) iπ 2n 1 0 42. = ∣ eiπ( 2n +1 )1 S1 iπ( 2n+ ) 1 − = 43. Inserting back (39) into the first te(rm +of t)he numerator in (42), 2 S . 1 iπ 2n 1 − = 44. Then, ( + ) 2i S1 n Z . 2n 1 π = { ∶ ∈ } ( +4 ) 45. S is a discrete multivalued sum. + 46. S1 is a continuous multivalued sum. Multivalued subtraction 47. S 1 1 1 ... 48. S+ = 1 + 1+ 1+ 1 ... 49. S+ = 1 + (S + + + ) 50. S+ = S+ 1+ 51. Si+n−ce S+ =is multivalued, let’s rewrite (50) as S S 1. $$ $ 52. In (51),+there are the following two cases: + − + = (i) S S , ii S$$ S$ ( ) + = + . $$ $ 53. (52i) and+ (≠51)+ lead to 0 1, which is absurd. 54. From (52) and (53), S = S . $$ $ + + 3 1 55. From (35) and (54), one≠possibility for (51) is S 2, and S 2. $$ $ + = + = Riemann’s rearrangement theorem 56. [6,7] 57. 1 n 1 1 1 1 1 1 R 1 ... +! + n 1 n 2 3 4 5 6 (− ) 58. = ∑= = − + − + − + R ln 2 = 5 59. 1 R ln 2 2 $ 60. = R ... ... $$$ 61. The positive terms of R sum up to =infinity (R ). 62. The negative terms of R sum up to minus infin→ity∞(R ). 63. Let x R. → −∞ 64. From (∈61) and (62), R can be rearranged to R x a1 a1 a2 a2 a3 a3 ... x. 65. Therefore, = + ( − ) + ( − ) + ( − ) + = R x x R . = { ∶ ∈ } Open Invitation Review, add content, and co-author this paper [8,9]. Join the Open Mathematics Collaboration. Send your contribution to [email protected]. Ethical conduct of research This original work was pre-registered under the OSF Preprints [10], please cite it accordingly [11]. This will ensure that researches are con- ducted with integrity and intellectual honesty at all times and by all means. 6 References [1] Lobo, Matheus P. “The Convergence of Grandi’s Series.” OSF Preprints, 4 Jan. 2020. https://doi.org/10.31219/osf.io/aq5xz [2] Lobo, Matheus P. “Superposition as a Mathematical Mapping Between Two Distinct Scales.” OSF Preprints, 31 Dec. 2019. https://doi.org/10.31219/osf.io/uksgx [3] Lobo, Matheus P. “Quantum Superposition as Entanglement.” OSF Preprints, 25 Dec. 2019. https://doi.org/10.31219/osf.io/m2ajq [4] Lobo, Matheus P. “Spin Is a Superposition of Circu- lar Charged Trajectories.” OSF Preprints, 30 June 2019. https://doi.org/10.31219/osf.io/tv2rb [5] Lobo, Matheus P. “Time Travel: Coexistence of Past, Present, and Future?.” OSF Preprints, 2 Sept. 2019. https://doi.org/10.31219/osf.io/7ruay [6] Wolfram MathWorld. Riemann Series Theorem. Retrieved 2020-3-19. https://mathworld.wolfram.com/RiemannSeriesTheorem.html [7] StackExchange: Mathematics. Retrieved 2020-3-19. https://math.stackexchange.com/questions/1795509 [8] Lobo, Matheus P. “Microarticles.” OSF Preprints, 28 Oct. 2019. https://doi.org/10.31219/osf.io/ejrct [9] Lobo, Matheus P. “Simple Guidelines for Authors: Open Jour- nal of Mathematics and Physics.” OSF Preprints, 15 Nov. 2019. https://doi.org/10.31219/osf.io/fk836 [10] COS. Open Science Framework. https://osf.io [11] Lobo, Matheus P. “The Infinity Theorem.” OSF Preprints, 4 Feb. 2020. https://doi.org/10.31219/osf.io/9zm6b 7 The Open Mathematics Collaboration Matheus Pereira Lobo (lead author, [email protected]),1,2 Djane da Silva Souza,1 José Carlos de Oliveira Junior1 1Federal University of Tocantins (Brazil); 2Universidade Aberta (UAb, Portugal) 8.
Load more

Recommended publications
  • Theorems Wednesday, May 9, 2018 12:53 AM
    Theorems Wednesday, May 9, 2018 12:53 AM Theorem 1.11: Greatest-Lower-Bound Property • Suppose is an ordered set with the least-upper-bound property • Suppose , and is bounded below • Let be the set of lower bounds of • Then exists in and Theorem 1.20: The Archimedean property of • Given , and • There is a positive integer such that Theorem 1.20: is dense in • If , and , then there exists a s.t. • We can always find a rational number between two real numbers Theorem 1.21: -th Root of Real Numbers • For every real , and positive integer • There is one and only one positive real number s.t. • Theorem 1.31: Properties of Complex Numbers • If and are complex numbers, then • • • • is real and positive (except when ) Theorem 1.33: Properties of Complex Numbers • If and are complex numbers, then • unless in which case • • • • (Triangle Inequality) Theorem 1.37: Properties of Euclidean Spaces • Suppose , then • Page 1 • if and only if • • (Schwarz's Inequality) • (Triangle Inequality) • (Triangle Inequality) Theorem 2.8: Infinite Subset of Countable Set • Every infinite subset of a countable set is countable Theorem 2.12: Union of Countable Sets • Let be a sequence of countable sets, then • Theorem 2.13: Cartesian Product of Countable Sets • Let be a countable set • Let be the set of all -tuples where ○ for ○ may not be distinct • Then is countable Theorem 2.14: Cantor's Diagonalization Argument • Let be the set of all sequqnecse whose digits are 0 and 1 • Then is uncountable Theorem 2.19: Every Neighborhood is an Open Set • Every neighborhood
    [Show full text]
  • The Modal Logic of Potential Infinity, with an Application to Free Choice
    The Modal Logic of Potential Infinity, With an Application to Free Choice Sequences Dissertation Presented in Partial Fulfillment of the Requirements for the Degree Doctor of Philosophy in the Graduate School of The Ohio State University By Ethan Brauer, B.A. ∼6 6 Graduate Program in Philosophy The Ohio State University 2020 Dissertation Committee: Professor Stewart Shapiro, Co-adviser Professor Neil Tennant, Co-adviser Professor Chris Miller Professor Chris Pincock c Ethan Brauer, 2020 Abstract This dissertation is a study of potential infinity in mathematics and its contrast with actual infinity. Roughly, an actual infinity is a completed infinite totality. By contrast, a collection is potentially infinite when it is possible to expand it beyond any finite limit, despite not being a completed, actual infinite totality. The concept of potential infinity thus involves a notion of possibility. On this basis, recent progress has been made in giving an account of potential infinity using the resources of modal logic. Part I of this dissertation studies what the right modal logic is for reasoning about potential infinity. I begin Part I by rehearsing an argument|which is due to Linnebo and which I partially endorse|that the right modal logic is S4.2. Under this assumption, Linnebo has shown that a natural translation of non-modal first-order logic into modal first- order logic is sound and faithful. I argue that for the philosophical purposes at stake, the modal logic in question should be free and extend Linnebo's result to this setting. I then identify a limitation to the argument for S4.2 being the right modal logic for potential infinity.
    [Show full text]
  • Cantor on Infinity in Nature, Number, and the Divine Mind
    Cantor on Infinity in Nature, Number, and the Divine Mind Anne Newstead Abstract. The mathematician Georg Cantor strongly believed in the existence of actually infinite numbers and sets. Cantor’s “actualism” went against the Aristote- lian tradition in metaphysics and mathematics. Under the pressures to defend his theory, his metaphysics changed from Spinozistic monism to Leibnizian volunta- rist dualism. The factor motivating this change was two-fold: the desire to avoid antinomies associated with the notion of a universal collection and the desire to avoid the heresy of necessitarian pantheism. We document the changes in Can- tor’s thought with reference to his main philosophical-mathematical treatise, the Grundlagen (1883) as well as with reference to his article, “Über die verschiedenen Standpunkte in bezug auf das aktuelle Unendliche” (“Concerning Various Perspec- tives on the Actual Infinite”) (1885). I. he Philosophical Reception of Cantor’s Ideas. Georg Cantor’s dis- covery of transfinite numbers was revolutionary. Bertrand Russell Tdescribed it thus: The mathematical theory of infinity may almost be said to begin with Cantor. The infinitesimal Calculus, though it cannot wholly dispense with infinity, has as few dealings with it as possible, and contrives to hide it away before facing the world Cantor has abandoned this cowardly policy, and has brought the skeleton out of its cupboard. He has been emboldened on this course by denying that it is a skeleton. Indeed, like many other skeletons, it was wholly dependent on its cupboard, and vanished in the light of day.1 1Bertrand Russell, The Principles of Mathematics (London: Routledge, 1992 [1903]), 304.
    [Show full text]
  • Mathematics Course: Analysis I
    Analysis I Carl Turner November 21, 2010 Abstract These are notes for an undergraduate course on analysis; please send corrections, suggestions and notes to [email protected] The author's homepage for all courses may be found on his website at SuchIdeas.com, which is where updated and corrected versions of these notes can also be found. The course materials are licensed under a permissive Creative Commons license: Attribution- NonCommercial-ShareAlike 3.0 Unported (see the CC website for more details). Thanks go to Professor G. Paternain for allowing me to use his Analysis I course (Lent 2010) as the basis for these notes. Contents 1 Limits and Convergence3 1.1 Fundamental Properties of the Real Line...........................3 1.2 Cauchy Sequences.......................................5 1.3 Series..............................................6 1.4 Sequences of Non-Negative Terms...............................7 1.5 Alternating Series and Absolute Convergence........................ 10 2 Continuity 13 2.1 Denitions and Basic Properties............................... 13 2.2 Intermediate Values, Extreme Values and Inverses..................... 15 3 Dierentiability 18 3.1 Denitions and Basic Properties............................... 18 3.2 The Mean Value Theorem and Taylor's Theorem...................... 20 3.3 Taylor Series and Binomial Series............................... 25 3.4 * Comments on Complex Dierentiability.......................... 26 4 Power Series 28 4.1 Fundamental Properties.................................... 28 4.2 Standard Functions....................................... 32 4.2.1 Exponentials and logarithms............................. 32 1 4.2.2 Trigonometric functions................................ 35 4.2.3 Hyperbolic functions.................................. 37 5 Integration 38 5.1 Denitions and Key Properties................................ 38 5.2 Computation of Integrals................................... 44 5.3 Mean Values and Taylor's Theorem Again, Improper Integrals and the Integral Test.
    [Show full text]
  • 1 Notes for Expansions/Series and Differential Equations in the Last
    Notes for Expansions/Series and Differential Equations In the last discussion, we considered perturbation methods for constructing solutions/roots of algebraic equations. Three types of problems were illustrated starting from the simplest: regular (straightforward) expansions, non-uniform expansions requiring modification to the process through inner and outer expansions, and singular perturbations. Before proceeding further, we first more clearly define the various types of expansions of functions of variables. 1. Convergent and Divergent Expansions/Series Consider a series, which is the sum of the terms of a sequence of numbers. Given a sequence {a1,a2,a3,a4,a5,....,an..} , the nth partial sum Sn is the sum of the first n terms of the sequence, that is, n Sn = ak . (1) k =1 A series is convergent if the sequence of its partial sums {S1,S2,S3,....,Sn..} converges. In a more formal language, a series converges if there exists a limit such that for any arbitrarily small positive number ε>0, there is a large integer N such that for all n ≥ N , ^ Sn − ≤ ε . (2) A sequence that is not convergent is said to be divergent. Examples of convergent and divergent series: • The reciprocals of powers of 2 produce a convergent series: 1 1 1 1 1 1 + + + + + + ......... = 2 . (3) 1 2 4 8 16 32 • The reciprocals of positive integers produce a divergent series: 1 1 1 1 1 1 + + + + + + ......... (4) 1 2 3 4 5 6 • Alternating the signs of the reciprocals of positive integers produces a convergent series: 1 1 1 1 1 1 − + − + − + ......... = ln 2 .
    [Show full text]
  • Measuring Fractals by Infinite and Infinitesimal Numbers
    MEASURING FRACTALS BY INFINITE AND INFINITESIMAL NUMBERS Yaroslav D. Sergeyev DEIS, University of Calabria, Via P. Bucci, Cubo 42-C, 87036 Rende (CS), Italy, N.I. Lobachevsky State University, Nizhni Novgorod, Russia, and Institute of High Performance Computing and Networking of the National Research Council of Italy http://wwwinfo.deis.unical.it/∼yaro e-mail: [email protected] Abstract. Traditional mathematical tools used for analysis of fractals allow one to distinguish results of self-similarity processes after a finite number of iterations. For example, the result of procedure of construction of Cantor’s set after two steps is different from that obtained after three steps. However, we are not able to make such a distinction at infinity. It is shown in this paper that infinite and infinitesimal numbers proposed recently allow one to measure results of fractal processes at different iterations at infinity too. First, the new technique is used to measure at infinity sets being results of Cantor’s proce- dure. Second, it is applied to calculate the lengths of polygonal geometric spirals at different points of infinity. 1. INTRODUCTION During last decades fractals have been intensively studied and applied in various fields (see, for instance, [4, 11, 5, 7, 12, 20]). However, their mathematical analysis (except, of course, a very well developed theory of fractal dimensions) very often continues to have mainly a qualitative character and there are no many tools for a quantitative analysis of their behavior after execution of infinitely many steps of a self-similarity process of construction. Usually, we can measure fractals in a way and can give certain numerical answers to questions regarding fractals only if a finite number of steps in the procedure of their construction has been executed.
    [Show full text]
  • Calculus Terminology
    AP Calculus BC Calculus Terminology Absolute Convergence Asymptote Continued Sum Absolute Maximum Average Rate of Change Continuous Function Absolute Minimum Average Value of a Function Continuously Differentiable Function Absolutely Convergent Axis of Rotation Converge Acceleration Boundary Value Problem Converge Absolutely Alternating Series Bounded Function Converge Conditionally Alternating Series Remainder Bounded Sequence Convergence Tests Alternating Series Test Bounds of Integration Convergent Sequence Analytic Methods Calculus Convergent Series Annulus Cartesian Form Critical Number Antiderivative of a Function Cavalieri’s Principle Critical Point Approximation by Differentials Center of Mass Formula Critical Value Arc Length of a Curve Centroid Curly d Area below a Curve Chain Rule Curve Area between Curves Comparison Test Curve Sketching Area of an Ellipse Concave Cusp Area of a Parabolic Segment Concave Down Cylindrical Shell Method Area under a Curve Concave Up Decreasing Function Area Using Parametric Equations Conditional Convergence Definite Integral Area Using Polar Coordinates Constant Term Definite Integral Rules Degenerate Divergent Series Function Operations Del Operator e Fundamental Theorem of Calculus Deleted Neighborhood Ellipsoid GLB Derivative End Behavior Global Maximum Derivative of a Power Series Essential Discontinuity Global Minimum Derivative Rules Explicit Differentiation Golden Spiral Difference Quotient Explicit Function Graphic Methods Differentiable Exponential Decay Greatest Lower Bound Differential
    [Show full text]
  • Attributes of Infinity
    International Journal of Applied Physics and Mathematics Attributes of Infinity Kiamran Radjabli* Utilicast, La Jolla, California, USA. * Corresponding author. Email: [email protected] Manuscript submitted May 15, 2016; accepted October 14, 2016. doi: 10.17706/ijapm.2017.7.1.42-48 Abstract: The concept of infinity is analyzed with an objective to establish different infinity levels. It is proposed to distinguish layers of infinity using the diverging functions and series, which transform finite numbers to infinite domain. Hyper-operations of iterated exponentiation establish major orders of infinity. It is proposed to characterize the infinity by three attributes: order, class, and analytic value. In the first order of infinity, the infinity class is assessed based on the “analytic convergence” of the Riemann zeta function. Arithmetic operations in infinity are introduced and the results of the operations are associated with the infinity attributes. Key words: Infinity, class, order, surreal numbers, divergence, zeta function, hyperpower function, tetration, pentation. 1. Introduction Traditionally, the abstract concept of infinity has been used to generically designate any extremely large result that cannot be measured or determined. However, modern mathematics attempts to introduce new concepts to address the properties of infinite numbers and operations with infinities. The system of hyperreal numbers [1], [2] is one of the approaches to define infinite and infinitesimal quantities. The hyperreals (a.k.a. nonstandard reals) *R, are an extension of the real numbers R that contains numbers greater than anything of the form 1 + 1 + … + 1, which is infinite number, and its reciprocal is infinitesimal. Also, the set theory expands the concept of infinity with introduction of various orders of infinity using ordinal numbers.
    [Show full text]
  • 1 X = 1 Ln 2 Ln 2
    Let’s consider that power series for ln x about x 1 x12 x 1 3 x 1 4 x 1 5 x 1 6 x 1 7 x 1 8 lnxx 1 ... 2 3 4 5 6 7 8 So if this is true, then what is ln2 ? 212 21 3 21 4 21 5 21 6 21 7 21 8 ln2 2 1 ... 2 3 4 5 6 7 8 12 1 3 1 4 1 5 1 6 1 7 1 8 ln2 1 ... 2 3 4 5 6 7 8 n1 1 1 1 1 1 1 1 1 ln2 1 ... ... = the alternating harmonic series! 2 3 4 5 6 7 8 n This series is Conditionally Convergent since the non-alternating harmonic series diverges (by integral test), but the alternating series converges (by alternating series test). Now we know that the alternating harmonic series converges to . We can also show that this series converges when 02x . 111111111111111 So ln2 1 ... 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 Let’s rearrange these terms: 1111111111111111 ln2 1 ... 2 4 3 6 8 5 10 12 7 14 16 9 18 20 11 22 Do you agree that this series will include all of the terms of the power series for ? Let’s group certain terms together: 1111111111 111 1 11 ln2 1 ... 2 4 3 6 8 5 10 12 7 14 16 9 18 20 11 22 11111111111 = ... 2 4 6 8 10 12 14 16 18 20 22 Look at each term in this series, compared with each term in our original series for ln2.
    [Show full text]
  • If Riemann's Zeta Function Is True, It Contradicts Zeta's Dirichlet Series, Causing
    If Riemann’s Zeta Function is True, it Contradicts Zeta’s Dirichlet Series, Causing “Explosion”. If it is False, it Causes Unsoundness. Ayal I. Sharon ∗y September 4, 2019 Abstract Riemann’s "analytic continuation" produces a second definition of the Zeta function, that Riemann claimed is convergent throughout half-plane s 2 C, Re(s) ≤ 1, (except at s = 1). This contradicts the original definition of the Zeta function (the Dirichlet series), which is proven divergent there. Moreover, a function can- not be both convergent and divergent at any domain value. In physics and in other mathematics conjectures and assumed-proven theorems, the Riemann Zeta func- tion (or the class of L-functions that generalizes it) is assumed to be true. Here the author shows that the two contradictory definitions of Zeta violate Aristotle’s Laws of Identity, Non-Contradiction, and Excluded Middle. Non-Contradiction is an axiom of classical and intuitionistic logics, and an inherent axiom of Zermelo- Fraenkel set theory (which was designed to avoid paradoxes). If Riemann’s defini- tion of Zeta is true, then the Zeta function is a contradiction that causes deductive "explosion", and the foundation logic of mathematics must be replaced with one that is paradox-tolerant. If Riemann’s Zeta is false, it renders unsound all theo- rems and conjectures that falsely assume that it is true. Riemann’s Zeta function appears to be false, because its derivation uses the Hankel contour, which violates the preconditions of Cauchy’s integral theorem. ∗Patent Examiner, U.S. Patent and Trademark Office (USPTO). This research received no funding, and was conducted in the author’s off-duty time.
    [Show full text]
  • Georg Cantor English Version
    GEORG CANTOR (March 3, 1845 – January 6, 1918) by HEINZ KLAUS STRICK, Germany There is hardly another mathematician whose reputation among his contemporary colleagues reflected such a wide disparity of opinion: for some, GEORG FERDINAND LUDWIG PHILIPP CANTOR was a corruptor of youth (KRONECKER), while for others, he was an exceptionally gifted mathematical researcher (DAVID HILBERT 1925: Let no one be allowed to drive us from the paradise that CANTOR created for us.) GEORG CANTOR’s father was a successful merchant and stockbroker in St. Petersburg, where he lived with his family, which included six children, in the large German colony until he was forced by ill health to move to the milder climate of Germany. In Russia, GEORG was instructed by private tutors. He then attended secondary schools in Wiesbaden and Darmstadt. After he had completed his schooling with excellent grades, particularly in mathematics, his father acceded to his son’s request to pursue mathematical studies in Zurich. GEORG CANTOR could equally well have chosen a career as a violinist, in which case he would have continued the tradition of his two grandmothers, both of whom were active as respected professional musicians in St. Petersburg. When in 1863 his father died, CANTOR transferred to Berlin, where he attended lectures by KARL WEIERSTRASS, ERNST EDUARD KUMMER, and LEOPOLD KRONECKER. On completing his doctorate in 1867 with a dissertation on a topic in number theory, CANTOR did not obtain a permanent academic position. He taught for a while at a girls’ school and at an institution for training teachers, all the while working on his habilitation thesis, which led to a teaching position at the university in Halle.
    [Show full text]
  • A Mathematical Model of Divine Infinity
    A Mathematical Model of Divine Infinity Prof. Eric Steinhart, Department of Philosophy, William Paterson University, Wayne NJ 07470. Email: <[email protected]>, <[email protected]>. Published in 2009 in Theology and Science 7 (3), 261 – 274. ABSTRACT: Mathematics is obviously important in the sciences. And so it is likely to be equally important in any effort that aims to understand God in a scientifically significant way or that aims to clarify the relations between science and theology. The degree to which God has any perfection is absolutely infinite. We use contemporary mathematics to precisely define that absolute infinity. For any perfection, we use transfinite recursion to define an endlessly ascending series of degrees of that perfection. That series rises to an absolutely infinite degree of that perfection. God has that absolutely infinite degree. We focus on the perfections of knowledge, power, and benevolence. Our model of divine infinity thus builds a bridge between mathematics and theology. KEYWORDS: God; mathematics; perfection; infinity; Cantor; transfinite recursion. 1. Introduction All will agree that science makes extensive use of mathematics. At least in physics, scientific progress seems to go hand in hand with increasing formalization. More sophisticated physical theories are also more highly mathematical. And the correlation of progress with increasing formalization also appears sciences like chemistry and biology as well. Mathematics is obviously important in the sciences. And so it is likely to be equally important in any effort that aims to understand God in a scientifically significant way or that aims to clarify the relations between science and theology. Nevertheless, mathematics has seen little application in theology.
    [Show full text]