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-1-1Definition of Intuitive Set Theory
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The two axioms which define intuitive set theory, Axiom of Combinatorial Sets and Axiom of Infinitesimals, are stated. Generalized Continuum Hypothesis is derived from the first axiom, and the infinitesimal is visualized using the latter.Sat, 03 Nov 2001 08:00:00 GMTKannan NambiarGeneralized Continuum Hypothesis and the Axiom of Combinatorial Sets
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Axiom of Combinatorial Sets is defined and used to derive Generalized Continuum Hypothesis.Sun, 07 Apr 2002 08:00:00 GMTKannan NambiarInstant Proof of Pythagoras' Theorem
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<p>Generalized Pythagoras theorem is discussed in terms of matrices.</p>Thu, 28 Mar 2002 08:00:00 GMTKannan NambiarLogsets and ZF Theory
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Logset, the inverse of the powerset operation, is introduced into set theory.Thu, 20 Jun 2002 07:00:00 GMTKannan NambiarTeaching Generalized Continuum Hypothesis
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Generalized Continuum Hypothesis is derived from a <br/>simple axiom called Axiom of Combinatorial Sets.Sun, 31 Mar 2002 08:00:00 GMTKannan NambiarThe Essence of Intuitive Set Theory
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Intuitive Set Theory (IST) is defined as the theory we get, when we add Axiom of Monotonicity and Axiom of Fusion to Zermelo-Fraenkel set theory. In IST, Continuum Hypothesis is a theorem, Axiom of Choice is a theorem, Skolem paradox does not appear, nonLebesgue measurable sets are not possible, and the unit interval splits into a set of infinitesimals.Sat, 03 Nov 2001 08:00:00 GMTKannan NambiarTwenty Problems of Information Technology
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Twenty significant and pressing problems of information technology are listed.Sat, 23 Feb 2002 08:00:00 GMTKannan NambiarMediaMathBlog
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A weblog in which the use of multimedia in distance education is discussed.Mon, 17 Jul 2006 07:00:00 GMTKannan NambiarFoundations of Computer Science
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NuMachine, as powerful as Turing machine, but more intuitive in its working is described. Adding three more derivation rules to Elementary Arithmetic of Godel and calling it Sentient Arithmetic (SA), the incompleteness theorems are proved within SA, without using any metalanguage. Intuitive Set Theory (IST), a theory in which we do not have to deal with cardinals higher than aleph-null, is described. In IST, there is no Skolem Paradox and there are no nonLebesgue measurable sets.Wed, 13 Jun 2001 07:00:00 GMTKannan NambiarWhite Hole, Black Whole, and The Book
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Intellectual space is defined as the set of all proofs of mathematical logic, contained in The Book of Erdos. Physical and intellectual spaces are visualized making use of concepts from Intuitive Set Theory.Tue, 26 Jun 2001 07:00:00 GMTKannan NambiarA Set Theory for Scientists and Engineers (youtube video)
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<p>Engineers know that they can land a man on the moon without using the Lebesgue integral and they will never encounter Skolem paradox in their nuclear reactor design. Intuitive Set Theory (IST) defined here, de-emphasizes concepts that are not required by scientists in their practical work.<br /><br />AXIOM OF COMBINATORIAL SETS: A set as important as the powerset of Cantor is what I call the combinatorial set of \aleph_0, which is defined as the set of all subsets of \aleph_0 with cardinality \aleph_0. Axiom of Combinatorial Sets (ACS) says that \aleph_1 is equal to the combinatorial set of \aleph_0. Even though, the combinatorial set is a subset of the powerset, it can be shown that powerset and combinatorial set have the same cardinality.<br /><br />AXIOM OF iNFINITESIMALS: First of all, let us note that corresponding to every real recursive number it is possible to visualize an infinitesimal attached to it. We will illustrate this with an example. Consider the number 2/3 written as an infinite binary sequence 0.101010... and its finite terminations 0.1, 0.101, 0.10101, ... which can be used to represent the intervals (1/2,2/3), (5/8,2/3), (21/32, 2/3), ... respectively. Note that the length of the interval decreases monotonically when the length of the termination increases and the cardinality of the set of points inside these intervals remain constant at 2^\aleph_0. From this, we can say that an infinitesimal is what we get when we visualize the interval corresponding to the entire nonterminating sequence, and this infinitely small interval contains 2^\aleph_0 points in it. The Axiom of Infinitesimals (AI) says that the unit interval is a set, with cardinality \aleph_0, of infinitesimals. We call an infinitesimal an relement and the elements in it figments, claiming that not even the axiom of choice can pick a figment from an relement.<br /><br />INTUITIVE SET THEORY: We define IST as the theory we get when AI and ACS are added to ZF theory. The discerning reader will easily recognize that the notion of a figment will not allow nonLebesgue measurable sets in IST. Also, the fact that \aleph_0 is the cardinality of the set of infinitesimals in a unit interval, provides us with a way to circumvent the Skolem paradox.<br /><br />IN A NUTSHELL: If only relements are allowed in set theory, it is enough for scientists for all practical purposes. If all elements of ZF theory are allowed, then set theorists can live happily in "Cantor's heaven".</p>Tue, 19 May 2009 11:30:12 GMTKannan NambiarAckermann Functions and Transfinite Ordinals
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An important part of Cantor's set theory, which forms the foundations of mathematics, is the concept of transfinite ordinals. A systematic way of writing the sequence of ordinals is given.Mon, 16 Jun 2008 05:51:20 GMTKannan NambiarDerivation of Continuum Hypothesis from Axiom of Combinatorial Sets
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Continuum Hypothesis is derived from an axiom called Axiom of Combinatorial Sets. The derivation is simple enough to be understood by any novice, with a passing acquintance of cardinals of Cantor.Tue, 02 Dec 2003 08:00:00 GMTKannan NambiarEight Wonders of the Mathematical World (youtube video)
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<p><span>The biggest mystery for me is that the sequence of symbols we scribble on paper is able to simulate our complex universe. From this point of view, The Book of Paul Erdos is nothing but the DNA molecule of the universe, from which we can decipher every mystery around us.</span></p><p> </p>Sun, 17 May 2009 02:48:08 GMTKannan NambiarElectrical Equivalent of Riemann Hypothesis
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Riemann Hypothesis is viewed as a statement about the power dissipated in an electrical network.Mon, 11 Aug 2003 07:00:00 GMTKannan NambiarGeometrical Equivalents of Goldbach Conjecture and Fermat Like Theorem
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Five geometrical eqivalents of Goldbach conjecture are given, calling one of them Fermat Like Theorem.Mon, 04 Nov 2002 08:00:00 GMTKannan NambiarInformation-Theoretic Equivalent of Riemann Hypothesis
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Riemann Hypothesis is viewed as a statement about the capacity of a communication channel as defined by Shannon.Thu, 10 Jul 2003 07:00:00 GMTKannan NambiarMetaMathBlog
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A weblog in which issues of metamathematics are discussed.Wed, 22 Jun 2005 07:00:00 GMTKannan NambiarReal Set Theory
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An axiomatic theory called Real Set Theory is defined in which Generalized Continuum Hypothesis and Axiom of Choice are theorems.Mon, 25 Jun 2001 07:00:00 GMTKannan NambiarSentient Arithmetic and Godel's Theorems
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Godel has proved that there are formulas in Elementary Arithmetic, which will introduce contradictions, irrespective of whether we assume the <br/>formula itself or its negation. His proof is in metalanguage. Sentient Arithmetic (SA) adds three more derivation rules to EA and shows that the proof for incompleteness of SA can be given in SA itself without using any metalanguage.Mon, 25 Jun 2001 07:00:00 GMTKannan NambiarShannon's Communication Channels and Word Spaces
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A complete analysis of Shannon's telegraph channel is given, making use of matrices with elements from a division ring. A notation is developed for representing the set of signals of a communication channel.Wed, 07 Nov 2001 08:00:00 GMTKannan NambiarThe Ecstasy of HyperLaTeX
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The most popular format for publishing mathematics on the Web is the PDF, and HyperLaTeX is the program which allows us to introduce hyper references in the PDF documents. This paper is a tutorial on using HyperLaTeX to produce PDF files.Fri, 30 Nov 2001 08:00:00 GMTKannan NambiarThe Mathematical Universe in a Nutshell
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The mathematical universe discussed here gives models of possible structures our physical universe can have.Wed, 31 Jul 2002 07:00:00 GMTKannan NambiarThe Mathematical Universe in a Nutshell: A Presentation
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The mathematical universe discussed here gives models of possible structures our physical universe can have.Sat, 01 Nov 2003 08:00:00 GMTKannan Nambiar