http://www.merlot.org:80/merlot/ A search of MERLOT materials Copyright 1997-2013 MERLOT. All rights reserved. Mon, 17 Jun 2013 23:21:40 PDT Mon, 17 Jun 2013 23:21:40 PDT http://www.merlot.org:80/merlot/images/merlot.gif http://www.merlot.org:80/merlot/ 44 34 http://www.merlot.org/merlot/viewMaterial.htm?id=75818 A simple and short proof of Dictator Theorem is given. Loosely stated, the theorem says that democracies are not possible, with the prevalent voting systems. http://www.merlot.org/merlot/viewMaterial.htm?id=75812 It is shown that Fractional Voting System can be used to circumvent Arrow's paradox. The paradox states that fair elections are not possible with the present voting systems. http://www.merlot.org/merlot/viewMaterial.htm?id=89637 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. http://www.merlot.org/merlot/viewMaterial.htm?id=89727 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. http://www.merlot.org/merlot/viewMaterial.htm?id=383787 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.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.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.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.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״. http://www.merlot.org/merlot/viewMaterial.htm?id=75820 A definition of information, which forms the basis of the current information technology, is given in terms of two axioms. http://www.merlot.org/merlot/viewMaterial.htm?id=574569 An author's Snapshot for Arrow's Paradox and the Fractional Voting System for the material found in MERLOT at http://www.merlot.org/merlot/viewMaterial.htm?id=75812. This snapshot shows an overview of the material. This was created in the MERLOT Content Builder. http://www.merlot.org/merlot/viewMaterial.htm?id=75790 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. http://www.merlot.org/merlot/viewMaterial.htm?id=574559 An author's Snapshot for Definition of Intuitive Set Theory material found in MERLOT at http://www.merlot.org/merlot/viewMaterial.htm?id=75790. This snapshot shows an overview of the material. This was created in the MERLOT Content Builder. http://www.merlot.org/merlot/viewMaterial.htm?id=80317 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.