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4434Infinite Reflections
http://www.merlot.org/merlot/viewMaterial.htm?id=280379
The author offers reflections on specific questions mathematicians and philosophers have asked about the infinite over the centuries. He examines why explorers of the infinite, even in its strictly mathematical forms, often find it to be sublime.MathPages
http://www.merlot.org/merlot/viewMaterial.htm?id=83263
This site contains several hundred articles concerned with mathematics and physics. General topics include Number Theory, Combinatorics, Geometry, Algebra, Calculus & Differential Equations, Probability & Statistics, Set Theory & Foundations, Reflections on Relativity, History, and Physics. The articles under each general heading are highly varied, many are quite advanced, and there is no apparent organizational scheme. For example, under Calculus & Differential Equations there is a proof that pi is irrational, a examination of the Limit Paradox, a discussion of Ptolemy's Orbit, and an historical review of the cycloid among many other articles. Visitors can browse by topics or search by keyword. (Anyone with information on the identity of the site author please contact the MERLOT submitter.)The Teacher's Guide
http://www.merlot.org/merlot/viewMaterial.htm?id=674464
Provides free worksheets, printouts, lesson plans, SMARTBoard templates, and more. Also provides free Math Interactive Sites.White Hole, Black Whole, and The Book
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.A Crash Course in the Mathematics of Infinite Sets
http://www.merlot.org/merlot/viewMaterial.htm?id=280696
This is pretty much what it says it is. This course is written in a lively and engaging style. It starts with elementary set theory and quickly builds up to a discussion of the Continuum Hypothesis with quite a few proofs along the way.A Set Theory for Scientists and Engineers (youtube video)
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״.Ackermann Functions and Transfinite Ordinals
http://www.merlot.org/merlot/viewMaterial.htm?id=316548
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.cuerpos geometricos
http://www.merlot.org/merlot/viewMaterial.htm?id=723689
cuerpos geometricosDefinition of Intuitive Set Theory
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.Definition of Intuitive Set Theory: A Snapshot
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.