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4434MathPages
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.)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 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.Definition 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.Derivation of Continuum Hypothesis from Axiom of Combinatorial Sets
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.Generalized Continuum Hypothesis and the Axiom of Combinatorial Sets
http://www.merlot.org/merlot/viewMaterial.htm?id=76571
Axiom of Combinatorial Sets is defined and used to derive Generalized Continuum Hypothesis.Logsets and ZF Theory
http://www.merlot.org/merlot/viewMaterial.htm?id=77049
Logset, the inverse of the powerset operation, is introduced into set theory.Real Set Theory
http://www.merlot.org/merlot/viewMaterial.htm?id=89722
An axiomatic theory called Real Set Theory is defined in which Generalized Continuum Hypothesis and Axiom of Choice are theorems.Sentient Arithmetic and Godel's Theorems
http://www.merlot.org/merlot/viewMaterial.htm?id=89720
Godel has proved that there are formulas in Elementary Arithmetic, which will introduce contradictions, irrespective of whether we assume the 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.