Elementary set theory

Georg Cantor (1845 – 1918)

The actual infinite arises in three contexts: first when it is realized in the most complete form, in a fully independent otherworldly being, in Deo, where I call it the Absolute Infinite or simply Absolute; second when it occurs in the contingent, created world; third when the mind grasps it in abstract as a mathematical magnitude, number or order type.

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The van Gogh of mathematics, tested the bounds of infinity and sanity, a man extraordinary gifted but with a tragic life story. His incredible originality dared challenge any convention. The early work was a mixture of mathematics and theology. Cantor corresponded with the Pope himself who fortunately liked his ideas and encouraged him in his theory of infinities and which was approved of by the catholic church.

Cantor was convinced by prominent members of the mathematics community to rewrite his work without the philosophy and theology and as purely mathematics. Pioneer of set theory and logic with incredible profound insights much contrary to intuition for example a hierarchy of infinities. In mathematics there are always surprises strange things and mysteries when working with infinity. It cannot be fathomed as such.

There was another famous mathematician a hindu, Ramanujan, who claimed he very explicitly saw in dreams divine revelations, that scrolls containing the most complicated mathematics used to enfold before his eyes.

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There is nothing original in this my presentation and no rigour. I want only to have the reader to understand the concepts and results in a strongly intuitive way. The content is usually regarded as advanced but is within reach of  most laymen and without too much effort either.

The ideas are easy to grasp and are simple and with no interdependence of theorems and the proofs require essentially no prerequisites. I don't provide proofs they are not hard you can look them up just about anybody can understand them with a bit of effort, and with only minimal prior knowledge.

It is remarkable in that most people could be able to understand this work and as elementary, but usually is treated as more advanced work. The examples would help a lot.

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There are three parts,

Elementary set theory

Some theorems on sets

Examples and Definitions

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A Standard Challenge,

Show that the cardinality of R^2 is the same as R, that is there exists a one-to-one onto function between the points on the Euclidean plane and the real line.

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References,

Lebesgue Integration & Measure, Alan J. Weir. A very simple approach: The completeness of the reals and null sets.

General Topology, Stephen Willard. For a very brief summary: Set theory.

Measure Theory and Integration, G. de Barra. Introductory material: Preliminaries.