# Foundations

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EATCS Bulletin , June 2002, vol. 77, pp. 167-179 G. J.

## Cantor and Cohen: Infinite investigators part I

June 2008 This is one half of a two-part article telling a story of two mathematical problems and two men: Georg Cantor, who discovered the strange world that these problems inhabit, and Paul Cohen (who died last year), who eventually solved them. The first of these problems — the axiom of choice — is the subject of this article, while the other article explores what is known as the continuum hypothesis. Each article is self-contained, so you don't have to read both to get the picture. Cantor: The infinite match-maker Georg Cantor was a German logician who, in the late 19th century, achieved a feat which scientists, philosophers, and theologians had previously only dreamed about: a detailed analysis of infinity.
The Whole Jolly Lot (now enriched with The list is presented here in reverse chronological order, so that new additions will appear at the top. This is not the order in which the theorem of the day is picked which is more designed to mix up the different areas of mathematics and the level of abstractness or technicality involved. The way that the list of theorems is indexed is described here . Every theorem number is linked to its entry in the delightful 'Prime Curios!' compendium at The Prime Pages .

## Theorem of the Day

Gödel's incompleteness theorems are two theorems of mathematical logic that establish inherent limitations of all but the most trivial axiomatic systems capable of doing arithmetic . The theorems, proven by Kurt Gödel in 1931, are important both in mathematical logic and in the philosophy of mathematics . The two results are widely, but not universally, interpreted as showing that Hilbert's program to find a complete and consistent set of axioms for all mathematics is impossible, giving a negative answer to Hilbert's second problem . The first incompleteness theorem states that no consistent system of axioms whose theorems can be listed by an " effective procedure " (e.g., a computer program, but it could be any sort of algorithm) is capable of proving all truths about the relations of the natural numbers ( arithmetic ). For any such system, there will always be statements about the natural numbers that are true, but that are unprovable within the system.

## Hammack Home

This book is an introduction to the standard methods of proving mathematical theorems. It has been approved by the American Institute of Mathematics' Open Textbook Initiative . Also see the Mathematical Association of America Math DL review , and the Amazon reviews . Click here for a pdf copy of the entire book, or get the chapters individually below. I: Fundamentals II: How to Prove Conditional Statements III: More on Proof IV: Relations, Functions and Cardinality