Zermelo-Fraenkel axiomatisation of set theory
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enPicking holes in mathematics
https://plus.maths.org/content/picking-holes-mathematics
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Marianne Freiberger </div>
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<p>In the 1930s the logician Kurt Gödel showed that if you set out proper rules for mathematics, you lose the ability to decide whether certain statements are true or false. This is rather shocking and you may wonder why Gödel's result hasn't wiped out mathematics once and for all. The answer is that, initially at least, the unprovable statements logicians came up with were quite contrived. But are they about to enter mainstream mathematics?</p>
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<p>Kurt Gödel</p><p><a href="https://plus.maths.org/content/picking-holes-mathematics" target="_blank">read more</a></p>https://plus.maths.org/content/picking-holes-mathematics#commentsmathematical realityaxiombinary treeGödel's Incompleteness Theoremgraphgraph theorylogicphilosophy of mathematicstreewhat is impossibleZermelo-Fraenkel axiomatisation of set theoryWed, 23 Feb 2011 10:00:00 +0000mf3445413 at https://plus.maths.org/contentSearching for the missing truth
https://plus.maths.org/content/searching-missing-truth
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Marianne Freiberger </div>
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<p>Many people like mathematics because it gives definite answers. Things are either true or false, and true things seem true in a very fundamental way. But it's not quite like that. You can actually build different versions of maths in which statements are true or false depending on your preference. So is maths just a game in which we choose the rules to suit our purpose? Or is there a "correct" set of rules to use? We find out with the mathematician Hugh Woodin.</p>
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<p>Many people like mathematics because it gives definite answers. Things are either true or false, and true things seem true in a very fundamental way. </p><div class="rightimage" style="width: 150px;"><img src="https://plus.maths.org/content/sites/plus.maths.org/files/articles/2011/woodin/woodin.jpg" alt="" width="150" height="224"
<p>Hugh Woodin.</p><p><a href="https://plus.maths.org/content/searching-missing-truth" target="_blank">read more</a></p>https://plus.maths.org/content/searching-missing-truth#commentsmathematical realitycontinuum hypothesisGödel's Incompleteness Theoreminfinitylogicphilosophy of mathematicsset theorywhat is impossiblewhat is infinityZermelo-Fraenkel axiomatisation of set theoryFri, 28 Jan 2011 19:09:07 +0000mf3445398 at https://plus.maths.org/contentCantor and Cohen: Infinite investigators part I
https://plus.maths.org/content/cantor-and-cohen-infinite-investigators-part-i
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Richard Elwes </div>
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What's the nature of infinity? Are all infinities the same? And what happens if you've got infinitely many infinities? In this article <b>Richard Elwes</b> explores how these questions brought triumph to one man and ruin to another, ventures to the limits of mathematics and finds that, with infinity, you're spoilt for choice. </div>
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<div class="pub_date">June 2008</div>
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<h1>The axiom of choice</h1>
<p><i>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 <a href="/issue47/features/elwes2">other article</a>
explores what is known as the continuum hypothesis.<p><a href="https://plus.maths.org/content/cantor-and-cohen-infinite-investigators-part-i" target="_blank">read more</a></p>https://plus.maths.org/content/cantor-and-cohen-infinite-investigators-part-i#comments47axiomaxiom of choicehistory of mathematicsinfinitylogicphilosophy of mathematicsRussell's Paradoxset theorywhat is infinityZermelo-Fraenkel axiomatisation of set theoryMon, 02 Jun 2008 23:00:00 +0000plusadmin2329 at https://plus.maths.org/contentCantor and Cohen: Infinite investigators part II
https://plus.maths.org/content/cantor-and-cohen-infinite-investigators-part-ii
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Richard Elwes </div>
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<b>Richard Elwes</b> continues his investigation into Cantor and Cohen's work. He investigates the <i>continuum hypothesis</i>, the question that caused Cantor so much grief. </div>
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<div class="pub_date">June 2008</div>
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<h1>The continuum hypothesis</h1>
<p><i>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. This article explores what is known as the continuum hypothesis, while <a href="/issue47/features/elwes1">the other article</a> explores the axiom
of choice.<p><a href="https://plus.maths.org/content/cantor-and-cohen-infinite-investigators-part-ii" target="_blank">read more</a></p>https://plus.maths.org/content/cantor-and-cohen-infinite-investigators-part-ii#comments47axiomcontinuum hypothesishilbert problemshistory of mathematicsinfinitylogicphilosophy of mathematicsset theorywhat is infinityZermelo-Fraenkel axiomatisation of set theorySun, 01 Jun 2008 23:00:00 +0000plusadmin2330 at https://plus.maths.org/contentMathematical mysteries: The Barber's Paradox
https://plus.maths.org/content/mathematical-mysteries-barbers-paradox
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Helen Joyce </div>
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<p>Suppose you walk past a barber's shop one day, and see a sign that says</p>
<p>"Do you shave yourself? If not, come in and I'll shave you! I shave anyone who does not shave himself, and noone else."<br />
This seems fair enough, and fairly simple, until, a little later, the following question occurs to you - does the barber shave himself?</p>
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<div class="pub_date">May 2002</div>
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<h2>A close shave for set theory</h2>
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<p>Suppose you walk past a barber's shop one day, and see a sign that says</p><p><a href="https://plus.maths.org/content/mathematical-mysteries-barbers-paradox" target="_blank">read more</a></p>https://plus.maths.org/content/mathematical-mysteries-barbers-paradox#comments20Barber's ParadoxlogicMathematical mysteriesphilosophy of mathematicsRussell's Paradoxset theoryTheory of Typeswhat is impossibleZermelo-Fraenkel axiomatisation of set theoryTue, 30 Apr 2002 23:00:00 +0000plusadmin4757 at https://plus.maths.org/content