Gödel's Incompleteness Theorem
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enThis is not a carrot: Paraconsistent mathematics
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Maarten McKubre-Jordens </div>
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Paraconsistent mathematics is a type of mathematics in which contradictions may be true.
In such a system it is perfectly possible for a statement <em>A</em> and its negation <em>not A</em> to both be true. How can this be, and be coherent? What does it all mean? </div>
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<p>Paraconsistent mathematics is a type of mathematics in which contradictions may be true.
In such a system it is perfectly possible for a statement <em>A</em> and its negation <em>not A</em> to both be true. How can this be, and be coherent? What does it all mean?
And why should we think mathematics might actually be paraconsistent? We'll look
at the last question first starting with a quick trip into mathematical history.</p><p><a href="http://plus.maths.org/content/not-carrot" target="_blank">read more</a></p>http://plus.maths.org/content/not-carrot#commentsmathematical realityGödel's Incompleteness Theoremhalting problemimpossible objectlogicphilosophy of mathematicsRussell's Paradoxwhat is impossibleWed, 24 Aug 2011 07:42:07 +0000mf3445522 at http://plus.maths.org/contentPicking holes in 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="http://plus.maths.org/content/picking-holes-mathematics" target="_blank">read more</a></p>http://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 http://plus.maths.org/contentSearching for the missing truth
http://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="http://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="http://plus.maths.org/content/searching-missing-truth" target="_blank">read more</a></p>http://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 http://plus.maths.org/contentEditorial
http://plus.maths.org/content/pluschat-19
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<li>Plus 100 —the best maths of the last century</li>
<li>More maths grads</li>
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<div class="pub_date">June 2007</div>
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<h2><i>Plus</i> 100 — The best maths of the last century</h2>
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<!-- END OF <p><a href="http://plus.maths.org/content/pluschat-19" target="_blank">read more</a></p>43computer scienceeditorialfour-colour theoremGödel's Incompleteness Theoremhistory of mathematicsmathematics educationplus birthdayThu, 31 May 2007 23:00:00 +0000plusadmin4899 at http://plus.maths.org/contentWe must know, we will know
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Rebecca Morris </div>
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<b>Runner up in the general public category</b>. Great minds spark controversy. This is something you'd expect to hear about a great philosopher or artist, but not about a mathematician. Get ready to bin your stereotypes as <b>Rebecca Morris</b> describes some controversial ideas of the great mathematician David Hilbert. </div>
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<div class="pub_date">December 2006</div>
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<p style="color:purple;"><b><i>This article is a runner-up in the general public category of the Plus new writers award 2006.</i></b></p>
<p><i>"It seems that there was a mathematician who had become a novelist. 'Why did he do that?' people in Göttingen marvelled. 'How can a man who was a mathematician write novels?' 'But that is completely simple,' Hilbert said. 'He did not have enough imagination for mathematics, but he had enough for novels.' "</i></p><p><a href="http://plus.maths.org/content/we-must-know-we-will-know" target="_blank">read more</a></p>http://plus.maths.org/content/we-must-know-we-will-know#comments41axiomEuclidean geometryGödel's Incompleteness Theoremhilbert problemshistory of mathematicslogicphilosophy of mathematicsFri, 01 Dec 2006 00:00:00 +0000plusadmin2295 at http://plus.maths.org/contentGödel and the limits of logic
http://plus.maths.org/content/goumldel-and-limits-logic
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John W Dawson </div>
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When Kurt Gödel published his <i>incompleteness theorem</i> in 1931, the mathematical community was stunned: using maths he had proved that there are limits to what maths can prove. This put an end to the hope that all of maths could one day be unified in one elegant theory and had very real implications for computer science. <b>John W Dawson</b> describes Gödel's brilliant work and troubled
life. </div>
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<div class="pub_date">June 2006</div>
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<p>Kurt Gödel. Photograph by Alfred Eisenstaedt, taken from the Gödel Papers courtesy of <a href='http://www.princeton.edu/main/'>Princeton University</a> and <a href='http://www.ias.edu/'>Institute for Advanced Study</a>.</p><p><a href="http://plus.maths.org/content/goumldel-and-limits-logic" target="_blank">read more</a></p>http://plus.maths.org/content/goumldel-and-limits-logic#comments39Gödel's Incompleteness Theoremhistory of mathematicslogicphilosophy of mathematicswhat is impossibleWed, 31 May 2006 23:00:00 +0000plusadmin2284 at http://plus.maths.org/contentOmega and why maths has no TOEs
http://plus.maths.org/content/omega-and-why-maths-has-no-toes
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Gregory Chaitin </div>
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Kurt Gödel, who would have celebrated his 100th birthday next year, showed in 1931 that the power of maths to explain the world is limited: his famous incompleteness theorem proves mathematically that maths cannot prove everything. <b>Gregory Chaitin</b> explains why he thinks that Gödel's incompleteness theorem is only the tip of the iceberg, and why mathematics is far too complex ever to be
described by a single theory. </div>
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<div class="pub_date">December 2005</div>
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<p><i>Over the millennia, many mathematicians have hoped that mathematics would one day produce a Theory of Everything (TOE); a finite set of axioms and rules from which every mathematical truth could be derived. But in 1931 this hope received a serious blow: Kurt Gödel published his famous Incompleteness Theorem, which states that in every mathematical theory, no matter how extensive, there will
always be statements which can't be proven to be true or false.</i></p><p><a href="http://plus.maths.org/content/omega-and-why-maths-has-no-toes" target="_blank">read more</a></p>http://plus.maths.org/content/omega-and-why-maths-has-no-toes#comments37binary codeGödel's Incompleteness Theoremphilosophy of mathematicsproofThu, 01 Dec 2005 00:00:00 +0000plusadmin2278 at http://plus.maths.org/contentThe origins of proof III: Proof and puzzles through the ages
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Jon Walthoe </div>
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For millennia, puzzles and paradoxes have forced mathematicians to continually rethink their ideas of what proofs actually are. <strong>Jon Walthoe</strong> explains the tricks involved and how great thinkers like Pythagoras, Newton and Gödel tackled the problems. </div>
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<div class="pub_date">September 1999</div>
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<p>In the Millennia since Euclid, people's conceptions of mathematical proof have been revolutionised. From the discovery of Calculus and the rise of abstract mathematics, to Gödel's amazing discovery. There have been many changes and a few surprises along the way.</p><p><a href="http://plus.maths.org/content/origins-proof-iii-proof-and-puzzles-through-ages" target="_blank">read more</a></p>http://plus.maths.org/content/origins-proof-iii-proof-and-puzzles-through-ages#comments9axiomcalculusdeductionGödel's Incompleteness Theoreminductionirrational numberparadoxproofrational numberRussell's ParadoxTue, 31 Aug 1999 23:00:00 +0000plusadmin2394 at http://plus.maths.org/content