Geometry.Net - Theorems_And_Conjectures: Paradox

Russell's Paradox
A nuanced article on this paradox of naive set theory, happened upon by Russell in 1901. Includes bibliography, links to related articles.
http://plato.stanford.edu/entries/russell-paradox/

Extractions: Please Read How You Can Help Keep the Encyclopedia Free Russell's paradox is the most famous of the logical or set-theoretical paradoxes. The paradox arises within naive set theory by considering the set of all sets that are not members of themselves. Such a set appears to be a member of itself if and only if it is not a member of itself, hence the paradox. Some sets, such as the set of all teacups, are not members of themselves. Other sets, such as the set of all non-teacups, are members of themselves. Call the set of all sets that are not members of themselves " R ." If R is a member of itself, then by definition it must not be a member of itself. Similarly, if R is not a member of itself, then by definition it must be a member of itself. Discovered by Bertrand Russell in 1901, the paradox has prompted much work in logic, set theory and the philosophy and foundations of mathematics.
Sorites Paradox
The sorites paradox is the name given to a class of paradoxical arguments, A common form of the sorites paradox presented for discussion in the
http://plato.stanford.edu/entries/sorites-paradox/

Extractions: Please Read How You Can Help Keep the Encyclopedia Free This phenomenon at the heart of the paradox is now recognised as the phenomenon of vagueness (see the entry on vagueness ). Once identified, vagueness can be seen to be a feature of syntactic categories other than predicates, nonetheless one speaks primarily of the vagueness of predicates. Names, adjectives, adverbs and so on are only susceptible to paradoxical sorites reasoning in a derivative sense. Sorites arguments of the paradoxical form are to be distinguished from multi-premise syllogisms (polysyllogisms) which are sometimes also referred to as sorites arguments. Whilst both polysyllogisms and sorites paradoxes are chain-arguments, the former need not be paradoxical in nature and the latter need not be syllogistic in form. 1. The Sorites In History
EPR Paradox - Wikipedia, The Free Encyclopedia
On the experiment to show that a measurement performed on one part of a quantum system can have an instantaneous effect on the result of a measurement
http://en.wikipedia.org/wiki/EPR_paradox

Extractions: In quantum mechanics , the EPR paradox is a thought experiment which demonstrates that the result of a measurement performed on one part of a quantum system can have an instantaneous effect on the result of a measurement performed on another part, regardless of the distance separating the two parts. This runs counter to the intuition of special relativity , which states that information cannot be transmitted faster than the speed of light . "EPR" stands for Albert Einstein Boris Podolsky , and Nathan Rosen , who introduced the thought experiment in a paper to argue that quantum mechanics is not a complete physical theory. It is sometimes referred to as the EPRB paradox for David Bohm , who converted the original thought experiment into something closer to being experimentally testable. The EPR paradox is a paradox in the following sense: if one takes quantum mechanics and adds some seemingly reasonable conditions (referred to as "locality" , "realism", and "completeness"), then one obtains a contradiction . However, quantum mechanics by itself does not appear to be internally inconsistent, nor â as it turns out â does it contradict relativity. As a result of further theoretical and experimental developments since the original EPR paper, most physicists today regard the EPR paradox as an illustration of how quantum mechanics violates
NickQueen.com
This is a weblog with opinions and commentary on news, religion and personal life.
http://www.patriot-paradox.com/

Extractions: September 16, 2005 Strange Place For Roadkill Wow is all I have to say: This is from Coast to Coast AM My job in the field of conservation results in hearing some strange stories and seeing some extremely strange photographs. Attached are three photos that definitely fall into the "strange" category. (the second two are below) These pictures were taken in the Northeast section of the United States in an extremely rural area. As shown, there is a whitetail doe stuck on top of a power pole. She had apparently been up there for a moderate amount of time. I've heard broad speculation regarding the explanation for this, but no one is certain. If you notice the back hooves are clipped (almost uniformly). Read the rest and see two more photos at the link. show comments right here » Wow. That's just really really weird. :oD Posted by Jewels at September 16, 2005 06:32 PM « hide comments
.-\|/-. Paradox [PDX] Portal (PSX - PS2 - GBC - DC - N64 - Amiga - PC) .-\|/-.
The playground of the paradox sections (PSX, PS2, GBC, DC, N64, Amiga and PC).
http://www.paradogs.com/
The EPR Paradox And Bell's Inequality Principle
A short article from the USENET Physics FAQ.
http://math.ucr.edu/home/baez/physics/Quantum/bells_inequality.html

Extractions: Original by John Blanton. In 1935 Albert Einstein and two colleagues, Boris Podolsky and Nathan Rosen (EPR) developed a thought experiment to demonstrate what they felt was a lack of completeness in quantum mechanics. This so-called "EPR paradox" has led to much subsequent, and still on-going, research. This article is an introduction to EPR, Bell's inequality, and the real experiments that have attempted to address the interesting issues raised by this discussion. One of the principal features of quantum mechanics is that not all the classical physical observables of a system can be simultaneously known with unlimited precision, even in principle. Instead, there may be several sets of observables which give qualitatively different, but nonetheless complete (maximal possible) descriptions of a quantum mechanical system. These sets are sets of "good quantum numbers," and are also known as "maximal sets of commuting observables." Observables from different sets are "noncommuting observables". A well known example is position and momentum. You can put a subatomic particle into a state of well-defined momentum, but then you cannot know where it is. It's not just a matter of your inability to measure, but rather, an intrinsic property of the particle. Conversely, you can put a particle in a definite position, but then its momentum is completely ill-defined. You can also create states of intermediate knowledge of both observables: if you confine the particle to some arbitrarily large region of space, you can define the momentum more and more precisely. But you can never know both, exactly, at the same time.
Epimenides
An analysis of several attempted resolutions of the Epimenides paradox (also known as the Liar paradox), showing how they all fail.
http://www.ocf.berkeley.edu/~sjblatt/notes/nottrue.html

Extractions: 2.5 So, IN FACT, Statement A is not not true. Your mind should be blown. You should not be saying, "That's puzzling." You should be saying, "My mind is exploding." Argument 1 proves that Statement A is not true. Argument 2 proves that Statement A is not not true. We have proved a contradiction, which is impossible.
Amos Storkey - Brain Teasers: Two Envelope Paradox - Solution
Claims that the two envelope paradox comes from the incorrect use of probabilities.
http://www.anc.ed.ac.uk/~amos/doubleswapsoln.html

Extractions: Hough Transform This is my solution to the Two Envelope Paradox . Many other people have said their bit. A google search returns a few. The best way, in my mind, to understand what is going on in this puzzle is to tackle it from a Bayesian perspective. First we note that there are the various possible sums of money which could be put in the envelopes. The lower value we denote by X , and the higher value is fully determined by the lower value, being twice its value. We define a prior probability distribution P(X) for the choice the host makes for the lower value in the envelopes. Then there is the issue as to which envelope you choose to open: the one that contains the lower value (denoted C=lower ), or the one that contains the higher (denoted C=higher ). As the amounts are hidden from you, you choose entirely at random, with equal probabilities for the two options. Hence the probability P(C) is 0.5 for both cases. The paradox deals with the case that the value of the content (denoted Y ) of the chosen envelope is revealed to be a certain value N say. We wish to ascertain, given this information, the posterior probability that the current envelope contains the higher or lower value. This is
Game's Home Page
A site about Parrando's paradox alternating two losing games yield a winning game.
http://seneca.fis.ucm.es/parr/GAMES
Corel: Paradox Forum - Tek-Tips
Corel paradox technical support forums and mutual help system for computer professionals. Selling and recruiting forbidden.
http://www.tek-tips.com/threadminder.cfm?pid=177
Paradox -- A First-Order Logic Model Finder
A tool that processes firstorder logic problems and tries to find finite-domain models for them; written by Koen Claessen and Niklas Sörensson.
http://www.cs.chalmers.se/~koen/paradox/

Extractions: Paradox Welcome to the homepage of Paradox! Paradox is a tool that processes first-order logic problems and tries to find finite-domain models for them. Paradox is written by Koen Claessen and Niklas Sörensson News Paradox won the SAT/Models class (generated most models) in the CASC 2003 competetion for first-order logic tools! The current stable version is Paradox v. 1.0-casc . You can download it below. Usage Paradox reads problems in TPTP and Otter syntax. It understands TPTP-style clauses and formulas, and it also supports the TPTP include command. Further, Paradox can read Otter-style clause sets, though only if you switch on prolog_style_variables. Paradox produces one of the following three results: CONTRADICTORY, when it has shown that there is no model, SATISFIABLE, when it has found a model, or INCONCLUSIVE, when it has given up. On request, a list of function and predicate tables over a finite domain can be produced when a model is found. Here is a list of options (which you get when typing paradox help Implementation Paradox is mainly implemented in Haskell , but a part of it is an incremental SAT-solver implemented in C++. They are linked together using Haskell's Foreign Function Interface. The techniques we use in Paradox are described in a
Visual Paradox Free 3D Wallpaper - New Wallpaper This Week
Quality 3D wallpapers that are updated weekly. This site also offers very clear instructions on how to use the wallpapers for both Windows 95/98 and
http://www.visualparadox.com/

Extractions: (all I ask is that while you are here, check out at least one banner ad.) Your computer background (wallpaper) says a lot about you to everyone who sees it, "oohs" and "ahhs" are a good sign you are impressing people. Look around and find the background wallpaper that will make the time at your computer more enjoyable, and get you some "oohs" and "ahhs" as well. Tip: add this site to your list of favorites, and swap your wallpaper everyday.
Fermi's Paradox And Extraterrestrial Visitation
Suggests that aliens might be preparing mankind for the eventuality of meeting species from other worlds.
http://www.ufoskeptic.org/contact.html

Extractions: ufoskeptic.org Fermi's Paradox and the Preparation for Contact Hypothesis The famous physicist, Enrico Fermi, was referring to such an argument when he asked: "Where are they?" Since he was a prominent (and very smart) scientist, his simple question has been given a duly profound name: Fermi's paradox. The paradox is resolved, of course, if the answer is: "Well, they are already here!" (and possibly have been for a very long time and perhaps have even been involved in the rather sudden emergence of homo sapiens sapiens, but that may be over the top, so ignore that last conjecture if it is too much, though I seem to recall that even Sagan considered that possibility at some point). Their being here but surreptitiously would answer Fermi and at the same time address a second common objection to the UFO phenomenon by scientists: that the observations indicate such utterly nonsensical, bizarre behaviour that it just could not possibly be real. Intelligent visitors just would not pull such disreputable antics as have been reported. It is tempting to argue, as some do, that since the visitors are not behaving like proper alien explorers, there are no visitors in the first place. (For some reason this logic brings to mind Galileo attempting to get the Cardinals to look through his telescope.) To counter this argument, I would like to suggest one possible rationale that might, in principle at least (no real world guarantees), account for the widespread, disjoint kind of behavior that the UFO phenomenon has manifested over 50 years. The reason for proposing this "preparation for contact hypothesis" (which is certainly no original idea of mine) is simply to facilitate a "suspension of disbelief" so that the UFO evidence can be confronted on its own terms without the mental roadblock that the phenomenon violates all common sense, or more precisely the common sense of modern western society, and that of the scientific community in particular, and hence cannot be real.
Paradox - Word Definition And Usage Examples Antimoon.com
Search this site. Home Definitions of English words "paradox" A paradox is a situation in which something seems both true and false.
http://tmsyn.wc.ask.com/r?t=an&s=hb&uid=24312681243126812&sid=343126
Visual Paradox - Galleries Of Free 3D Wallpaper
Weekly updated free hires, high quality, 3D computer desktop wallpaper backgrounds and other graphics. Original art by professional artist Brian Kissinger.
http://www.visualparadox.com/gallery.htm
The Troublesome Paradox - Per Lundgren
Online version of book seeking publication by Per Lundgren. Author attempts to argue that a consequence of Goedel's incompleteness theorem is that we should overturn our current approach to scientific method.
http://www.yesgoyes.com/
The Top 10 Examples Of Paradox And How To Resolve Them
The Top 10 Examples of paradox and How to Resolve Them. Category Spirituality, Awareness, Path, Energy, Flow, Learning, Consciousness (BC38)
http://tmsyn.wc.ask.com/r?t=an&s=hb&uid=24312681243126812&sid=343126
Amos Storkey - Brain Teasers: Two Envelope Paradox
Amos Storkey Mathematical Puzzles The Two Envelope paradox. An introduction to the problem and a pointer to the solution.
http://www.anc.ed.ac.uk/~amos/doubleswap.html

Extractions: Hough Transform This is one of my favourite brain teasers. It was first introduced to me some years ago in a Cambridge pub garden during (if my memory serves me well) the Bayesian Methods workshop at the Neural Networks and Machine Learning session at the Newton Institute. There are many interpretations, but needless to say, I prefer the Bayesian ones. You are taking part in a game show. The host introduces you to two envelopes. He explains carefully that you will get to choose one of the envelopes, and keep the money that it contains. He makes sure you understand that each envelope contains a cheque for a different sum of money, and that in fact, one contains twice as much as the other. The only problem is that you don't know which is which. The host offers both envelopes to you, and you may choose which one you want. There is no way of knowing which has the larger sum in, and so you pick an envelope at random (equiprobably). The host asks you to open the envelope. Nervously you reveal the contents to contain a cheque for 40,000 pounds. The host then says you have a chance to change your mind. You may choose the other envelope if you would rather. You are an astute person, and so do a quick sum. There are two envelopes, and either could contain the larger amount. As you chose the envelope entirely at random, there is a probability of 0.5 that the larger check is the one you opened. Hence there is a probability 0.5 that the other is larger. Aha, you say. You need to calculate the expected gain due to swapping. Well the other envelope contains either 20,000 pounds or 80,000 pounds equiprobably. Hence the expected gain is 0.5x20000+0.5x80000-40000, ie the expected amount in the other envelope minus what you already have. The expected gain is therefore 10,000 pounds. So you swap.
Paradox Security Systems Ltd - Headquarters
Known for being an innovative research and development company, paradox Security Systems is a recognized worldwide force in the security industry.
http://www.paradox.ca/
Definition Of "paradox", 22C096, University Of Iowa
Definition of "paradox" Not to be confused with "antinomy." From Webster's New World Dictionary. paradox
http://tmsyn.wc.ask.com/r?t=an&s=hb&uid=24312681243126812&sid=343126

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