Conservapedia - User contributions [en]

Banach-Tarski Paradox

2010-01-04T23:21:05Z

Aroth: Added an interpretation section

The '''Banach-Tarski paradox''' uses the [[Axiom of Choice]] to take a [[sphere]], split it into finitely many pieces, and reassemble them into two new spheres. The new spheres both have the same volume as the original, suggesting paradoxically that you can create two identical copies out of one, despite our physical intuition that this is impossible. This paradox has been mathematically proven to be consistent with [[Zermelo-Fraenkel]] set theory.

Note that, since the pieces must be non-measurable, this cannot be done with an actual object.

==Interpretation==
Some view the Banach-Tarski paradox as an absurd result, and evidence that the [[Axiom of Choice]] is false. However, it can also be viewed as a mathematical affirmation of the biblical story of the loaves and the fishes (as Jesus would not have had to violate the conservation of mass, but merely partition loaves and fishes into non-measurable subsets).

==External Links==
*[http://mathworld.wolfram.com/Banach-TarskiParadox.html Banach-Tarski Paradox] at MathWorld

[[Category:Set Theory]]
[[Category:geometry]]

Banach-Tarski Paradox

2010-01-04T23:15:10Z

Aroth: Correcting a typo; Banach-Tarski requires splitting the sphere into infinitely many pieces. Axiom of choice is not necessary to select elements from a finite collection of sets.

The '''Banach-Tarski paradox''' uses the [[Axiom of Choice]] to take a [[sphere]], split it into infinitely many pieces, and reassemble them into two new spheres. The new spheres both have the same volume as the original, suggesting paradoxically that you can create two identical copies out of one, despite our physical intuition that this is impossible. This paradox has been mathematically proven to be consistent with [[Zermelo-Fraenkel]] set theory.

Note that, since the pieces must be non-measurable, this cannot be done with an actual object.

==External Links==
*[http://mathworld.wolfram.com/Banach-TarskiParadox.html Banach-Tarski Paradox] at MathWorld

[[Category:Set Theory]]
[[Category:geometry]]

Godel's Incompleteness Theorems

2010-01-03T17:38:16Z

Aroth: Added a relevant paragraph about undecidability and faith. This paragraph has been in the undecidability article for some time.

'''Gödel's Incompleteness Theorems''' are two theorems published in 1931 by [[Kurt Gödel]] that show that all sufficiently strong first-order theories can never yield answers to all mathematical questions--they are incomplete.

'''Gödel's First Incompleteness Theorem''': If T is any [[consistency|consistent]], [[sound]], [[recursion|recursively enumerable]] first-order theory containing the axioms of [[Peano Arithmetic]], there is a sentence G_T such that T cannot prove G_T and such that T cannot prove ~G_T. This means that T is not complete: there is some statement (G_T) that cannot be proved or refuted in the theory T.

The statement G_T says "G_T is not provable in T". This statement is encoded in [[number theory]]--symbols of [[first-order language|first-order logic]] are assigned numbers in such a way so that statements in logic can be viewed as statements about numbers. Further, this statement refers to itself; that this is possible is the key to Godel's argument.

Recursively enumerable (r.e.) theories are theories that can be written down in a mechanical way. The restriction to recursively enumerable theories is required, since it is easy to construct non-r.e. theories that are complete--for example, the theory that contains exactly all true statements of number theory is complete, but cannot be written down in a mechanical way.

The first incompleteness theorem is itself a theorem of Peano Arithmetic. A corollary of this fact, noted independently by [[John von Neumann]] and Gödel, is the:

'''Gödel's Second Incompleteness Theorem''': If T is any consistent, sound, recursively enumerable first-order theory containing the axioms of Peano Arithmetic, then T cannot prove its own consistency. In particular, Peano Arithmetic cannot prove its own consistency. This means that the consistency of the axioms of Peano Arithmetic and stronger mathematical theories must be justified in other ways, usually by appealing to the fact that the axioms are obviously true or by appealing to the fact that no one has yet derived an inconsistency.

==Incompleteness and God==

Incompleteness is sometimes used to refute the existence of an [[omnipotent]] god: If God were omnipotent, he would act as an [[oracle]] (in the mathematical sense) for any mathematical theory. That is, he could decide for any statement S whether S or not-S were true, which contradicts incompleteness. This argument is a more formal version of the "Can God make a rock even He could not move" [[paradox]].

On the other hand, he same Godel that formulated the Incompleteness Theorems used those same tools ([[Model Theory]]) to forge an onthological proof of God's existence.<ref>[http://www.scar.utoronto.ca/~sobel/OnL_T/OnGodel(toKoons).pdf Godel's Onthological Proof that God exists]</ref>

Similarly, the existence of undecidable statements proves the illogic of atheistic attempts to demand proof for the existence of God. Because we know that there exist true statements that we can never logically prove, there are necessarily things that must be believed on the basis of faith, rather than logic. The atheistic mantra that "the burden of proof lies with the believer" ignores this classic result in mathematical logic, and exposes their ignorance. It is notable that [[Kurt Godel]] who demonstrated the existence of undecidability in mathematical logic was a devoutly religious man.
[[Category:Logic]][[Category:Mathematics]]

Essay:Conservatives of the Decade

2009-12-29T02:27:29Z

Aroth: Added Andy. Starting Conservapedia, the CBP, and promoting both to a national audience on Colbert certainly merit the nomination.

It's time; in fact, we have only three days left to identify '''the top ten [[conservatives]] of the decade'''. The criterion would presumably be the best and longest lasting influence. For example, [[Ronald Reagan]] would be a leading candidate for "conservative of the decade" of the 1980s.

The [[best of the public]] is welcome in this list of nominations:

== Nominations ==

*[[Michele Bachman]]
*[[Sarah Palin]]
*[[John Bolton]]
*[[Antonin Scalia]]
*[[Clarence Thomas]]
*[[Samuel Alito]]
*[[John Roberts]]
*[[Ron Paul]]
*[[Joe Wilson]]
*[[John Thune]]
*[[Marco Rubio]]
*[[Hannah Giles]] and [[James O'Keefe]]
*supporters of [[Proposition 8]]
*[[homeschoolers]] who captured all the finalist positions in the [[National Bible Bee]]
*[[Sean Hannity]]
*[[Bill Oreilly]]
*[[Mike Huckabee]]
*[[Mitt Romney]]
*[[Charles H Bronson]]
*[[Art Robinson]]
*[[Charlton Heston]]
*[[Jon Voight]]
*[[Dick Cheney]]
*[[Rush Limbaugh]]
*[[Glenn Beck]]
*[[Rudy Giuliani]]
*[[Tea Party Movement]]
*[[Ann Coulter]]
*[[Michelle Malkin]]
*[[David Petraeus]]
*[[Karl Rove]]
*[[Andrew Schlafly]]

(add more, or suggest ranking; maybe we should name a "Conservative of the Year" also)

== Quotes of the Decade ==

*"You lie!" (Joe Wilson about Barack Obama)
*"You know babies have finger nails!" (reference to an unborn child in the movie ''Juno'')
*"from my cold, dead hands" (Charlton Heston in 2000 dramatically referring to the only way [[liberal]]s could take his rifle from him)

== Story of the Decade ==

*The taking, for the benefit of a corporation and higher tax revenue, of the long-held homes of Susanne Kelo and others in New London, Connecticut, decided against the homeowners by a 5-4 U.S. Supreme Court. Forcibly ousted and their homes destroyed, the residents moved elsewhere at great disruption to their lives. Then, four years later, Pfizer [[Kelo v. New London|abandoned the project]] for its own reasons, leaving a useless eyesore in the wreckage.

[[Category:Essays]]

Evolutionary algorithm

2009-10-01T23:35:49Z

Aroth: Updated popularity section to point out that evolutionary algorithms have fallen out of favor since the early 90s. Work on them has essentially stopped in academia.

An '''Evolutionary algorithm''' (EA) is a stochastic numerical analysis that takes its inspiration from the [[theory of evolution]], particularly the optimization power of [[natural selection]]. The main thing that sets an evolutionary algorithm apart from other stochastic methods is the use of a [[fitness function]] to select for optimal solutions. New solutions are a created by allowing existing ones to breed with each other. Also many algorithms use random alterations in the coded solution similar to the biological principle of [[mutation]]. The fitness program selects solutions that better solve the problem and increases the frequency of that solution and its descendants in the over all population of solutions.

==Types of Evolutionary algorithms==
There are several different approaches to evolutionary computation the most frequently used fall into a few general categories:
*[[Genetic algorithm]] - This is the most popular type of EA, it involves using strings of numbers as your solution set. It uses [[recombination]], [[mutation]], and [[selection]] to find optimal solution sets.
*[[Genetic programming]] - Instead of strings of numbers, genetic programming uses whole programs and the fitness function is based on their ability to quickly and accurately solve the problem.
*[[Evolutionary programming]] - Similar to genetic programming, but the logical structure of the programs do not change, rather the numerical constructs do.
*[[Evolution strategy]] - Uses numerical vectors as a solution.

==Popularity and usage==

Evolutionary algorithms are general purpose optimizers because they do not require any assumptions about the landscape of the fitness function. They are used in a wide range of problems in diverse fields and have proven to be a highly effective numerical analysis method.

However, in the last decade, research on evolutionary algorithms has fallen off sharply, and they have not lived up to their initial promise. Although they are a reasonable search technique in a wide variety of problems, they are not the best search technique in almost any field. Algorithms such as [[simulated annealing]], and fast [[integer programming]] solvers have largely superceded evolutionary algorithms in modern use. Evolutionary algorithms can be seen as an experimental test of Darwin's theory of evolution, and their eventual failure can be seen as a rejection of that theory.

==See Also==
[[Monte Carlo method]]

[[Neural networks]]

[[Genetic algorithm]]

==References==
<references />
[[category:mathematics]]
[[Category:Computer Science]]

Undecidable

2009-09-15T01:49:38Z

Aroth:

A statement in formal [[logic]] is called '''undecidable''' if there is no [[proof]] or [[disproof]] of the statement in formal logic. A common misconception is that undecidable statements have no truth value, but this statement is not true. For example, many set theorists now believe that the [[continuum hypothesis]] (which is known to be undeciable in [[Zermelo-Fraenkel]] set theory) is actually false.

As a more concrete example, suppose we had a "theory of colored shapes" <math>T</math> where the objects were colored shapes (red triangles, blue squares, etc), and the possible [[atom]]ic sentences were of the form "shape <math>S</math> is a square." Since we do not have the language to state "shape <math>S</math> is red" then any statement of this form is undecidable in <math>T</math>, though it could be true or false in some larger theory <math>T'</math>.

==Undecidability and Faith==
The existence of undecidable statements proves the illogic of atheistic attempts to demand proof for the existence of God. Because we know that there exist true statements that we can never logically prove, there are necessarily things that must be believed on the basis of faith, rather than logic. The atheistic mantra that "the burden of proof lies with the believer" ignores this classic result in mathematical logic, and exposes their ignorance. It is notable that [[Kurt Godel]] who demonstrated the existence of undecidability in mathematical logic was a devoutly religious man.

==Famous Undecidable Statements==

* [[Axiom of Choice]]
* [[Banach-Tarski paradox]]
* [[Continuum hypothesis]]
* Existence of large [[cardinal]]s
* [[Halting problem]]
* [[König's lemma]]
* [[Lefschetz principle]]
* [[Liar's paradox]]
* [[Ramsey theory]] involving infinite sets
* [[Russell's paradox]]

[[category:logic]]

Talk:Undecidable

2009-09-15T01:49:00Z

Aroth:

I'd like to create a small section that outlines the argument that mathematical undecidability provides a logical basis for religious faith. The argument is simply that we can prove rigorously that there exist true statements that have no logical proof. Therefore the atheistic tendency to demand evidence for religious matters, like the existence of god, is not backed up by formal logic: if we know that there are true things that cannot be proven, then it is logical to sometimes have faith without proof.

What do you think? I don't know if anyone reads this page, but I wanted to throw this out there before making the edit. [[User:Aroth|Aroth]] 18:05, 14 September 2009 (EDT)

:Sounds reasonable to me. You could also link to the page [[Kurt Godel]], to which I just added a brief discussion of Godel's ontological argument. Godel believed not that undecidability provides a logical basis for religious faith, but that a logical argument for faith may be explicitly constructed along the lines of Anselm's ontological argument. --[[User:MarkGall|MarkGall]] 20:33, 14 September 2009 (EDT)

:: Great. I added a short section and linked to Godel's page. [[User:Aroth|Aroth]] 20:50, 14 September 2009 (EDT)

::: Looks great! I do wonder about why Zeno's paradox is listed as undecidable... I think the matter of infinite series is now fairly well understood. I'd be willing to believe that something related is undecidable, but there must be a precise, logical formulation of exactly the statement of the paradox: "how can infinitely many events string together into a single event?" is not something whose logical meaning is clear to me. I'm removing it for now, but if I'm mistaken, please add back! --[[User:MarkGall|MarkGall]] 21:02, 14 September 2009 (EDT)

:::: Good call. The Church-Turing thesis also doesn't belong there (its not even a mathematical statement...) [[User:Aroth|Aroth]] 21:49, 14 September 2009 (EDT)

Talk:Undecidable

2009-09-15T01:48:50Z

Aroth:

Talk:Undecidable

2009-09-15T00:50:38Z

Aroth:

Undecidable

2009-09-15T00:50:03Z

Aroth: Added section on undecidability and faith, as discussed on the talk page.

A statement in formal [[logic]] is called '''undecidable''' if there is no [[proof]] or [[disproof]] of the statement in formal logic. A common misconception is that undecidable statements have no truth value, but this statement is not true. For example, many set theorists now believe that the [[continuum hypothesis]] (which is known to be undeciable in [[Zermelo-Fraenkel]] set theory) is actually false.

As a more concrete example, suppose we had a "theory of colored shapes" <math>T</math> where the objects were colored shapes (red triangles, blue squares, etc), and the possible [[atom]]ic sentences were of the form "shape <math>S</math> is a square." Since we do not have the language to state "shape <math>S</math> is red" then any statement of this form is undecidable in <math>T</math>, though it could be true or false in some larger theory <math>T'</math>.

==Undecidability and Faith==
The existence of undecidable statements proves the illogic of atheistic attempts to demand proof for the existence of God. Because we know that there exist true statements that we can never logically prove, there are necessarily things that must be believed on the basis of faith, rather than logic. The atheistic mantra that "the burden of proof lies with the believer" ignores this classic result in mathematical logic, and exposes their ignorance. It is notable that [[Kurt Godel]] who demonstrated the existence of undecidability in mathematical logic was a devoutly religious man.

==Famous Undecidable Statements==

* [[Axiom of Choice]]
* [[Banach-Tarski paradox]]
* [[Church-Turing thesis]]
* [[Continuum hypothesis]]
* Existence of large [[cardinal]]s
* [[Halting problem]]
* [[König's lemma]]
* [[Lefschetz principle]]
* [[Liar's paradox]]
* [[Ramsey theory]] involving infinite sets
* [[Russell's paradox]]
* [[Zeno's paradox]]

[[category:logic]]

Talk:Undecidable

2009-09-14T22:05:55Z

Aroth:

Talk:Undecidable

2009-09-14T22:05:41Z

Aroth: Created page with 'I'd like to create a small section that outlines the argument that mathematical undecidability provides a logical basis for religious faith. The argument is simply that we can pr...'

Computer science

2009-09-13T17:38:24Z

Aroth: A brief description of theoretical computer science

'''Computer science''' is the study of various aspects of [[computers]]. It is a common course of study and degrees at the college level and beyond.

Studies can range from the very practical, such as learning [[programming language]]s, to the highly theoretical.

Students who plan to study computer science should take as many [[mathematics]] courses as possible. While not all of computer science is directly mathematical, the mental discipline that comes from studying math is very useful in studying computers.

==Common programming languages in Computer Science==
*[[Java]]
*[[C++|C/C++]]
*[[C#]]
*[[Matlab]]

==Theoretical Computer Science==
[[Theoretical computer science]] is the mathematical study of what can and cannot in principle be computed under various constraints. For example, [[Alan Turing]] proved that it is impossible to write a computer program that can determine whether other computer programs will ever ''halt'', or whether they will run forever. Other problems which can be solved in principle seem impossible to solve efficiently. For example, the famous P vs. NP problem asks whether there exists a polynomial time algorithm to solve any NP-complete problem, like the [[travelling salesman problem]]. There is a $1,000,000 prize for the solution to the P vs. NP problem.

[[Category:Information technology]]
[[Category:Computer Science]]

User:Aroth

2009-09-13T17:33:05Z

Aroth: Created page with 'Hello. My name is Arnold Rothe, and I am a long time resident of Western Pennsylvania. I am a computer scientist by training, and teach AP-computer science to both public and hom...'

Hello. My name is Arnold Rothe, and I am a long time resident of Western Pennsylvania. I am a computer scientist by training, and teach AP-computer science to both public and home-schooled students.

I'm a firm believer that accuracy and truth are core conservative principles, and hope to help contribute these ideals to Conservapedia!

{{userboxtop|aroth}}
{{User Christian1}}
{{User Not Obama}}
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