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Topic: List of axioms

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	Axiom - Wikipedia
		The word axiom comes from the Greek word αξιωμα (axioma), which means that which is deemed worthy or fit or that which is considered self-evident.
		As the word axiom is understood in mathematics, an axiom is not a proposition that is self-evident.
		An axiom is an elementary basis for a formal logic system that together with the rules of inference define a logic.
	wikipedia.findthelinks.com /ax/Axiom.html (654 words)

	PlanetMath: axiom
		Axioms and postulates are the basic assumptions underlying a given body of deductive knowledge.
		In the modern understanding, a set of axioms is any collection of formally stated assertions from which other formally stated assertions follow by the application of certain well-defined rules.
		A set of axioms should be consistent; it should be impossible to derive a contradiction from the axiom.
	planetmath.org /encyclopedia/Axiom.html (1184 words)

	List of axioms - Wikipedia, the free encyclopedia
		In epistemology, the word axiom is understood differently; see axiom and self-evidence.
		Individual axioms are almost always part of a larger axiomatic system.
		These are the de facto standard axioms for contemporary mathematics or set theory.
	en.wikipedia.org /wiki/List_of_axioms (133 words)

	PlanetMath: PA
		Peano Arithmetic (PA) is the restriction of Peano's axioms to a first order theory of arithmetic.
		Note that this replaces the single, second-order, axiom of induction with a countably infinite schema of axioms.
		A full list of the axioms of PA looks like this (although the exact list of axioms varies somewhat from source to source):
	planetmath.org /encyclopedia/PeanoArithmeticFirstOrder.html (141 words)

	What is the nature of the axioms of Objectivism? - Objectivism Online Forum
		The axioms are self-evident statements that cannot be denied in an opposing argument.
		Note that the axioms are not just chucked out there willy-nilly, so that the axiom that existence is (or entails) an identity logically depends on there being existence.
		Axioms are about or part of epistemology, and epistemology is about the relationship between a consciousness and the nature of existence.
	forum.objectivismonline.net /index.php?showtopic=4534 (3023 words)

	Modal Logic
		Density corresponds to the axiom (C4): □□A→□A, the converse of (4), so for example, the system KC4, which is K plus (C4) is adequate with respect to models where the frame is dense, and KDC4, adequate with respect to models whose frames are serial and dense, and so on.
		The relationship between conditions on frames and corresponding axioms is one of the central topics in the study of modal logics.
		A list of some of the more commonly discussed conditions on frames and their corresponding axioms along with a map showing the relationship between the various modal logics can be found in the next section.
	plato.stanford.edu /entries/logic-modal (7308 words)

	Untitled Document (Site not responding. Last check: 2007-10-29)
		Introduction and instructions: In this report, you will create a list of axioms that will be used to justify the procedure your group developed for computing the area of a polygonal figure.
		List any conjectures as to the theorems that you used; you do not need to offer proofs of your conjectures at this point.
		Use your set of axioms and your procedure for finding the area of a polygon to write justifications for your formulas for calculating the area of a 1) triangle, 2) parallelogram, and 3) trapezoid.
	www.math.ohiou.edu /~connor/prrep2.htm (405 words)

	Philosophy is Bullshit: David Hume
		This is the Prime Axiom of the axiom sets of all religions, and of course always a handy one to have if you wish to be able to derive the truth of your beliefs (ooo, oxymoron city) from your principle axioms:-).
		The only set of axioms that are applied to the physical world and are laid out with anything approaching rigor or those utilized by the scientific community, by the natural philosophers, who have established an open, easily understood basis for according a proposition concerning observable reality a degree of belief.
		The commonality of these axioms in social groups provides the basis of a shared debate, but also hides a tremendous amount of fundamental inconsistency, both within the base axioms used ``on average'' by the group and between specific instances or human understandings of these axioms by group members.
	www.phy.duke.edu /~rgb/Beowulf/axioms/axioms/node4.html (5347 words)

	Metamath Proof Explorer Home Page
		When an axiom or theorem with a distinct variable condition is referenced in a proof, the distinct variable conditions attached the theorem being proved must satisfy those of the referenced axiom or theorem after substitutions are made into the referenced axiom or theorem.
		Although in some sense the traditional axiom schemes are more compact than Metamath's ax-4 through ax-16, their goal is simply to provide recipes for generating actual axioms, from which we then prove actual theorems.
		This list of axioms was determined by scanning back through the proof tree until axioms were reached, and in fact required a considerable amount of CPU power.
	metamath.planetmirror.com /mpegif/mmset.html (9063 words)

	[No title] (Site not responding. Last check: 2007-10-29)
		The distinct variable constraints are a list, with each constraint a list of variables (thus, each constraint corresponds to a $d command in Metamath).
		The hypotheses are a list, with each hypothesis a valid term (thus, each hypothesis corresponds to a $e command in Metamath).
		Each hypothesis is a two-element list, the first of which is a label, the second of which is the corresponding term.
	www.ghilbert.org /design (2612 words)

	The Left Hand of the Electron -Euclid's Fifth
		To see what this means, consider that you would want your list of axioms to be complete, that is, they should suffice to prove all the theorems that are useful in the particular field of knowledge being studied.
		Well, any list of consecutive numbers can be divided into odd numbers and even numbers, so that we might conclude that in any such list of consecutive numbers, the total of all numbers present must be greater than the total of even numbers.
		It is possible to start with any set of axioms, provided they are not self-contradictory, and work up a system of theorems consistent with those axioms and with each other, even though they are not consistent with what we think of as the real world.
	www.fortunecity.com /emachines/e11/86/l-hand10.html (3867 words)

	Arithmetic Rules (Site not responding. Last check: 2007-10-29)
		You should expect there to be enough axioms to be sufficient to derive theorems that are "useful" to the mathematician.
		That is, you should expect the axioms to be consistent with one another -- that is, you shouldn't be able to derive the contradiction of one of the axioms as a logical consequence of the other axioms.
		This last fact sets up a choice: where one proposed axiom can be derived from another, then either one can be tossed out of the list of axioms, and reformulated as a "theorem".
	mcraefamily.com /MathHelp/BasicArithmetic.htm (1588 words)

	Block Tableau Proof (Site not responding. Last check: 2007-10-29)
		is a list of axioms, definitions, and theorems.
		To improve the readability of proofs, the required axioms, definitions, and previously proved theorems are added to the block when required.
		If a formula which would normally be introduced after branching for the sole purpose of closing that branch, it may be introduced prior to branching and then eliminated as part of the application of the appropriate simplification rule.
	cs.wwc.edu /~aabyan/Articles/GodelOntological/node6.html (251 words)

	Ontolingua Theory KIF-LISTS
		The sentence {tt (item $tau_1$ $tau_2$)} is true if and only if the object denoted by $tau_2$ is a non-empty list and the object denoted by $tau_1$ is either the first item of that list or an item in the rest of the list.
		The function {tt reverse} produces a list in which the order of items is the reverse of that in the list supplied as its single argument.
		The value of {tt subst} is the object or list obtained by substituting the object supplied as first argument for all occurrences of the object supplied as second argument in the object or list supplied as third argument.
	www-ksl.stanford.edu /htw/dme/thermal-kb-tour/kif-lists.html (950 words)

	The Nature of Axioms
		It would seem that the selection of a theory's axioms has to be made with great care because every idea in the theory gets its start from those axioms.
		Simply put, the axioms of today's theory of autism seem to have been copied automatically and unconsciously from beliefs that are widely viewed as true in our society.
		By presenting my list of autism's axioms, I make the suggestion that the more axioms we have and the more complex they are, the more our reason gets mixed up.
	web1.greatbasin.net /~sprang/axioms.htm (1192 words)

	[No title] (Site not responding. Last check: 2007-10-29)
		For example, ;; the axioms below should be read: ;; ;; p ;; ~p \/ ~q \/ r ;; ~s \/ q ;; ~t \/ q ;; t ;; ;; as in the example on page 150 of the text.
		(defvar *axioms) (setq axioms* '((p) ((not p) (not q) r) ((not s) q) ((not t) q) (t))) ;; -------------------------------------------------------------------------- ;; Function PROVE attempts to prove a statement using propositional resolution.
		It keeps a list of all axioms and inferred facts.
	www-users.cs.umn.edu /~gini/aiprog/rich-knight/resolve.lisp (223 words)

	Incomplete and Non-Monotonic Requirements (Site not responding. Last check: 2007-10-29)
		Godel's incompleteness theorem of arithmetic says that there is no finite list of axioms that completely describe integer arithmetic.
		If new axioms are added to the theory, the already existing theorems remain valid and the theory is extended into a new theory with new theorems added to the established theorems.
		A new theory is created which is not a simple extension of the old theory, but a collection of new theorems and some of the established theorems.
	www-db.stanford.edu /~burback/watersluice/node10.html (291 words)

	Vision Technology Management - Axioms Page
		These axioms have evolved over time and I am sure many of you have heard them before.
		Established by technicians long ago and now used heavily on the net, this axiom is the least followed by most computer users (including me).
		This axiom is so often referred to that there is now an FTP site at MIT dedicated to RTFM FAQs located at ftp://rtfm.mit.edu/pub/faqs.
	www.visiontm.com /axioms.html (589 words)

	Print
		The so-called "formalist program" aimed to find a master list of axioms, from which all of mathematics could be derived by rigid logical deduction.
		Given any system of axioms that produces no paradoxes, there exist statements about numbers which are true, but which cannot be proved using the given axioms.
		Whatever definition you propose (say, "two slices of bread with peanut butter in between"), there are still lots of non-peanut-butter-sandwiches that fall within its scope (say, two pieces of bread laid side by side with a stripe of peanut butter spread on the table between them).
	www.slate.com /toolbar.aspx?action=print&id=2114561 (952 words)

	Axiomatic Set Theory. Zermelo-Fraenkel Axioms
		The axioms C1 and C2[F] (for all formulas F that do not contain x) and the axiom of choice define a formal set theory C which corresponds almost 100% to Cantor's intuitive set theory (of the "pre-paradox" period of 1873-94).
		The axiom of infinity completes the list of comprehension axioms, which are necessary for reconstruction of common mathematics, i.e.
		The set theory adopting the axiom of extensionality (C1), the separation axiom schema (C21), the pairing axiom (C22), the union axiom (C23), the power-set axiom (C24), the replacement axiom schema (C25), the axiom of infinity (C26) and the axiom of regularity (C3), is called Zermelo-Fraenkel set theory, and is denoted by ZF.
	linas.org /mirrors/www.ltn.lv/2001.03.27/~podnieks/gt2.html (7448 words)

	Schema for transfinite induction and ordinal arithmetic (from set theory) -- Britannica Concise Encyclopedia - The ... (Site not responding. Last check: 2007-10-29)
		Intuitively, the ninth axiom or schema (see 9) is the assertion that if the domain of a function is a set then so is its range.
		That this is a powerful schema (in respect to the further inferences that it yields) is suggested by the fact that the third axiom can be derived from it and that, when applied in conjunction with the sixth axiom, the axiom of pairing can be deduced.
		For infinite cardinals, the arithmetic is uninteresting since, as a consequence of the eighth axiom, the sum and product of two such cardinals are each equal to the maximum of the two.
	www.britannica.com /ebc/article-24039 (1667 words)

	Betweenness Axioms
		There are numerous Propositions proved in the text based on the Betweenness Axioms.
		We shall list most of them, but shall prove only a few.
		From Incidence Axiom 3 there is at least one point E not on the line
	www.math.uncc.edu /~droyster/math3181/notes/hyprgeom/node29.html (830 words)

	Whatweknow.html
		Most mathematicians would probably regard the axiom of choice as 'obviously true,' while others may regard it as a somewhat questionable assertion which might even be false(and I am myself inclined towards this second viewpoint).
		A list of axioms is said to be consistent if there is there is no statement S for which S and
		A mathematical model for a list of axioms is a well-defined set which assigns "meaning" for the undefined terms in the axioms, in a manner such that the axioms are "true".
	www.umsl.edu /~siegel/SetTheoryandTopology/Whatweknow.html (505 words)

	Project on Proof for Computer Science 261
		Axioms are written as Horn clauses in the propositional calculus,
		Theorems may be written as axioms in conjunctive normal form, with the theorem's conclusion negated.
		that takes a conclusion and a list of axioms, such as your answers for the homework based on the resolution handout, and indicates either that the result is provable or that no proof is possible.
	www.math.grin.edu /~walker/courses/261.sp98/proj-proof.html (554 words)

	Origami Geometric Constructions
		This list of axioms encompasses everything you can do with a SE&C. That is, anything you do with a SE&C can be broken down into a sequence of the above operations.
		Using this axiom list, one can begin to talk about things that cannot be done using a SE&C. For example, students are traditionally told that certain things are impossible to do with a SE&C, like trisecting an arbitrary angle, or doubling the volume of a cube (i.e., constructing the cube root of 2).
		Huzita's axioms (O1)-(O4) are rather simple, and it's easy to see that they are operations that can also be done by a SE&C. (O5) and (O6), on the other hand, take some getting used to.
	www.merrimack.edu /~thull/omfiles/geoconst.html (1641 words)

	Axiom scheme of replacement
		The general axiom scheme for building up complex sets like the ordinals is called replacement.
		These axioms could be defined by a single finite expression, but they are usually defined as an easily generated sequence.
		By restricting new sets to those obtained by applying well defined functions to the elements of existing sets it was felt that one could avoid such contradictions.
	www.mtnmath.com /whatrh/node53.html (231 words)

	[No title]
		empty(list(h, r)) === false friend item head(const list & l); // precondition: list is not empty // axioms: // 3.
		head(list(h, r)) === h friend list rest(const list & l); // precondition: list is not empty // axioms: // 4.
		e) Provide axioms for the method palindrome(l) - true iff l is the same forwards or backwards.
	www.cs.haverford.edu /curriculum/courses/cmsc206/midterm/sample.txt (347 words)

	Blogotomy - A Manager's Journal: My Programming Methodology (Site not responding. Last check: 2007-10-29)
		I admit it, I do have a longer list of axioms, but it makes a much more interesting article if my entire programming methodology is just one axiom, so humor me and allow me to list the following as "sub-axioms".
		This isn't the whole list, I am just skimming the top of my head, but it's a good start.
		One of the difficulties of this list is that some of the sub-axioms are contradictory.
	blogs.osafoundation.org /blogotomy/000405.html (499 words)

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