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Topic: Continuum postulate

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continuum hypothesis

Continuum mechanics

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generalized continuum hypothesis

Continuum mathematics

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	continuum postulate (Site not responding. Last check: 2007-10-29)
		Continuum mechanics is a branch of physics that deals with solids and fluids (i.e., liquids and gases).
		Continuum mechanics makes the assumption that these materials are continuous: the fact that matter is made of atoms is ignored.
		Continuum mechanics thus uses tensors, which are mathematical objects that are independent of coordinate system.
	www.yourencyclopedia.net /Continuum_postulate (218 words)

	Axiom - Wikipedia, the free encyclopedia
		This set of axioms turns out to be incomplete, and many more postulates are necessary to rigorously characterize his geometry (Hilbert used 23).
		The axioms are referred to as "4 + 1" because for nearly two millennia the fifth (parallel) postulate ("through a point outside a line there is exactly one parallel") was suspected of being derivable from the first four.
		Ultimately, the fifth postulate was found to be independent of the first four.
	en.wikipedia.org /wiki/Postulate (1598 words)

	PlanetMath: axiom (Site not responding. Last check: 2007-10-29)
		Axioms and postulates are the basic assumptions underlying a given body of deductive knowledge.
		A great lesson learned by mathematics in the last 150 years is that it is useful to strip the meaning away from the mathematical assertions (axioms, postulates, propositions, theorems) and definitions.
		The development of hyperbolic geometry taught mathematicians that postulates should be regarded as purely formal statements, and not as facts based on experience.
	planetmath.org /encyclopedia/Postulate.html (1184 words)

	Postulates of Special Relativity (Site not responding. Last check: 2007-10-29)
		The postulate that spacetime forms a 4-dimensional continuum is a generalization of the classical Galilean concept that space and time form separate 3 and 1 dimensional continua.
		The postulate of a 4-dimensional spacetime continuum is retained in general relativity.
		This postulate is the nub of special relativity, and much of the content of these pages is concerned with exploring its paradoxical consequences, starting with the next page, The Paradox of Special Relativity.
	casa.colorado.edu /~ajsh/sr/postulate.html (690 words)

	Chapter 24. Euclidean and Non-Euclidean Continuum. Einstein, Albert. 1920. Relativity: The Special and General Theory
		We express this property of the surface by describing the latter as a continuum.
		The surface is not a Euclidean continuum with respect to the rods, and we cannot define Cartesian co-ordinates in the surface.
		Thus it is that mathematicians long ago solved the formal problems to which we are led by the general postulate of relativity.
	www.bartleby.com /173/24.html (827 words)

	Axiom
		For example, in some rings, the operation of multiplication is commutative, and in some it is not; those rings in which it is are said to satisfy the "axiom of commutativity of multiplication." Another name for an axiom is postulate.
		Probably the most famous very early set of axioms is the 4+1 postulates of Euclid.
		4+1 since the fifth postulate (through a point outside a line there is exactly one parallel) was suspected to be derivable from the first 4 for nearly two millennia.
	www.brainyencyclopedia.com /encyclopedia/a/ax/axiom.html (738 words)

	The continuum hypothesis (Site not responding. Last check: 2007-10-29)
		The role of the continuum hypothesis in set theory is similar to the role of the parallel postulate in plane geometry.
		On the other hand, the parallel postulate cannot be proved either, since it is false in various noneuclidean models of geometry which do satisfy all the other axioms.
		Therefore the parallel postulate, or its negation, may be taken as a new axiom.
	mathcircle.berkeley.edu /BMC3/infinity/node12.html (192 words)

	Continuum, Mu-Ency at MROB
		Roughly speaking, a continuum is a type of connected set that can be divided into smaller and smaller pieces infinitely many times and any such pieces, if they are obtained after a finite number of steps, have the same order as the original set.
		Examples of continuums are a straight line, a plane, a circle, a disc, the set of real numbers, and the set of complex numbers.
		The Generalized Continuum Hypothesis states that if N is the order of set S and M is the order of the power set of S, there exist no sets that have more elements than N and fewer elements than M. This would mean that the order of a power set of a continuum is Aleph-2.
	www.mrob.com /pub/muency/continuum.html (491 words)

	A Cartography of the Ecstatic and Meditative States (Site not responding. Last check: 2007-10-29)
		Along the perception-hallucination continuum of increasing arousal of the sympathetic nervous system (ergotropic arousal), man--the self-referential system--perceptually-behaviorally (cortically) interprets the change (drug-induced or "natural") in his subcortical activity as creative, psychotic, and ecstatic experiences (2).
		However, in the next aroused state on the continuum, acute schizophrenic [or rather "hyperphrenic" (5)] state, further increase in data content may not be matched by a corresponding increase in the rate of data processing.
		We have seen that the departure from the physical dimension during a voyage on the perception-meditation continuum is accompanied by a gradual loss of freedom, which is manifested in the increasing inability to verify the experience through voluntary motor activity (53).
	www.mimbres.com /holp/holpath/cartog.htm (6914 words)

	Special Relativity
		It is shown that assumptions regarding absolutism or relativism cannot be substantiated within the context of the two postulates and that a re-interpretation of the constancy of the speed of light effectively accounts for the null result of the Michelson-Morley experiment.
		Among these are the unsatisfactory interpretation of the first postulate, the arbitrary limitations imposed on the means to determine synchronism and simultaneity, and the total exclusion of one-dimensional aspects in the justification for, and calculation of space-time modifications.
		In an Euclidean (isotropic) space-time continuum, the former would result in compound velocities for a frame of reference moving relative to the beam unless contraction-dilation effects were introduced.
	wbabin.net /babin/webdoc1.htm (2867 words)

	L.L. Thurstone: The Measurement of Psychological Value (Site not responding. Last check: 2007-10-29)
		All measurement consists in allocating the object measured to a point in an abstract continuum, as when a sack of flour and a child are assigned to their respective points in the abstract continuum of weight.
		Postulate a continuum of appetition-aversion which may also be referred to here as the psychological continuum.
		The psychological continuum which in the older psychophysics was a cognitive one, as well as the affective continuum of the present discussion, is of course an abstract continuum that serves purposes of description just as the more common but equally abstract continua of weight, volume, temperature, length, or area.
	spartan.ac.brocku.ca /~lward/Thurstone/Thurstone_1929.html (4933 words)

	Continuum mechanics (Site not responding. Last check: 2007-10-29)
		Continuum mechanics is a branch of physics (specifically materials science) that deals with solids and fluids (i.e., liquids and gases).
		The fact that matter is made of atoms and that it commonly has some sort of heterogeneous microstructure is ignored in the simplifying approximation that physical quantities, such as energy and momentum, can be handled in the infinitesimal limit.
		Some of these differential equations are specific to the materials being investigated, while others capture fundamental physical laws, such as conservation of mass or conservation of momentum.
	www.sciencedaily.com /encyclopedia/continuum_mechanics (353 words)

	MPI MIS Leipzig - Abstracts for the talks
		Thus, among the classical principles of determinism and local action, a new postulate is formulated, that is an extension of these two to take the variables belonging to the species assemblage into account and that is already known, in other forms, in the literature of mixtures with continuous diversity: The postulate of global action.
		The current contribution is concerned with the formulation of continuum thermodynamic field models for materials with microstructure for which the distribution of (micro)structure at each material point and its evolution relative to the continuum as a whole influences the macroscopic behaviour.
		Once these are established, one may proceed by analogy with the case of the standard continuum to formulate kinematics, balance relations, and constitutive relations, for the extended one.
	www.mis.mpg.de /conferences/diversity/abstracts.html (1339 words)

	Theory of Gravity Singularity - Singularity Pair Bonding
		The space-time continuum is the potential for motion and dimension to exist.
		It is the space-time continuum, and all of its mass that is moving.
		Their motion in gravity vectors distort the continuum and they now have mass, and exist within our frame of reference as motion.
	www.freewebs.com /spinstate/relativitysingularity.htm (1699 words)

	Nano time (Site not responding. Last check: 2007-10-29)
		Clarification of the time postulate (TP) of the Nanostate, especially as implicit in articles 3, 4, 5, of the Nanostate Protocol vol 1 (NPv1), published 1900 hrs pm BST 270898.
		These are classically understood as a non-countable infinity, and as infinitely greater in number than the (countably infinite) rationals, represented for example in digital form as the aggregate of all possible terminating or recurring decimals.
		Users will need to be assured that the Nanostate TP does not pretend to be in respect of the fundamental givens of the physical universe, but to offer a postulational basis for real transactional, political and aesthetic intervention within honorifically defined ostensible constants of geopolitical space.
	www.nanostate.org /time.htm (272 words)

	Axiom (Site not responding. Last check: 2007-10-29)
		For example, in some rings, theoperation of multiplication is commutative, and in some it is not; those rings in which it is are said to satisfy the"axiom of commutativity of multiplication." Another name for an axiom is postulate.
		This turns out to be incomplete, and many more postulates are necessary tocompletely characterize his geometry (Hilbert used 23).
		4+1 since the fifth postulate (through a point outside a linethere is exactly one parallel) was suspected to be derivable from the first 4 for nearly two millennia.
	www.therfcc.org /axiom-1912.html (672 words)

	[No title]
		In virtue of the latter property we speak of a " continuum," and owing to the fact that there are three co-ordinates we speak of it as being " three-dimensional." Similarly, the world of physical phenomena which was briefly called " world " by Minkowski is naturally four dimensional in the space-time sense.
		The" world" is in this sense also a continuum; for to every event there are as many "neighbouring" events (realised or at least thinkable) as we care to choose, the co-ordinates x[1], y[1], z[1], t[1] of which differ by an indefinitely small amount from those of the event x, y, z, t originally considered.
		Thus in reality, the description of the time-space continuum by means of Gauss co-ordinates completely replaces the description with the aid of a body of reference, without suffering from the defects of the latter mode of description; it is not tied down to the Euclidean character of the continuum which has to be represented.
	www.gutenberg.org /dirs/etext04/relat10.txt (19921 words)

	The Radiation Continuum Model of Light and the Galilean Invariance of Maxwell' (Site not responding. Last check: 2007-10-29)
		By modifying the second postulate to state: "The observed velocity of light is c from all frames of reference," the radiation continuum model (RCM) of electromagnetic radiation is developed.
		The second postulate as stated by Einstein does not deserve the status of a postulate, as it is at once overly restrictive and ultimately phenomenological--the value of c is based on experimental measurement rather than an analysis of first principles.
		The modified second postulate of RCM theory, however, says nothing about the actual propagation of EM energy (although RCM theory itself does), but only of the relative speed with which it must interact with matter to be detected.
	renshaw.teleinc.com /papers/galinv3/galinv3.stm (4971 words)

	sciforums.com - Light ray invariance.....why?
		He made the observation that nobody had ever observed anything like what one would predict to be observed in that case, and therefore postulated that it is impossible to enter the rest frame of light, or, equivalently, that light has no rest frame.
		The main reason was that, from Maxwell's equations, the force on a charged particle moving through a magnetic field, depends on the speed of the charged particle, relative to the magnetic field.
		So the light cones describe the "Time continuum" not duration, or length but the time continuum.....thus the centre of the light cones is teh centre of change or time......
	www.sciforums.com /showthread.php?t=39937 (2486 words)

	[No title]
		For this Elastic Continuum the equilibrium equations of elasticity are identical with vector wave equation of Maxwell's electromagnetic theory.
		The Elastic Continuum Theory or ECT in short, essentially deals with the study of formation, existence and mutual interactions of stress/strain wave packets in the Elastic Continuum, leading to their co-relation with the formation, existence and mutual interactions of elementary particles in the ‘Vacuum’ or the ‘Empty Space’.
		For this ‘Elastic Continuum’ the equilibrium equations of elasticity are found to be identical with the vector wave equation of Maxwell’s electromagnetic theory.
	www.geocities.com /gssandhu_1943 (2328 words)

	EVIDENCE OF A CONFINED UNIVERSE (SPHERICAL OR ELLIPTICAL), AS DEDUCTED BY EINSTEIN.
		We shall proceed to show that both the general postulate of relativity and the fact of the small stellar velocities are compatible with the hypothesis of a spatially finite universe; through certainly, in order to carry through this idea, we need a generalizing modification of the field equations of gravitation.
		According to the general theory of relativity the metrical character (curvature) of the four-dimensional space-time continuum is defined at every point by the matter at that point and the state of that matter.
		Therefore, on account of the lack of uniformity in the distribution of matter, the metrical structure of this continuum must necessarily be extremely complicated.
	www.geocities.com /fdocc3/circumscribed.htm (3591 words)

	Set Theory
		Rather, sets are introduced either informally, and are understood as something self-evident, or, as is now standard in modern mathematics, axiomatically, and their properties are postulated by the appropriate formal axioms.
		For instance, it is desirable to have the “set of all integers that are divisible by number 3,” the “set of all straight lines in the Euclidean plane that are parallel to a given line”, the “set of all continuous real functions of two real variables” etc.
		The Axiom of Choice, which postulates the existence of a certain set (the choice set) without giving specific instructions how to construct such a set, is of different nature than the other axioms, which all formulate certain construction principles for sets.
	plato.stanford.edu /entries/set-theory (3302 words)

	Atoms
		Dreischner and Weizsäcker proposed a "Postulate of Ultimate Objects" -- all objects consist of ultimate 2-state sub-objects: ultimate in the sense that they are atoms, in the original sense of the word, i.
		Weizsäcker observes that the postulate is trivial, since the mathematical objects used to express the states of a quantum object (n-dimensional Hilbert spaces), are always describable as sub-spaces of products of 2-dimensional spaces.
		But then the postulate is not so trivial after all since it depends on the mathematical framework we are using, and hence on our metaphysical assumptions.
	www.cs.yorku.ca /~peter/MH/atoms.html (666 words)

	[No title]
		The argument is that even the second order axioms of geometry leave the parallel postulate undecided, while in the semantical sense, the continuum hypothesis is decided by the axioms of Zermelo set theory.
		Of course this argument is not conclusive, but it does underline that there can be no purely formal justification for a distinction between arithmetical statements and such statements as CH as far as the consequences of their being undecidable are concerned.
		It's true that one reason why people are more reluctant to add conjectures of number theory than statements like the continuum hypothesis to axiom systems is that they are less confident that they are independent of the base theory they are starting with.
	www.math.niu.edu /~rusin/known-math/00_incoming/undecideable (1004 words)

	Wikinfo \| Axiom (Site not responding. Last check: 2007-10-29)
		rings, the operation of multiplication is commutative, and in some it is not; those rings in which it is are said to satisfy the "axiom of commutativity of multiplication." Another name for an axiom is postulate.
		These turn out to be fairly incomplete, actually, and many more postulates are necessary to completely characterize his geometry (Hilbert used 23).
		I say 4+1 since the 5th postulate (through a point outside a line there is exactly one parallel) was suspected to be derivable from the first 4 for nearly two millennia.
	www.wikinfo.org /wiki.php?title=Axiom (721 words)

	The Thurstone's Law--Principle and Method (Site not responding. Last check: 2007-10-29)
		The psychological continuum can be considered to be a continuum of subjective or psychological magnitudes.
		The postulate is made that the frequencies with which discriminal processes are associated with any given stimulus form a normal distribution on the psychological continuum.
		The scale value of the stimulus on the psychological continuum is taken as the value of its modal discriminal process.
	eies.njit.edu /~zxl8078/research/Thurstone.html (2483 words)

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