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Topic: History of the separation axioms

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	Zermelo-Fraenkel Axioms -- from Wolfram MathWorld
		The Zermelo-Fraenkel axioms are the basis for Zermelo-Fraenkel set theory.
		Axiom of Choice: Every family of nonempty sets has a choice function.
		axioms of choice or foundation in Zermelo set theory, but does include the axiom of replacement.
	mathworld.wolfram.com /Zermelo-FraenkelAxioms.html (316 words)

	Separation axiom - Wikipedia, the free encyclopedia
		The separation axioms are denoted with the letter "T" after the German "Trennung", which means separation.
		The separation axioms are about the use of topological means to distinguish disjoint sets and distinct points.
		The separation axioms all say, in one way or another, that points or sets that are distinguishable or separated in some weak sense must also be separated in some stronger sense.
	en.wikipedia.org /wiki/Separation_axiom (1480 words)

	Eliohs - Macaulay - History
		History has its foreground and its background: and it is principally in the management of its perspective that one artist differs from another.
		Their history, from the Trojan to the Persian war, is covered with an obscurity broken only by dim and scattered gleams of truth.
		The majesty of history seems to resemble the majesty of the poor King of Spain, who died a martyr to ceremony because the proper dignitaries were not at hand to render him assistance.
	www.eliohs.unifi.it /testi/800/macaulay/macaulay_history.html (14865 words)

	The Ontology and Cosmology of Non-Euclidean Geometry
		This could be called the axiom of hetero-curvature, and it would make true non-Euclidean geometry possible, since lines with non-Euclidean relations to each other would be straight in the common meaning of the term understood by Euclid or Kant.
		It is necessary to keep in mind that these axioms are answers to questions concerning reality that would be asked in physics or metaphysics and are logically entirely separate from the status of geometry in logic or mathematics or from our psychological powers of visual imagination.
		So again we have an object lesson in the history of science, that a careful examination of the implications of a theory is sometimes neglected by professional science.
	www.friesian.com /curved-1.htm (6358 words)

	Haines Brown, The Contradiction of World History
		In the sense that the nation was private culture raised to the political level, any conception of history in terms of private interest and power inevitably reduced world history to the history of the world's nations, and the outcome of history legitimated the private sphere that rose to world dominance.
		That requires a conception of world history in which there is progress is the result of struggle for an improbable outcome, not of the decisions, however wise, of the ruling class, who are capable only of greater efficiency and functionality, which only further deepens the contradiction of the existing system.
		A world history worthy of the twenty-first century must be a world history that has a determinant relation with the natural and social sciences, not one that flees them to embrace literary indulgence, a self-absorbed fixation on roots, or to flee science altogether in flights of irrationality.
	www.hartford-hwp.com /archives/10/039.html (21223 words)

	CATHOLIC ENCYCLOPEDIA: Masonry (Freemasonry)
		Moral and religious definitions, axioms and propositions have as regular and certain dependence upon each other as any in physics or mathematics." "Let me recommend you to pursue such knowledge and cultivate such dispositions as will secure you the Brotherly respect of this society and the honour of your further advancement in it".
		In nearly all the countries of Europe the separation between State and Church and the laicization or neutralization of the popular instruction and education, were and are still demanded by all parties of the Left with redoubled impetuosity.
		History affords numerous instances of acts which have been justified by subsequent events, and none of us, whether Masons or not, are inclined to condemn the plots hatched between Paul Revere, Dr. J.
	www.newadvent.org /cathen/09771a.htm (14836 words)

	Pragmatism by Philip Wiener in Dictionary of the History of Ideas
		Examples of pragmatic history appear in patriotic histories and the biographies of heroes and spiritual leaders that are supposed to teach rulers, statesmen, and moralists how to be guided by the experience of the past.
		Kant's separation of phenomena from the metaphysical unknowable "thing-in-itself" ( Ding an sich) led to the positivistic element in empiricistic pragmatism.
		From the standpoint of technique, the axioms enable one to prove with the aid of acceptable rules of inference a body of theorems or consequences deducible from the axioms, thus reducing a large number of theorems to what is contained in a small number of axioms.
	www.pragmatism.org /companion/pragmatism_wiener.htm (12971 words)

	Nqthm-1992 Logic and Reference Guide
		A history h is a finite sequence of axiomatic acts such that either (a) h is empty or (b) h is obtained by concatenating to the end of a history h' an axiomatic act that is ``admissible'' under h'.
		The axioms of a history h is the union of the axioms added by each act in h together with the axioms described in this chapter.
		If h is a history, fs is a tolerable functional substitution, thm is a theorem of h, and the fs instance of an fs renaming of every axiom of h can be proved in h, then thm\fs is a theorem of a definitional/constrain extension of h.
	www.cs.utexas.edu /users/boyer/logic-reference.html (16647 words)

	Hegel's History of Philosophy (Site not responding. Last check: 2007-10-03)
		Philosophy and exact science were not yet separated, and it was only later on that this separation first took place.
		(Axiom.) “We now have a Notion of God, but His objective reality is neither formally nor eminenter contained within us, and it can thus be only in God Himself.” (20) Consequently we see that with Descartes this Idea is an hypothesis.
		Descartes thus separates extension from God, remains constant to this separation, unites the universe, matter, with God in such a way as to make Him its creator and first cause: and he has the true perception that conservation is a continuous creation, in so far as creation as activity is asserted to be separated.
	www.marxists.org /reference/archive/hegel/works/hp/hpdescar.htm (7979 words)

	Phrenology, Mesmerism, and Spiritualism
		Charles Dickens practiced mesmerism on his own wife(70).3 In 1838 approximately 170 of the 1,000 members of the newly formed phrenological societies were physicians and surgeons(29).4 As for the working class, spiritualism provided a way to deny the permanent separation caused by the massive numbers of deaths brought about by their miserable living conditions.
		However Lavater did set forth some axioms that served as the direct forerunners of Gall's phrenology as well as what could be termed the Victorian obsession with death.
		The axioms set forth in spiritualism allowed for a type of solution to the problem of death.
	www.gober.net /victorian/reports/mesmersm.html (4009 words)

	Hausdorff space - Encyclopedia (directory) (Site not responding. Last check: 2007-10-03)
		The conditions imposed are the most significant separation axioms.
		space, or separated space, iff, given any distinct points x and y, there are a neighbourhood U of x and a neighbourhood V of y that are disjoint.
		The terms "Hausdorff", "separated", and "preregular" can also be applied to such variants on topological spaces as uniform spaces, Cauchy spaces, and convergence spaces.
	www.egran.com /Hausdorff_space.html (821 words)

	History of Delevopmental Genetics IV
		One of the reasons that molecular biology has taken so long to enter embryology is the longstanding fear among embryologists that genetics--whether it be classical genetics or molecular biology--is trying to take over their discipline and bring with it all its reductionism and lack of appreciation for the complexity and species differences.
		Although experimental embryology had successfully separated itself from the earlier traditions of developmental anatomy, it remained a phenotypic science, and it identified itself as a science concerned with cytoplasmic changes.
		Waddington is sufficiently aware of history to relate all this back to T. Morgan's 1934 statement that the initial protoplasmic regions determine which of the genes are active.
	zygote.swarthmore.edu /gene5a.html (8894 words)

	54: General topology
		Thus a general theme in topology is to test the extent to which the axioms force the kind of structure one expects to use and then, as appropriate, introduce other axioms so as to better match the intended application.
		Since the axioms of topology are stated in terms of subsets of X, it should be no surprise that one branch of topology is closely related to set theory, particularly "descriptive set theory".
		The distinction between this and the previous paragraph is that additional axioms are assumed about a new construct provided at the outset, rather than additional axioms about the topology; thus the questions asked about these structures can be about either the topology or about the new construct.
	www.math.niu.edu /~rusin/known-math/index/54-XX.html (2455 words)

	[No title]
		Motivation for KP KP is largely concerned with issues of constructibility, as its history would suggest.
		The negation of this axiom asserts that all ordinals are finite.
		This is similar to the first formulation of the axiom of choice, except it is restricted to well-founded relations.
	www.afn.org /~afn07474/kppaper.html (1654 words)

	Set Theory. Zermelo-Fraenkel Axioms. Russell's Paradox. Infinity. By K.Podnieks
		The axioms C1, 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).
		An alternative, extremely convenient form of the separation schema can be obtained by using the notion of classes: the formula F defines a class A, hence, the axiom C21[F] says that the intersection A^x (of the class A and the set x) is a set: A^x=z.
		The set theory adopting the axiom of extensionality (C1), the axiom 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.
	www.ltn.lv /~podnieks/gt2.html (8377 words)

	Baptist History and Heritage: 20th century AD
		Stances for religious liberty and the separation of church and state have long been among the hallmarks of Baptists.
		This article will briefly review the doctrinal reasons for and the history of Baptist efforts toward and statements on religious liberty and separation of church and state in America.
		In describing why he chose to unite with Baptists, evangelist Billy Graham wrote, "I share with Baptists a strong belief in the separation of church and state" and went on briefly to describe Baptist contributions to that end.
	www.findarticles.com /p/articles/mi_m0NXG/is_2_34/ai_94160859 (1440 words)

	Regular space - Enpsychlopedia (Site not responding. Last check: 2007-10-03)
		X is a regular space iff, given any closed set F and any point x that does not belong to F, there are a neighbourhood U of x and a neighbourhood V of F that are disjoint.
		The point x, represented by a dot to the left of the picture, and the closed set F, represented by a closed disk to the right of the picture, are separated by their neighbourhoods U and V, represented by larger open disks.
		There are many situations where another condition of topological spaces (such as normality, paracompactness, or local compactness) will imply regularity if some weaker separation axiom, such as preregularity, is satisfied.
	www.grohol.com /psypsych/Regular_space (1067 words)

	Articles - Topological space (Site not responding. Last check: 2007-10-03)
		Topological spaces can be broadly classified according to their degree of connectedness, their size, their degree of compactness and the degree of separation of their points and subsets.
		Some of these terms are defined differently in older mathematical literature; see History of the separation axioms.
		A space is separable if it has a countable dense subset.
	www.x-moto.net /articles/Topological_space (2493 words)

	Math History - Age of Liberalism
		Pearson publishes the first in a series of 18 papers, written over the next 18 years, which introduce a number of fundamental concepts to the study of statistics.
		Zermelo uses the axiom of choice to prove that every set can be well ordered.
		He bases set theory on seven axioms : Axiom of extensionality, Axiom of elementary sets, Axiom of separation, Power set axiom, Union axiom, Axiom of choice and Axiom of infinity.
	lahabra.seniorhigh.net /pages/teachers/pages/math/timeline/mLiberalism.html (1522 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).
		He proposed to restrict the comprehension axiom schema by adopting only of those axioms, which are really necessary for reconstruction of common mathematics.
		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 (7478 words)

	Introduction to the works of Euclid
		An axiom or common notion is an assertion, the truth of which is taken for granted as being blatantly obvious, and which is applicable -- by analogy, at least -- in all sciences.
		The specification takes separately the thing that is sought and makes clear precisely what it is. The construction adds what is lacking in the given for finding what is sought.
		It is believed that in this axiom, he is asserting that superposition is an acceptable method of proving the equality of two figures.
	www.obkb.com /dcljr/euclid.html (9124 words)

	THE HISTORY OF NEW EVANGELICALISM
		We disagree with Ashbrook in a number of areas pertaining to the church and his Protestant view of history, yet I am convinced that this material needs to be in the hands of our readers, and I am grateful that we have been given permission to reprint the following excerpt from his book.
		That is not the case with Dr. Ockenga's repudiation of separation.
		In the past history of gospel meetings, testimonies were given by those who were famous for great hymns, missionary service or notable sacrifice for Christ.
	wayoflife.org /~dcloud/fbns/historyevangelical.htm (5598 words)

	Commentary
		They traveled by carriage but as the snow increased during their trip, blanketing the area with possibly as much as two feet of white powder, they were forced to traverse the final, rough eight miles on horseback (11).
		Lincoln named those principles "the definitions and axioms of free society." They have done more to define the values and goals of the nation than any other act or statement.
		It is this Jefferson who stands at the radiant center of his own history, and who makes for the present a symbol that unites the nation's birth with its inexorable ideal.
	www.jeffersonlegacy.org /commentary.html (9052 words)

	The Constitution For The United States, Its Sources and Its Applications - History
		It is highly self-protective, and is skillfully designed to break up and dissipate radical attacks on any of its fundamental axioms, while at the same time it permits a large freedom of movement to those who are entrenched behind it.
		Torn between the hazards of lending his reputation to a gathering perhaps doomed to failure and the chance that the public would view his reluctance to attend with a critical eye, the general finally agreed to make the trip.
		The determined Madison had for several years insatiably studied history and political theory searching for a solution to the political and economic dilemmas he saw plaguing America.
	www.barefootsworld.net /consti15.html (13217 words)

	Why should evolutionism be called a fairytale rather than a theory? (Site not responding. Last check: 2007-10-03)
		Often persuasive pseudo-scientists will state the meanings of these axioms in ways that really don't spell out the implications of what the so-called axioms are.
		So, to sum it up, sometimes SSWAAFT s are called the axioms of science.
		People prefer to believe what they prefer to be true, and the unproved axioms are a great method for justifying preposterous beliefs.
	www.seekfind.net /dinosaurs/evolution/The_SSWAAFT_of_evolution.html (3415 words)

	Separation axiom (Site not responding. Last check: 2007-10-03)
		The separation axioms are denoted with the letter "T" after the German "Trennung"
		It will be a common theme among the separation axioms to have one version of an axiom that requires T0 and one version that doesn't
		The T0 axiom is special in that it cannot only be added to a property (so that regular plus T0 is T3) but also subtracted from a property (so that Hausdorff minus T0 is preregular)
	www.baapoo.com /index.php/Separation_axiom (1457 words)

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