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Topic: Actual infinity

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	Actual infinity - Wikipedia, the free encyclopedia
		Hence, in the philosophy of mathematics, the abstraction of actual infinity is the acceptance of infinite entities, such as the set of all natural numbers or an arbitrary sequence of rational numbers, as given objects.
		To reject the abstraction of actual infinity, as in intuitionism (see also intuitionistic logic), requires the reconstruction of the foundations of set theory and calculus as constructivist set theory and constructivist analysis respectively.
		One mathematician who accepted actual infinities was Georg Cantor, who decided that it is possible for natural and real numbers to be definite sets, and that if we reject the axiom of Euclidean finiteness (that states that actualities, singly and in aggregates, are necessarily finite), then we are not involved in any contradiction.
	en.wikipedia.org /wiki/Actual_infinity (278 words)

	Infinity - Wikipedia, the free encyclopedia
		In popular usage, infinity is usually thought of as something like "the largest possible number" or "the furthest possible distance" : hence naive questions such as "what is the next number after infinity?" or "if you travel to infinity, what happens if you then go a bit further?".
		In mathematics, "infinity" is often used in contexts where it is treated as if it were a number (i.e., it counts or measures things : "an infinite number of terms") but it is clearly a very different type of "number" than the integers or reals.
		Infinity is relevant to, or the subject matter of, limits, aleph numbers, classes in set theory, Dedekind-infinite sets, large cardinals, Russell's paradox, hyperreal numbers, projective geometry, extended real numbers and the absolute Infinite.
	en.wikipedia.org /wiki/Infinity (4071 words)

	Infinity (Site not responding. Last check: 2007-10-12)
		Infinity is the concept of limitlessness and unboundedness in size, number or extent.
		In mathematics, if a function grows beyond all bounds when the argument approaches a certain value, we say that the limit is infinity (written as ∞); this is also an example of potential infinity.
		In mathematics, actually infinite sets were first considered by Georg Cantor around 1873 and met with much resistance.
	www.ebroadcast.com.au /lookup/encyclopedia/in/Infinite_set.html (397 words)

	CATHOLIC ENCYCLOPEDIA: Infinity
		Infinity is a concept of the utmost importance in Christian philosophy and theology.
		Infinity implies that an infinite being cannot lack any reality in the line in which it is infinite, and that it cannot be surpassed by anything else in that particular perfection; but this does not necessarily mean that no other being can have perfections.
		How imperatively thought demands that infinity be ascribed to the self-existent Being is best shown by the fact, that all those who have at any time identified, and especially those who nowadays identify God and the world -- in short, all Monists -- almost universally speak if the infinity of their God.
	www.newadvent.org /cathen/08004a.htm (3244 words)

	[No title]
		Aristotle determined that actual infinity is the total parts of a segment resulting from the completed subdivisions of repeated halving and potential infinity as the process, evolving over time, of continued subdivision by repeated halving (Vilenkin 9).
		Aristotle emphasized that space is not an actual infinity, but potential infinity, using the example that a line does not consist of an actual infinity of points because it is a continuum, with only the potential for infinite divisions, a potential which can never be fully realized (“Aristotle’s Solution”).
		Leibniz stated it was possible to ‘master’ infinity using the principle of continuity and the constancy of law to transform characteristics of the finite to the infinite (Katasonov).
	web.pdx.edu /~emhorst/ResearchPaper.doc (3577 words)

	Archimedes Palimpsest
		We may divide all of infinity into two major types, one to be called “potential infinity”, the other to be called “actual infinity”.
		By actual infinity we mean something such as, say, the number of points on a line.
		For this reason they have completely avoided the concept of actual infinity, concentrating instead on potential infinity (whose rigorous treatment was perfected by Archimedes in his measurement of curves).
	www.archimedespalimpsest.org /scholarship_netz2.html (1131 words)

	A History of Infinity
		Famous intellectuals throughout the course of time have made many quotes about the idea of infinity, however to read them all, one would still wonder what the actual definition of infinity should be, because the ideas presented are as varied as the concepts that these famous people put forth in their lifeís works.
		Although the Greeks avoided infinity and did not have the notation to use it in their mathematics, they were able to devise the ratio of the circumference of a circle to its diameter (or as we call it today pi).
		Infinity was a very large part of this new math and appeared in the form of the infinitesimal (very small).
	www.missioncollege.org /depts/math/clouse2/wendi.htm (3202 words)

	Some remarks on cosmology
		Many of the arguments against an actual infinity can be traced back to Aristotle, although the Christian philosopher John Philoponus seems to have been the first to apply them (in 529 A.D.) to a demonstration of the finite age of the universe.
		Infinity is a property of the set as a whole, not of any particular member of the set.
		Although there may be aspects of infinity that appear incomprehensible to us, it seems to me that, in the absence of water-tight logical disproofs of actual infinity, the better course is to attribute this per ceived deficiency to human finiteness rather than to confer undue constraints upon God and His attributes.
	www.geocentricity.com /ba1/no76/cosmol.html (2631 words)

	INFINITY AND THE PAST
		This argument is based on a fallacy of equivocation with respect to the phrases “actual infinity” and “potential infinity.” Whitrow’s proof that the past if infinite is “actually infinite” is based on a different sense of “actually infinite” than that belonging to his proof that an “actually infinite past” is impossible.
		However, in his proof that an actually infinite past is impossible, Whitrow uses “actually infinite” in the sense of a series of events some of which are separated from E by an infinite number of intermediate events.
		For if the past is an “actual infinity” in the sense of being an infinity of events that have really occurred, it does not follow that it is also an “actual infinity” in the sense that some past events are separated from the present event by an infinite number of intermediate events.
	www.qsmithwmu.com /infinity_and_the_past.htm (5159 words)

	Is The Past Really Finite?
		An actual infinite, by its definition, cannot be added to (for it is the number of all possible numbers).
		His arguments about the actual infinity may be unquestionably true, but if he cannot span these gaps, they have no relevant use in his cosmological argument.
		One is to evaluate Craig's philosophical arguments on the concept of the actual infinity in a broad and comprehensive fashion; this is the method used by Quentin Smith in his paper Infinity and the Past.
	ccwf.cc.utexas.edu /~govind/stories/finitude.html (3433 words)

	Questions of whether an actual infinite exists, and in what capacity, have woven their way through the history of ...
		Despite the fact that infinities are capable of being introduced in complex theories and attributed properties that limit their functions (such as subtraction and division in cardinal arithmetic), it is the individuals themselves, the infinities, that are defined in such a way as to reject instantiation, not the consistent mathematical systems.
		Transfinite arithmetic is not an ontological argument for infinities.
		If infinity is taken to have the predicate un-completeable totality, we certainly have record of objects that, given their form constitute ‘always incomplete,’ the natural numbers being the most obvious.
	www.tektonics.org /guest/kalam.htm (6714 words)

	Reviews of The New Mormon Challenge
		An "actual infinity" can mean: (a) an infinite set of events that are all actual at once, or (b) an infinite set of events some of which have been but are no longer actual.
		My argument does not attempt to establish that infinities actually do or do not occur in the real world; but merely demonstrates that it is logically possible that the world has always existed.
		The distinction between an actual and a potential infinite is made by virtually every proponent of the kalam infinity arguments.
	www.fairlds.org /New_Mormon_Challenge/TNMC01.html (12337 words)

	Infinity
		In mathematics and philosopy we find two concepts of infinity: potential infinity, which is the infinity of a process which never stops, and actual infinity which is supposed to be static and completed, so that it can be thought of as an object.
		Thus we cannot use the concept of actual infinity at all.
		Infinity is not possible possible in nature, only in an nonapplied math..., Comment by jason larkin
	pespmc1.vub.ac.be /INFINITY.html (986 words)

	Infinity and Jesus' Humanity
		The term, actual infinite, is confusing, because the word, actual, makes it seem as if it were actually possible for there to be an actual infinite.
		In other words, it is not possible for there to be an actual infinite set of physical objects or temporal durations in the universe.
		Clearly, it is not appropriate to use a potential or an actual infinite with respect to Deity, because these types of infinites refer to physical or temporal entities, such as sets or as divisible continuums.
	www.muhammadanism.com /Jesus/JesusInfinite.htm (1974 words)

	Potential vs. Completed Infinity
		Completed infinity, or actual infinity, is an infinity that one actually reaches; the process is already done.
		There are no points for plus or minus infinity on the line, but it is natural to attach those "numbers" to the endpoints of the semicircle.
		But more complicated notions such as infinity, less grounded in physical reality, are harder to explain; it is harder to be sure that we are successfully conveying a concept from the inside of one head to the inside of another.
	www.math.vanderbilt.edu /~schectex/courses/thereals/potential.html (1976 words)

	Actual and Potential Infinity - Eidos Forums
		I would be suprised if there are more votes for Actual Infinity than there is for Potential since I suspect that most (90% perhaps) of people here will say they live in a Potential Infinite Universe.
		Infinity is merely a term applied to numbers, distances or quantities of such great magnitude (or minutiae) that the mind cannot grasp.
		Actually, I believe that debate on the nature of infinity is pointless, since it is an interesting but ultimately unproveable concept.
	forums.eidosgames.com /showthread.php?t=8153 (2238 words)

	God and Mathematical Infinity
		As mentioned earlier, we are indebted to two late nineteenth century thinkers for systematizing and rationalizing the treatment of actual infinity in mathematics.
		The chief claim is that mathematical ideas about actual infinity are often justified and, more to the point, that they are useful in religious thought.
		Another insight is that the infinity of God requires the divinity of Christ, since the only possible atonement for sins committed against the Infinite is by way of an infinite redeemer.
	www.asa3.org /ASA/PSCF/1998/PSCF3-98Kneale.html (2999 words)

	[No title]
		> a past event has actually happened but now it´s no longer actual > itself (that´s why it´s called "past") No, it's called the "past" because it is in the set of all events that could potentially be seen as influencing (but not being influenced by) the events you currently call "present".
		If an actual infinity could be shown to already exist, then you wouldn't bother defending your position.
		Similarly, if an actual infinity could be shown to be logically impossible, then I wouldn't bother defending my position.
	humanknowledge.net /Correspondence/Paul_Holbach/Infinity/2002-03-23.txt (1258 words)

	Consistency and Representations: The Case of Actual Infinity. (Site not responding. Last check: 2007-10-12)
		Consistency and Representations: The Case of Actual Infinity.
		Demonstrates how research-based knowledge of students' incompatible solutions to various representations of the same problem could be used to raise their awareness of inconsistencies in their reasoning.
		Reports that students' decisions as to whether two given infinite sets have the same number of elements depend on the specific representation of the infinite sets in the problem.
	www.eric.ed.gov /sitemap/html_0900000b800145af.html (84 words)

	The Ancients and Infinity
		Aristotle allows the use of potential infinity, but rejects the use of actual infinities.
		I think he was trying to find a way to salvage the use of infinity in arguments without getting mired in the absurd.
		However, the mere fact that Euclid named the fifth postulate as a postulate seems to indicate that Euclid understand that the fifth postulate was logically independent of the first four postulates.
	descmath.com /diag/ancients.html (3303 words)

	Infinity: You Can't Get There From Here -- Platonic Realms MiniText
		Infinity, of course, infected our imaginations, and for some of us it cropped up in our conscious thoughts every now and then in new and interesting ways.
		Cantor completely contradicted the Aristotelian doctrine proscribing actual, “completed” infinities, and for his boldness he was rewarded with a lifetime of controversy, including condemnation by many of the most influential mathematicians of his time.
		To a present day mathematician, infinity is both a tool for daily use in his or her work, and a vast and intricate landscape demanding to be explored.
	www.mathacademy.com /pr/minitext/infinity/index.asp (3732 words)

	Must the past have a beginning (Site not responding. Last check: 2007-10-12)
		To derive the impossibility of the actual infinite from premise 2 (together with some other premises), while at the same time deriving premise 2 from the claim that the real world can contain only finite sets, would be to argue in a very tight – and vicious – circle.
		But in that case, the object of God’s knowledge will contain an actual infinity of logically distinct items – and this will be so whether or not they all somehow ‘depend’ on his awareness.
		Craig has recently opined that ‘any normal adult whose intuitions have not been jaded by the common textbook assertions that actual infinities are wholly unobjectionable’ will find his argument against the actual infinite ‘extremely plausible.’[47] I do not share this assessment of Craig’s various arguments against the actual infinite.
	stripe.colorado.edu /~morristo/actual-infinite.html (9442 words)

	A Neighborhood of Infinity: An Actual Application of Fractal Dimension
		A Neighborhood of Infinity: An Actual Application of Fractal Dimension
		According to this story, and others, a trove of 32 paintings, purported to be Jackson Pollocks, was discovered last year.
		This is a paragraph of text that could go in the sidebar.
	sigfpe.blogspot.com /2006/03/actual-application-of-fractal.html (174 words)

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