Infinity In Deutsch Beispielsätze für "infinity"
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A different form of "infinity" are the ordinal and cardinal infinities of set theory—a system of transfinite numbers first developed by Georg Cantor.
This modern mathematical conception of the quantitative infinite developed in the late 19th century from works by Cantor, Gottlob Frege , Richard Dedekind and others—using the idea of collections or sets.
Dedekind's approach was essentially to adopt the idea of one-to-one correspondence as a standard for comparing the size of sets, and to reject the view of Galileo derived from Euclid that the whole cannot be the same size as the part however, see Galileo's paradox where he concludes that positive square integers are of the same size as positive integers.
An infinite set can simply be defined as one having the same size as at least one of its proper parts; this notion of infinity is called Dedekind infinite.
The diagram to the right gives an example: viewing lines as infinite sets of points, the left half of the lower blue line can be mapped in a one-to-one manner green correspondences to the higher blue line, and, in turn, to the whole lower blue line red correspondences ; therefore the whole lower blue line and its left half have the same cardinality, i.
Cantor defined two kinds of infinite numbers: ordinal numbers and cardinal numbers. Ordinal numbers characterize well-ordered sets, or counting carried on to any stopping point, including points after an infinite number have already been counted.
Generalizing finite and ordinary infinite sequences which are maps from the positive integers leads to mappings from ordinal numbers to transfinite sequences.
Cardinal numbers define the size of sets, meaning how many members they contain, and can be standardized by choosing the first ordinal number of a certain size to represent the cardinal number of that size.
The smallest ordinal infinity is that of the positive integers, and any set which has the cardinality of the integers is countably infinite. If a set is too large to be put in one-to-one correspondence with the positive integers, it is called uncountable.
Cantor's views prevailed and modern mathematics accepts actual infinity as part of a consistent and coherent theory.
This hypothesis can neither be proved nor disproved within the widely accepted Zermelo—Fraenkel set theory , even assuming the Axiom of Choice.
Cardinal arithmetic can be used to show not only that the number of points in a real number line is equal to the number of points in any segment of that line, but also that this is equal to the number of points on a plane and, indeed, in any finite-dimensional space.
These curves can be used to define a one-to-one correspondence between the points on one side of a square and the points in the square.
The structure of a fractal object is reiterated in its magnifications. Fractals can be magnified indefinitely without losing their structure and becoming "smooth"; they have infinite perimeters—some with infinite, and others with finite surface areas.
One such fractal curve with an infinite perimeter and finite surface area is the Koch snowflake. Leopold Kronecker was skeptical of the notion of infinity and how his fellow mathematicians were using it in the s and s.
This skepticism was developed in the philosophy of mathematics called finitism , an extreme form of mathematical philosophy in the general philosophical and mathematical schools of constructivism and intuitionism.
In physics , approximations of real numbers are used for continuous measurements and natural numbers are used for discrete measurements i.
Concepts of infinite things such as an infinite plane wave exist, but there are no experimental means to generate them.
The first published proposal that the universe is infinite came from Thomas Digges in Living beings inhabit these worlds.
Cosmologists have long sought to discover whether infinity exists in our physical universe : Are there an infinite number of stars?
Does the universe have infinite volume? Does space "go on forever"? This is an open question of cosmology. The question of being infinite is logically separate from the question of having boundaries.
The two-dimensional surface of the Earth, for example, is finite, yet has no edge. By travelling in a straight line with respect to the Earth's curvature one will eventually return to the exact spot one started from.
The universe, at least in principle, might have a similar topology. If so, one might eventually return to one's starting point after travelling in a straight line through the universe for long enough.
The curvature of the universe can be measured through multipole moments in the spectrum of the cosmic background radiation.
To date, analysis of the radiation patterns recorded by the WMAP spacecraft hints that the universe has a flat topology. This would be consistent with an infinite physical universe.
However, the universe could be finite, even if its curvature is flat. An easy way to understand this is to consider two-dimensional examples, such as video games where items that leave one edge of the screen reappear on the other.
The topology of such games is toroidal and the geometry is flat. Many possible bounded, flat possibilities also exist for three-dimensional space.
The concept of infinity also extends to the multiverse hypothesis, which, when explained by astrophysicists such as Michio Kaku , posits that there are an infinite number and variety of universes.
These are defined as the result of arithmetic overflow , division by zero , and other exceptional operations. Some programming languages , such as Java  and J ,  allow the programmer an explicit access to the positive and negative infinity values as language constants.
These can be used as greatest and least elements , as they compare respectively greater than or less than all other values. They have uses as sentinel values in algorithms involving sorting , searching , or windowing.
In languages that do not have greatest and least elements, but do allow overloading of relational operators , it is possible for a programmer to create the greatest and least elements.
In languages that do not provide explicit access to such values from the initial state of the program, but do implement the floating-point data type , the infinity values may still be accessible and usable as the result of certain operations.
In programming, an infinite loop is a loop whose exit condition is never satisfied, thus theoretically executing indefinitely.
Perspective artwork utilizes the concept of vanishing points , roughly corresponding to mathematical points at infinity , located at an infinite distance from the observer.
This allows artists to create paintings that realistically render space, distances, and forms. Escher is specifically known for employing the concept of infinity in his work in this and other ways.
Variations of chess played on an unbounded board are called infinite chess. Cognitive scientist George Lakoff considers the concept of infinity in mathematics and the sciences as a metaphor.
The symbol is often used romantically to represent eternal love. Several types of jewelry are fashioned into the infinity shape for this purpose.
From Wikipedia, the free encyclopedia. For the symbol, see Infinity symbol. For other uses of "Infinity" and "Infinite", see Infinity disambiguation.
Mathematical concept. Further information: Infinity philosophy. Main article: Infinity symbol. Main articles: Cardinality and Ordinal number.
Main article: Cardinality of the continuum. Main article: Dimension vector space. Math Vault. Retrieved Perspectives on Science.
Archived from the original on 15 February The Enlightenment set up the conditions for knowledge creation which disrupted the static societies that previously existed.
These conditions are the valuing of creativity and the free and open debate that exposed ideas to criticism to reveal those good explanatory ideas that naturally resist being falsified due to their having basis in reality.
Deutsch points to previous moments in history, such as Renaissance Florence and Plato's Academy in Golden Age Athens , where this process almost got underway before succumbing to their static societies' resistance to change.
The source of intelligence is more complicated than brute computational power, Deutsch conjectures, and he points to the lack of progress in Turing test AI programs in the six decades since the Turing test was first proposed.
What matters for knowledge creation, Deutsch says, is creativity. New ideas that provide good explanations for phenomena require outside-the-box thinking as the unknown is not easily predicted from past experience.
To test this Deutsch suggests an AI behavioural evolution program for robot locomotion should be fed random numbers to see if knowledge spontaneously arises without inadvertent contamination from a human programmer's creative input.
If it did Deutsch would concede that intelligence is not as difficult a problem as he currently thinks it is. Deutsch sees quantum superpositions and the Schrödinger equation as evidence for his many worlds quantum multiverse , where everything physically possible occurs in an infinite branching of alternate histories.
Deutsch argues that a great deal of fiction is close to a fact somewhere in the multiverse. Deutsch explains that interference offers evidence for this multiverse phenomenon where alternate histories affect one another without allowing the passage of information, as they fungibly intertwine again shortly after experiencing alternate events.
According to Deutsch, our perspective on any object we detect with our senses is just a single universe slice of a much larger quantum multiverse object.
Deutsch speculates on the process of human-culture development from a genetic basis through to a memetic emergence. This emergence led to the creation of static societies where innovation occurs, but most of the time at a rate too slow for individuals to notice during their lifetimes.
It was only at the point where knowledge of how to purposefully create new knowledge through good explanations was acquired that the beginning of infinity took off during the enlightenment.
Relax on a palm-fringed beach lying a short boat hop away or by the resort's infinity -edge swimming pool.
From TIME. Of course this function also says that as time t goes to infinity so does the cumulative amount of oil. From Wired. In love, differences disappear and the human soul accomplishes its object in perfection, exceeding its own boundaries and traversing the threshold of infinity.
I witnessed the merging of water and sky, the infinity of the universe, the changing of perspective. From The Atlantic.
It offers not just another scenic vista but at once invites both physical exertion and reverie, plus reminders of a universe stretching to infinity.
From OregonLive. As anyone who has lost a loved one knows, the difference between zero and one is an infinity.
But other great mathematicians realized that he had opened up fantastic vistas with his ideas about infinity , and they're pretty nearly universally accepted today.
From NPR. Translations of infinity in Chinese Traditional. Need a translator? Translator tool. What is the pronunciation of infinity?
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