Nevertheless, Cantor's remark would also serve nicely to express the surprise that so many mathematicians after him have experienced on first encountering a result that is so counter-intuitive. The evidence strongly suggests that Cantor was quite confident in the result itself and that his comment to Dedekind refers instead to his then-still-lingering concerns about the validity of his proof of it. Measure theory provides a more nuanced theory of size that conforms to our intuition that length and area are incompatible measures of size.
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This demonstrates that the "size" of sets as defined by cardinality alone is not the only useful way of comparing sets. "I see it but I don't believe," Cantor wrote to Richard Dedekind after proving that the set of points of a square has the same cardinality as that of the points on just an edge of the square: the cardinality of the continuum. See Hilbert's paradox of the Grand Hotel for more on paradoxes of enumeration. Since there is a bijection between the two sets involved, this follows in fact directly from the definition of the cardinality of a set. The answer is no, because the squares are a proper subset of the naturals: every square is a natural number but there are natural numbers, like 2, which are not squares of natural numbers.īy defining the notion of the size of a set in terms of its cardinality, the issue can be settled.The answer is yes, because for every natural number n there is a square number n 2, and likewise the other way around.Are there as many natural numbers as squares of natural numbers when measured by the method of enumeration? It had been discussed by Galileo Galilei and Bernard Bolzano, among others. But not every intuition regarding the size of finite sets applies to the size of infinite sets, leading to various apparently paradoxical results regarding enumeration, size, measure and order.īefore set theory was introduced, the notion of the size of a set had been problematic. Just as for finite sets, the theory makes further definitions which allow us to consistently compare two infinite sets as regards whether one set is "larger than", "smaller than", or "the same size as" the other. Instead of relying on ambiguous descriptions such as "that which cannot be enlarged" or "increasing without bound", set theory provides definitions for the term infinite set to give an unambiguous meaning to phrases such as "the set of all natural numbers is infinite". A special case of Cantor's theorem proves that the set of all real numbers R cannot be enumerated by natural numbers. These sets have in common the cardinal number | N| = ℵ 0 with the usual order 0 | S|. Examples of countably infinite sets are the natural numbers, the even numbers, the prime numbers, and also all the rational numbers, i.e., the fractions. Every infinite set which can be enumerated by natural numbers is the same size (cardinality) as N, and is said to be countable. As this assumption cannot be proved from first principles it has been introduced into axiomatic set theory by the axiom of infinity, which asserts the existence of the set N of natural numbers. Set theory as conceived by Georg Cantor assumes the existence of infinite sets. 4.1 Early paradoxes: the set of all setsīasics Cardinal numbers.
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Is crossplot cutoffs help to generate 3D lithofacies cubes, which is used to extract facies geobodies. Integration between Inversion results and Ip vs. Is crossplot, clear separation occurs in the P-impedance so post stack inversion is enough to be applied. Spectral decomposition unravels the seismic signal into its constituent frequencies.Ī crossplot between P-wave Impedance (Ip) and S-wave Impedance (Is) derived from well logs (P-wave velocity, S-wave velocity and density) can be used to discriminate between gas-bearing sand, water-bearing sand, and shale. Spectral decomposition of a seismic signal aims to characterize the time-dependent frequency response of subsurface rocks and reservoirs for imaging and mapping of bed thickness and geologic discontinuities. Seismic data, being non-stationary in nature, have varying frequency content in time.
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A detailed study for channel connectivity and lithological discrimination was established to delineate the gas charged geobody. The main challenge of seismic reservoir characterization is to discriminate between Gas sand, Water sand and Shale, and extracting the gas-charged geobody from the seismic data.
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Solar gas discovery is one of the Turbidities Slope channels within the shallow Pliocene level that was proved by Solar-1 well. Nile delta province is rapidly emerging as a major gas province commercial gas accumulations have been proved in shallow Pliocene channels of El-Wastani Formation.