Infinity:The Continuum Hypothesis
Theconcept of infinity has been perceived by different mathematicians asan important concept since it has varied applications in mathematics.It is almost impossible to study mathematics without encountering theconcept of infinity, which shows the importance of the concept. Fromthe early mathematics history, two basic concepts of infinity thathave been in discussion concerns the potential and actual infinity.According Fischbein (2001), actual infinity comprises what theintelligence of individuals finds difficult to understand andimpossible to grasp. This may include the infinity of the world, theinfinity of real numbers as given, and the infinity of the number ofpoints in a given segment among other examples. Fischbein (2001)continues to argue that the concept of potential infinity deals witha dynamic kind of infinity. For example, after every natural number,there exists another natural number. There is a rich historyconcerning the origin of the concept of infinity, which has beenassociated with different individuals. This is an indication that theconcept did not just come, but it became developed by great thinkerswho were really concerned with the physical as well as naturaloccurrences. It was out of the physical and natural experiences thatgreat thinkers were in a position to come up with the idea ofinfinity, and developed it further through critical reasoning. Also,the concept of infinity has different applications in mathematics, aswell as in other spheres of life. In this report, the historicalsignificance of the concept will be discussed, where significantdevelopments of the concept will be discussed as well the individualsthat contributed to the development of the concept. Furthermore, theapplications of the concept in mathematics as well as in real worldwill be discussed in the report.
TheHistorical Significance of Infinity
Theinfinity is considered to have a great history and ancient Greekswere amidst the first individuals to consider infinite. AncientGreeks considered the concept of dividing an item in terms of halvesforever. Alternatively, the atomists had the believe that mattercomprised of tiny indivisibles while others had a different ideasince they felt that items could be broken into smaller onescontinuously. The Eleatic School of Philosophers was among the firstknown to determine the area of a circle through cutting it intovarious triangles and measuring the area of every triangle (Whinston,2009). They noticed that as the number of triangles increased, thesize of each triangle decreased and the estimation was closer to theactual area. In order to obtain the actual area, they noticed thatone needed to take infinitely many infinite small triangles. A majorquestion that these philosophers faced was how an infinite number ofnothing could add up to something such as a circle.
Theconcept of infinity became forced upon the Greeks by threetraditional observations in the physical world, which they hadexperience about space has no bound, time has no end, and time andspace can be subdivided unendingly. The most important of theseobservations that brought about the notion of infinity was theunending subdivision of space and time (Allen, n.d). The unendingsubdivision of space and time led to enquiry that wanted infinityclarified and as a result Zeno introduced his paradoxes throughmixing finite reasoning with infinite processes.
Therewere different contributions concerning the concept of infinity byvarious contributors, which led to the development of the concept andits application in mathematics. The following paragraphs will discussdifferent contributions to the concept that were developed by variousindividuals
Oneof the most important contributors to the idea of infinity wasPythagoras. The chief contribution of Pythagoras to the idea ofinfinity stood in the discovery of irrational numbers these arenumbers that proceed up to infinity. Pythagoras had the feeling thatthere was a finite amount of natural numbers (Allen, n.d). However,Aristotle argued against anything being infinite but believed thatthere was a potential infinity. In his argument Aristotle positedthat for any finite group, there exists a larger finite group andonly a finite number has ever been conceived.
Theconcept of infinity can also be associated with philosopher Zeno.Zeno emerged as the spokesperson for the Eleatic School ofPhilosophers. He posited that science cannot wrestle with realityunless it considers the ways infinity appears almost everywhere innature. He used paradoxes in showing how something could move throughinfinitely many points in finite time. In his arguments, Zenosupported the idea that it was possible to have something continuingto infinity (Whinston, 2009).
Justlike Aristotle, Euclid contributed to the concept of infinity bysupporting the idea of potential infinity. In 300 B.C.E, Euclidproved that there were infinitely many prime numbers, which indicatedthat there was potential infinity. However, Euclid did not considerthe existence of actual infinity (Allen, n.d).
TheBabylonians contributed to the concept of infinity through theintroduction of a number system having place value. This wasimportant because it helped in writing larger and larger numberswithout limitations. The Kabbalah writings emerged as the firstwriting to suggest that there were different types of infinity, as anendless collection of discrete things as well as a continuum and theidea did not come again until with Georg cantor.
Theconcept of infinity expanded further in the beginning of the 7thcentury, when Indian mathematicians added zero to be part of thenumber system. The Indian mathematicians had problems working withzero because they struggled in making it follow the arithmetic rules.The biggest problem, which the mathematicians had involved dividingby zero the Indian mathematicians argued that dividing by zeroresulted in an infinite quantity.
Besides,the contributions of Galileo were of importance to the concept ofinfinity because he set forth the difference between actual infinityand potential infinity. He argued that there are many perfect squaresas there are whole numbers. Also, in his contributions, he assertedthat the totality of all numbers is infinite and the number ofsquares is usually infinite.
Thecontributions of Georg Cantor were also of immense importance in thedevelopment of the concept of infinity. He introduced the concept ofa completed set. According to this concept, integers can beconsidered as a set in themselves and as a completed infinitemagnitude. He also defined transfinite number. He argued that ifomega (ω) could denote the first transfinite number, the nexttransfinite numbers could be ω + 1, ω + 2, and so on. The sequenceω, ω + 1, ω + 2,…..was considered a completed set. Through this,Cantor built a broad hierarchy of ordinal numbers. He discoveredthat some ordinals although infinite in size, are however smallercompared to other ordinals. In addition, Cantor also definedinfinite.
Theconcept of infinity has been applied in different mathematicalcontexts, either in form of theories or theorems that help in solvingmathematical problems. The following paragraphs will discuss the useof the infinity concept in mathematics
Calculusis one area of mathematics that makes use of limits. Since it usesthe idea of limits, it is difficult to separate calculus with theconcept of infinity. When solving mathematics involving limits incalculus, infinity, which is denoted as ∞ is used to representlarge numbers. It is important to note that numbers may be large,small or normal size normal size numbers comprise numbers thatindividuals have a common and clear feeling for (Clegg, 2013). Whensolving mathematics involving limits, the following concepts ofinfinity are important
Thiscomes from the idea that the inverse of a small number is usually alarge number. However, it is important to have a consideration forthe negative or positive sign when using the infinity sign. It isfirst important to establish whether the small number is negative orpositive. 0–represents small negative numbers while 0+represents small positive numbers. This implies that there isnegative infinity and positive infinity as follows
Incalculus, since positive infinity and negative infinity do notconstitute ordinary numbers, they are not treated as such and do notobey the common arithmetic rules. Also, in calculus, the applicationof infinity is exceedingly critical since continuity is a requirementin calculus, which is a concept of infinity. In addition, incalculus, -∞ and +∞ may indicate the boundary under whichprocesses like integration are to be carried out.
Theuse of limits is not limited only to calculus, but is also applied inFourier analysis. Limits play a significant role in studying improperintegrals improper integrals are integrals whose values may or maynot exist because their integrands may be discontinuous or the lengthof the interval of the integration is infinite (Clegg, 2013).Improper integrals and infinite series are fundamental concepts inFourier analysis. In defining the infinite series, the infinity ideais used. A Fourier series is an infinite series that is presented inthe form
Where,L is a positive number. Through the use of the idea of limits, it ispossible to determine whether the infinite series converges ordiverges. Therefore, the concept of infinity can be appliedmathematically in the Fourier analysis.
Inreal analysis, -∞ and +∞ may be added to a topological space ofreal numbers so as to generate a two-point compactification of realnumbers. When the algebraic properties become added, extended realnumbers are produced. Also in topology, -∞ and +∞ may be treatedto be the same, which can result in one-point compactification ofreal numbers to generate a real projective line (Clegg, 2013).Furthermore, ∞ can be added to a complex plane to be a topologicalspace, offering a one-point compactification of a complex plane. Inthis case, the resulting plane is usually a one-dimensional Riemannsurface. In topology, infinite-dimensional spaces are also used,which depict the concept of infinity.
Anotherarea of mathematics, where the concept of infinity can be applied isthe set theory. In set theory there is cardinal infinity and ordinalinfinity. From Georg Cantor’s transfinite concept, ordinal numberscan be associated with well-ordered sets, including points consideredfollowing an infinite number. Alternatively cardinal numbers usuallydefine the sets’ size implying how many members are include andmay be standardized through selecting the initial ordinal number of agiven size so as to represent a cardinal number of the desired size.
Theconcept of infinity has also been applied in mathematics involvingalgebra. In algebraic mathematics, infinity concept may be used todefine a range over which values extend or to indicate the boundaryfor instance, negative infinity and positive infinity may be used inindicating that the boundary of the values being considered fallsunder the range of -∞ and +∞. For instance, a function may bedefined by values that range from -∞ and +∞.
Theinfinite concept can be applied not only in mathematics, but also inother fields such as physics. In physics, the concept of infinitythat has applications is the Fourier series (Whinston, 2009). TheFourier series comprises of an infinite series. Communication is onearea that individuals encounter with in daily life withoutcommunication, life may be difficult since it may not be easy tounderstand or know what another person want. In physics, the Fouriertheory utilizes the concept of infinite series in trying to makepeople understand how a signal behaves the moment it passes throughamplifiers, filters and communication channels. Therefore, Fourieranalysis that utilizes the infinite concept is considered to play arole in communication in real-life situations.
Fourieranalysis can also be associated with real-life application, where itcan be used in the area of astronomy. At times, it is not feasible toobtain all the data required from a common telescope and in such ascenario one may be required to make use of radio waves or radarrather than using light (Ye, 2011). The radar signals obtained areusually treated as any other common time-varying voltage signal andmay also be processed digitally. Since the Fourier analysis makes useof the infinite series, this can be considered as a real-worldexample that makes use of the infinity concept.
Geologyalso entails another area, where Fourier transformations are used. Inthe research for seismic, Fourier transformations have been indicatedto have significance role in distinguishing amid the natural seismicoccurrences and nuclear test explosions due to their capacity toproduce varied frequency spectra. Thus, the use of Fouriertransformations in this area is a representation of an example whereinfinity concept has an application.
Inoptics, Fourier transformation is used in the diffraction of lightafter it goes through narrow slits. This idea can also be applied inmicrowave diffraction, acoustics, and x-ray (Ye, 2011). This uses theelectromagnetic theory, where the intensity of light is usuallyrelative to the square of the electric field that exists at any pointin the space. Because the Fourier transformation makes use ofinfinite series, then this area can be considered to use the conceptof infinity.
Anotherreal-world application of the concept of infinity is in harmonicanalysis. It is feasible to express any period function in form of aninfinite series of cosine and sine functions provided that the rightconditions apply. The infinite series is then applied in analyzingthe initial periodic function and filters applied to it. Forinstance, a sound recording device may have its bass amplified, oreven removed when using this technique. Thus, since there is use ofinfinite series in this situation, then this can be considered toapply the concept of infinity.
Also,another real-world application of the concept of infinity is roundingoff values (Ye, 2011). Calculations form part and parcel of people’slives and sometimes individuals are encountered with the need toround off values so as to have a value that is close to the accuratevalue. In so doing, it is important to use the infinite series. Thisusually applies to recurring decimals for example, 1/3 = 0.333….butthis can be written in the form 3/10 + 3/100 + 3/1000 +…..The morefractions that one would sum up would tend towards attaining anaccurate or close to an accurate answer.
Furthermore,another area of real-world application of the concept of infinityconcerns the movements that individuals make daily. On a daily basis,individuals make varied movements from one place to another. Thismovement may be through walking, travelling using vehicles, cycling,or even through being airlifted. The distance that individuals coverin a day is usually an approximation of the distance, which isactually covered. Therefore, since the distance covered is usually anapproximation of the actual distance moved, then the concept ofinfinity applies in this case.
Infinityis a mathematical concept, which cannot go unmentioned in the fieldof mathematics due to its broad application in the field. There aretwo forms of infinity the actual infinity and potential infinity.Actual infinity comprises what the intelligence of individuals findsdifficult to understand and impossible to grasp. This may includethe infinity of the world, the infinity of real numbers as given, andthe infinity of the number of points in a given segment among others.On the other hand, potential infinity deals with a dynamic kind ofinfinity. For example, after every natural number, there existsanother natural number. There is a rich history concerning the originof the concept of infinity, which has been associated with differentindividuals. This is an indication that the concept did not justcome, but it became developed by great thinkers who were reallyconcerned with the physical as well as natural occurrences. Theconcept of infinity became forced upon the Greeks by threetraditional observations in the physical world, which they hadexperience about space has no bound, time has no end, and time andspace can be subdivided unendingly. The most important of theseobservations that brought about the notion of infinity was theunending subdivision of space and time. The Eleatic School ofPhilosophers was among the first known to determine the area of acircle through cutting it into various triangles and measuring thearea of every triangle. They noticed that as the number of trianglesincreased, the size of each triangle decreased and the estimation wascloser to the actual area. Calculus is one area of mathematics thatmakes use of limits. Since it uses the idea of limits, it isdifficult to separate calculus with the concept of infinity. Otherareas include set theory, topology, and Fourier analysis. Inaddition, it can be argued that the Cantor’s continuum hypothesiscannot be decided because its truthfulness or falsehood has not beendecided. Indeed, the Cantor’s continuum hypothesis became presentedas part of Hilbert’s problems.
Allen,D.G. (n.d). Historyof Infinity.Texas: Texas A&M University.
Clegg,B. (2013). Abrief history of infinity: The quest to think the unthinkable.London: Constable & Robinson.
Whinston,A. (2009). AFinite History of Infinity. Oregon: Portland State University.
Ye,F. (2011). Strictfinitism and the logic of mathematical applications.Dordrecht [etc.: Springer.