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# Infinite Sets Essay Example

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## Infinite Sets

Numbers with three rules: (1) Initial rule, which says, there exists an initial number ‘0’; (2) Successor rule that is for every number n, there exists a successor number; (3) Limit rule, which says for any endless series of numbers, there exists a limit number greater than every number in that series (Steinhart, 2009). In addition, Cantor defined infinite set as a set, which have one-to-one correspondence with a proper subset of itself. This theory provided a criterion to distinguish finite and infinite sets. For example, the set of natural numbers are infinite as it has one-to-one correspondence with the set of even natural numbers (1 is matched with 2, 2 with 4 and so on). Also, he presents the concept of cardinality of sets that is the size and magnitude of infinite sets also called as transfinite cardinals.Building on the Cantor’s work, a German mathematician David Hilbert explained different kinds of infinities through the paradox of a grand hotel which has infinitely many rooms, labeled as 1, 2, 3…and so on (Mamolo & Zazkis, 2008). He explains first type of infinity as a situation when a guest enters the hotel to check in and the receptionist makes an announcement to shift. all guests to the next room near to them. For example, occupant of room #1shift.s to room #2, occupant in room #2 shifts to room #3, and so on. In this way, room #1 becomes empty and could be given to the new guest. The hotel does not have the last room, therefore, infinity plus one is still infinity.Hilbert explains another situation when a bus with infinite members enters the grand hotel. To accommodate infinite passengers of that bus, the receptionist announces to move into the room with twice the number of their current room. For instance, occupant of room #1 shifts to room #2, occupant of room #2 shifts to room #4, and so on. In this way, all the rooms labeled with odd numbers are empty now to accommodate new guests; so, infinity plus infinity is still infinity.Comparing the infinite cubic toy blocks with that of infinite rooms in the Hilbert’s hotel, it can be argued that if the infinite cubes in the Deluxe set are stacked one on top of the other, it will reach Alpha Centauri which is the third brightest star seen from the planet earth. As the grand hotel has the longest infinite corridor, the toy cubes will also reach to the infinite height, which is only possible in our imagination. Moreover, as the infinite passengers can be easily accommodated in the

Works Cited

Mamolo, A. & Zazkis, R. Paradoxes as a window to infinity, Research in Mathematics Education, 10(2), 2008: 167-182, DOI: 10.1080/14794800802233696

Steinhart, E.C. A Mathematical Model of Divine Infinity, Theology and Science, 7(3), 2009: 261-274, DOI: 10.1080/14746700903036528

Tapp, C. Infinity in Mathematics and Theology, Theology and Science, 9(1), 2011: 91-100, DOI: 10.1080/14746700.2011.547009

Tirosh, D. Finite and infinite sets: definitions and intuitions, International Journal of Mathematical Education in Science and Technology, 30(3), 1999: 341-349, DOI: 10.1080/002073999287879

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