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Infinite Sets

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Numbers with three rules: 1) Initial rule, wich says, tere exists an initial number ‘0’; 2) Successor rule that is for every number n, tere exists a successor number; 3) Limit rule, wich says for any endless series of numbers, tere exists a limit number greater than every number in that series (Steinhart, 2009). I addition, Cntor defined infinite set as a set, wich have one-to-one correspondence with a proper subset of itself. Tis theory provided a criterion to distinguish finite and infinite sets. Fr example, te set numbers are infinite as it has one-to-one correspondence with the set of even natural numbers (1 is matched with 2, 2with 4 and so on).

Aso, h presents the concept of cardinality of sets that is the size and magnitude of infinite sets also called as transfinite cardinals. Bilding on the Cantor’s work, aGerman mathematician David Hilbert explained different kinds of infinities through the paradox of a grand hotel which has infinitely many rooms, lbeled as 1, 2 3…and so on (Mamolo & Zazkis, 2008). H explains first type as a situation when a guest enters the hotel to check in and the receptionist makes an announcement to shift.

al guests to the next room near to them. Fr example, ocupant of room #1shift. sto room #2, ocupant in room #2 shifts to room #3, ad so on. I this way, rom #1 becomes empty and could be given to the new guest. Te hotel does not have the last room, terefore, ifinity plus one is still infinity. Hlbert explains another situation when a bus with infinite enters grand hotel.

T accommodate infinite passengers of that bus, te receptionist announces to move into the room with twice the number of their current room. Fr instance, ocupant of room #1 shifts to room #2, ocupant of room #2 shifts to room #4, ad so on. I this way, al the rooms labeled with odd numbers are empty now to accommodate new guests; s, ifinity plus infinity is still infinity. Cmparing the infinite cubic toy blocks with that of infinite rooms in the Hilbert’s hotel, i can argued that the infinite cubes in the Deluxe set are stacked one on top of the other, i will reach Alpha Centauri which is the third brightest star seen from the planet earth.

A the grand hotel has the longest infinite corridor, te toy cubes will also reach to the infinite height, wich is only possible in our imagination. Mreover, a the infinite passengers can be easily accommodated in the. ..

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