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Most everyone is familiar with the infinity symbol the one that looks like the number eight tipped over on its side Infinity sometimes crops up in everyday speech as a superlative form of the word many But how many is infinitely many How big is infinity Does infinity really existYou cant count to infinity Yet we are comfortable with the idea that there are infinitely many numbers to count with no matter how big a number you might come up with someone else can come up with a bigger one that number plus one plus two times two and many others There simply is no biggest number You can prove this with a simple proof by contradictionProof Assume there is a largest number n Consider n1 n1nTherefore the statement is false and its contradiction there is no largest integer is true This theorem is valid based on the Validity of Proof by ContradictionIn 1895 a German mathematician by the name of Georg Cantor introduced a way to describe infinity using number sets The number of elements in a set is called its cardinality For example the cardinality of the set 3 8 12 4 is 4 This set is finite because it is possible to count all of the elements in it Normally cardinality has been detected by counting the number of elements in the set but Cantor took this a step farther Because it is impossible to count the number of elements in an infinite set Cantor said that an infinite set has No elements By this definition of No No1No He said that a set like this is countable infinite which means that you can put it into a 1-1 correspondenceA 1-1 correspondence can be seen in sets that have the same cardinality For example 1 3 5 7 9has a 1-1 correspondence with 2 4 6 8 10 Sets such as these are countable finite which means that it is possible
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