Teacher resources and professional development across the curriculum

Teacher professional development and classroom resources across the curriculum

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3 How Big is Infinity?



Cantor called the cardinality of all the sets that can be put into one-to-one correspondence with the counting numbers aleph, or "Aleph Null."

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Three is the common property of the group of sets containing three members. This idea is called "cardinality," which is a synonym for "size." The set {a,b,c} is a representative set of the cardinal number 3.


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This means that for any two magnitudes, one should always be able to find a fundamental unit that fits some whole number of times into each of them (i.e., a unit whose magnitude is a whole number factor of each of the original magnitudes)—an idea known as commensurabilty.

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In some ways, the opposite of a multitude is a magnitude, which is continuous. In other words, there are no well defined partitions.

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A "countable" infinite set is one that can be put into one-to-one correspondence with the set of natural numbers.

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In any ratio of two whole numbers, expressed as a fraction, we can interpret the first (top) number to be the "counter," or numerator—that which indicates how many pieces—and the second (bottom) number to be the "namer," or denominator—that which indicates the size of each piece.

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The multitude concept presented numbers as collections of discrete units, rather like indivisible atoms.

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Irrational numbers cannot be written as a ratio of natural numbers.

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Rational numbers arise from the attempt to measure all quantities with a common unit of measure.

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The cardinality of sets that cannot be put into one-to-one correspondence with the counting numbers, such as the set of real numbers, is referred to as c. The designations A_0 and c are known as "transfinite" cardinalities.

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