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Glossary

3 How Big is Infinity?

 

Aleph-Null

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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Cardinality

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.

cardinality



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Commensurability

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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Continuous

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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Countable

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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Denominator

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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Discrete

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

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Irrational

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

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Rational

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

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Transfinite

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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