Types of numbers:
Natural numbers: The positive integers. The number of natural numbers is sometimes designated as n.
Cardinal Numbers: Cardinal numbers consider numbers as sets or collections of things that are equinumerous or can be put in a one to one correspondence. The basic experience involved here is comparing a set to some paradigmatic set to see if they are equivalent.
Ordinal Numbers: Ordinal numbers involve the concept of a well ordered set, or a set based upon some sequence or ordering relation, such as counting, the successor relation. The basic experience involved here is going through a collection by counting to determine its number.
For finite numbers, ordinals and cardinals are equivalent.
Rational numbers: Numbers that can be expressed as a ratio of integers. These can be represented by a repeating infinite decimal expansion. E. g. 1/3= 0.33333. . . . These comprise a denumerable infinity.
Irrational numbers: Numbers that cannot be represented as a ratio of integers. These can be represented by a non-repeating infinite decimal expansion.
Real numbers: All the rationals and the irrationals. Can be conceived of as all the points on the number line.
These comprise a non-denumerable infinity. The number of real numbers is sometimes designated as c, for the number of points on the continuum.
Infinite numbers:
Infinite ordinals:
w - Infinity. The first infinite ordinal. The ordinal number corresponding to n, or the natural numbers. The limit of the series {1, 2, 3, 4, . . .}. This can also be seen as 1+1+1+1 . . . . From this you can see that
1 + w= w But w + 1= w+1
w2 - Two infinities. The limit of { w+1, w+2, w+3, w+4, . . . }
w2 - Infinity Infinities. The limit of { w1, w2, w3, w4, . . . } This can also be seen as w+w+w+w . . . .
So w+w2=w2
w3 - An Infinity of Infinities of Infinities. A really, really big number. (Normal Words fail us here.) The limit of { w21, w22, w23, w24, . . . }.
ww - Infinity to the Infinitieth power. The limit of { w, w2, w3, w4 . . .}. This can also be seen as wwww . . . .
So www=ww
But all of these, and any more arrived at in this way, are really the same cardinal number as w.
Infinite Cardinal numbers:
The cardinal number corresponding to w is 0. (Aleph null) . All denumerable infinities are have this cardinal number. it is the cardinal number of n, The natural numbers.
The cardinal number associated with the Real numbers ( by the Continuuum Hypothesis) or the first non-denumerable infinity is 1 (Aleph one).
© 2006 David Banach
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