157.193.53.59: /* Examples */

2014-12-19T18:48:59Z

Examples

157.193.11.175 at 20:31, 26 January 2014

2014-01-26T20:31:43Z

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<noinclude>{{../../prev next subpage|5|7}}</noinclude>
{{Selected picture
| image = Omega-exp-omega-labeled.svg
| caption = spiral figure representing both finite and transfinite ordinal numbers
| text = This spiral diagram represents all '''[[ordinal number]]s''' less than <math>\omega^\omega</math>. The first (outermost) turn of the spiral represents the finite ordinal numbers, which are the regular [[Natural number|counting numbers]] starting with [[zero]]. As the spiral completes its first turn (at the top of the diagram), the ordinal numbers approach [[infinity]], or more precisely <math>\omega</math>, the first [[transfinite number|transfinite]] ordinal number (identified with the set of all counting numbers, a "[[Countable set|countably infinite]]" set, the [[cardinality]] of which corresponds to the first transfinite [[cardinal number]], called <math>\aleph_0</math>). The ordinal numbers continue from this point in the second turn of the spiral with <math>\omega+1</math>, <math>\omega+2</math>, and so forth. (A special [[ordinal arithmetic]] is defined to give meaning to these expressions, since the + symbol here does not represent the addition of two [[real number]]s.) Halfway through the second turn of the spiral (at the bottom) the numbers approach <math>\omega+\omega</math>, or <math>\omega\cdot2</math>. The ordinal numbers continue with <math>\omega\cdot2+1</math> through <math>\omega\cdot2+\omega=\omega\cdot3</math> (three-quarters of the way through the second turn, or at the "9 o'clock" position), then through <math>\omega\cdot4</math>, and so forth, up to <math>\omega\cdot\omega=\omega^2</math> at the top. (As with addition, the multiplication and exponentiation operations have definitions that work with transfinite numbers.) As one would expect, the ordinals continue in the third turn of the spiral with <math>\omega^2+1</math> through <math>\omega^2+\omega</math>, then through <math>\omega^2+\omega^2=\omega^2\cdot2</math>, up to <math>\omega^2\cdot\omega=\omega^3</math> at the top of the third turn. Continuing in this way, the ordinals increase by one power of <math>\omega</math> for each turn of the spiral, approaching <math>\omega^\omega</math> in the middle of the diagram, as the spiral makes a countably infinite number of turns. This process can actually continue through <math>\omega^{\omega^\omega}</math> and <math>\omega^{\omega^{\omega^\omega}}</math>, and so on, approaching the [[first uncountable ordinal]] number, which (assuming the [[axiom of choice]]) corresponds to only the ''second'' transfinite cardinal number, <math>\aleph_1</math>, the cardinality (according to the [[continuum hypothesis]]) of the set of [[real numbers]].
| credit = [[User talk:Pop-up casket|Pop-up casket]] & [[User:Fool|Fool]]
| link = Ordinal numbers
| page = picture
| framecolor = transparent
}}

Near polygon - Revision history

157.193.53.59: /* Examples */

157.193.11.175 at 20:31, 26 January 2014