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Template:Sequence of the Day for October 1

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Intended for: October 1, 2011

Timetable

  • First draft entered by Alonso del Arte on August 1, 2011
  • Draft reviewed by Alonso del Arte on August 7, 2011 ✓ and Daniel Forgues on September 30, 2018
  • Draft to be approved by September 1, 2011
Yesterday's SOTD * Tomorrow's SOTD

The line below marks the end of the <noinclude> ... </noinclude> section.



A001057: Canonical enumeration of integers: interleaved positive and negative integers with zero prepended. (A permutation of the integers.)

{ 0, 1, −1, 2, −2, 3, −3, 4, − 4, 5, −5, 6, − 6, 7, −7, ... }
Questions of infinity can too easily lead to confusing, paradoxical thoughts. The cardinality of the set of integers
| ℤ |
is the same as the cardinality
0
of the set of nonnegative integers, since we have a bijection* (one-to-one and onto correspondence) between the two sets.

The order type** of the nonnegative integers is
ω
, and since it is a well ordered set,*** it is also its ordinal number.**** The set of integers
is not well ordered, so the order type of the integers
ω + ω
(order type of negative integers + order type of nonnegative integers)  is not an ordinal number. What is the order type of A001057?

_______________

* Weisstein, Eric W., Bijection, from MathWorld—A Wolfram Web Resource.
** Weisstein, Eric W., Order Type, from MathWorld—A Wolfram Web Resource.
*** Weisstein, Eric W., Well Ordered Set, from MathWorld—A Wolfram Web Resource.
**** Weisstein, Eric W., Ordinal Number, from MathWorld—A Wolfram Web Resource.

The order type
ω
is the reverse of the order type
ω
. (
ω
is not an ordinal number since it is not a well ordered order type.)

Harold Simmons, Introduction to order types and ordinals.