Tag Archives: Daviette Stansbury

Ordinals

This is about the ordinal numbers, which (except for the finite ones) are less well known than the real numbers, although theoretically simpler.

The numbers of either kind compose a linear order: they can be arranged in a line, from less to greater. The orders have similarities and differences:

  • Of real numbers,

    • there is no greatest,

    • there is no least,

    • there is a countable dense set (namely the rational numbers),

    • every nonempty set with an upper bound has a least upper bound.

  • Of ordinal numbers,

    • there is no greatest,

    • every nonempty set has a least element,

    • those less than a given one compose a set,

    • every set has a least upper bound.

One can conclude in particular that the ordinals as a whole do not compose a set; they are a proper class. This is the Burali-Forti Paradox.

Diagram of reals as a solid line without endpoints; the ordinals as a sequence of dots, periodically coming to a limit Continue reading