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:
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Of real numbers,
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there is no greatest,
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there is no least,
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there is a countable dense set (namely the rational numbers),
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every nonempty set with an upper bound has a least upper bound.
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Of ordinal numbers,
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there is no greatest,
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every nonempty set has a least element,
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those less than a given one compose a set,
- every set has a least upper bound.
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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.
