Explanation: 1.5: Comparing the Size of Sets and Their Subsets

(related to Subsection: Common Notions (all Books))

(Common Notion 5 from Book 1 of Euclid's “Elements”)

And the whole [is] greater than the part.

Modern Formulation (Proposal)1

The cardinality of a set is greater than the cardinality of its subset.

Proofs: 1 2 3

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Adapted from (subject to copyright, with kind permission)

  1. Fitzpatrick, Richard: Euclid's "Elements of Geometry"


  1. Today we know that this modern formulation is not true! It holds only for finite sets. In the time of Euclid, mathematicians and philosophers had only a vivid concept of "the whole" and "its part" which is totally different from contemporary set-theoretical concepts.