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Historical Note on Cardinality $3$
Jules Henri Poincaré and others questioned the validity and usefulness of this definition of Cardinality $3$, in that one needs to have counted the elements of $S$ before starting.
Hence the intuitive notion of $3$-ness is argued to be prior to this definition.
- 2010: Raymond M. Smullyan and Melvin Fitting: Set Theory and the Continuum Problem (revised ed.) ... (previous) ... (next): Chapter $3$: The Natural Numbers: $\S 1$ Preliminaries