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So, despite Wittgenstein's qualms, it must not only be conceded to Frege that equinumerosity - the notion expressed by the phrase "just as many" - is one-one correlatability (2) It must also be granted that numerical predicates can be explained by reference to the notion of equinumerosity (cf.
For in conjunction with the general rule that equinumerosity is symmetric and transitive, the equinumerosity of the concepts F and numeral up to "n" indeed implies that any concept must be equinumerous either to both or to neither of them.
The interaction of these two devices allows a first-order representation of arithmetic in which numbers are selected as representatives of equinumerosity classes of concepts (in the style of Frege and Russell).