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Dedekind's path to logicism had a turning point when he was able to construct a model satisfying the [[Axiom|axioms]] characterizing the [[real number]]s using certain sets of [[Rational number|rational numbers]]. This and related ideas convinced him that arithmetic, algebra and analysis were reducible to the natural numbers plus a "logic" of classes. Furthermore by 1872 he had concluded that the naturals themselves were reducible to sets and mappings. It is likely that other logicists, most importantly Frege, were also guided by the new theories of the real numbers published in the year 1872.
The philosophical impetus behind Frege's logicist programme from the [[The Foundations of Arithmetic|Grundlagen der Arithmetik]] onwards was in part his dissatisfaction with the [[Epistemology|epistemological]] and [[Ontology|ontological]] commitments of then-extant accounts of the natural numbers, and his conviction that Kant's use of truths about the natural numbers as examples of [[A_priori_and_a_posteriori#Relation_to_the_analytic-synthetic|synthetic a priori truth]] was incorrect.
This started a period of expansion for logicism, with Dedekind and Frege as its main exponents. However, this initial phase of the logicist programme was brought into crisis with the discovery of the classical paradoxes of [[set theory]] (Cantor 1896, Zermelo and Russell 1900–1901). Frege gave up on the project after Russell recognized and communicated [[Russell's paradox|his paradox]] identifying an inconsistency in Frege's system set out in the Grundgesetze der Arithmetik. Note that [[naive set theory]] also suffers from this difficulty.
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