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    Axioms implicitly define mathematical concepts without a ... — Carmelics
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    Challenges→The implicit definition of sets is problematic because it permits defining objects that appear to be proper sets but are internally inconsistent.

    Axioms implicitly define mathematical concepts without a positive predicate that fully characterizes set membership.

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    Objects defined under this implicit framework may initially appear to satisfy th...The implicit definition of sets is problematic because it permits defining objec...Upon scrutiny, some such objects reveal internal inconsistencies.

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    The idea that mathematical concepts are defined implicitly by a set of axioms was proposed by Hilbert but is not uncontroversial (see entry on the Frege-Hilbert controversy). The fact that the definition is implicit entails that we only have examples of what sets are without the possibility to formulate any positive predicate that defines them. Elements of a set are not necessarily physical, nor abstract, nor spatial or temporal, nor simple, nor real. The only prerequisite is the possibility t

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