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Frege’s Theorem and Foundations for Arithmetic > Proof of Equinumerosity Lemma (Stanford Encyclopedia of Philosophy)

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In this proof of the Equinumerosity Lemma, we utilize the following abbreviation, where ϕ 𝜙 is any formula in which the variable y 𝑦 may or may not be free and ϕ ν υ 𝜙 𝜐 𝜈 is the result of replacing the free occurrences of υ 𝜐 in ϕ 𝜙 by ν 𝜈 : x = ι y ϕ = a b b r ϕ & ∀ z ( ϕ z y → z = x ) 𝑥 = 𝜄 𝑦 𝜙 = 𝑎 𝑏 𝑏 𝑟 𝜙 & ∀ 𝑧 ( 𝜙 𝑦 𝑧 → 𝑧 = 𝑥 ) We may read this as follows: x 𝑥 is identical to the object y 𝑦 which is such that ϕ 𝜙 if and only if both x 𝑥 is such that ϕ 𝜙 and everything which is such that ϕ 𝜙 is identical to x 𝑥 . This abbreviation is employed below to simplify the definition of new relations. Given this new notation, we will use only the following simple consequence of this definition: Principle of Descriptions: x = ι y ϕ → ϕ x y 𝑥 = 𝜄 𝑦 𝜙 → 𝜙 𝑦 𝑥 In other words, if x 𝑥 is the object y 𝑦 such that ϕ ( y ) 𝜙 ( 𝑦 ) , then x 𝑥 is such that ϕ 𝜙 . The use of this principle will be obvious in what follows. Proof o

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