The Compactness of First-order Languages

John W. Dawson
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John W. Dawson: Penn State York

Chapter Chapter 12 in Why Prove it Again?, 2015, pp 171-186 from Springer

Abstract: Abstract A first-order formal language ℒ $$\mathcal{L}$$ (with identity) is characterized by disjoint (possibly empty) sets C ℒ $$\mathbf{C}_{\mathcal{L}}$$ of constant symbols, R ℒ n $$\mathbf{R}_{\,\,\mathcal{L}}^{n}$$ of n-place relation symbols (n ≥ 1) and F ℒ n $$\mathbf{F}_{\,\,\mathcal{L}}^{n}$$ of n-place function symbols (n ≥ 1) — collectively constituting the non-logical symbols of the language — together with the following logical symbols: (i) a denumerable set V ℒ $$V _{\mathcal{L}}$$ of variable symbols (ii) the unary connective ¬ $$\neg $$ (denoting negation) (iii) the binary connective ∨ $$\vee $$ (denoting disjunction) (iv) the binary connective ∧ $$\wedge $$ (denoting conjunction) (v) the existential quantifier ∃ $$\exists $$ (vi) the universal quantifier ∀ $$\forall $$ (vii) the binary relation symbol = (denoting identity).

Keywords: Function Symbol; Finite Subset; Atomic Formula; Compactness Theorem; Relation Symbol (search for similar items in EconPapers)
Date: 2015
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DOI: 10.1007/978-3-319-17368-9_12

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