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Set Theory (MATH 6730) HOMEWORK 1 (Due on February 6, 2017) In all problems below, ` refers to the proof system described on pp. mid 7 – mid 8 of the handout “Background in Logic”; see also p. 10 for some metatheorems for that proof system. In your solution to Problem k you are allowed to use the statements proved in Problems 1, . . . , k − 1. Problems: 1. Prove the following consequence of Existential Initialization: Let Γ ∪ {ϕ} be a set of LC -formulas, and let d be a constant symbol not in C. If Γ ∪ {Subf xd (ϕ)} is an inconsistent set of formulas in the language LC∪{d} , then Γ ∪ {∃x ϕ} is an inconsistent set of formulas in the language LC . 2. Let Γ be a set of LC -formulas. (i) Show that the following conditions on any LC -formulas ϕ and ψ are equivalent: (a) Γ ∪ {ϕ} ` ψ and Γ ∪ {ψ} ` ϕ; (b) Γ ` ϕ ↔ ψ. Definition. If these equivalent condition hold, we will say that ϕ and ψ are provably Γ-equivalent. Two formulas are provably equivalent if they are provably ∅-equivalent. (ii) Verify that “provable Γ-equivalence” is an equivalence relation on the set of all LC -formulas. (iii) Show that for any LC -formulas ϕ, ψ and for any variable x: • ϕ → ∀x ψ and ∀x (ϕ → ψ) are provably equivalent if x is not free in ϕ; • ∃xϕ → ψ and ∀x (ϕ → ψ) are provably equivalent if x is not free in ψ. 3. Let ϕ be an LC -formula, and let x, y be variables. (i) Prove that • ∀x ∀y ϕ and ∀y ∀x ϕ are provably equivalent; • ∃x ∃y ϕ and ∃y ∃x ϕ are provably equivalent. (ii) Show that ` Subf xt (ϕ) → ∃x ϕ if either t is a constant symbol or t is a variable such that no quantifier ∀t in ϕ has a free occurrence of x in its scope. 4. Let χ be an LC -formula. (i) Give a rigorous definition of what it means that χ occurs as a subformula of an LC -formula. (Use recursion on ϕ.) (ii) Use this definition to prove the following statement: Let Γ be a set of LC -sentences, and let χ0 be a formula provably Γ-equivalent to χ. If ϕ is a formula in which χ occurs as a subformula, and one such occurrence is replaced by χ0 to obtain ϕ0 , then ϕ0 is a formula which is provably Γ-equivalent to ϕ. 5. Prove that {Cmpr, Pair} ` Pair] by formalizing our informal proof for this statement. 6. Prove that {Cmpr, Pset} ` Pset] . 1 2 7. Prove that the formula Uni ∀A ∃B ∀x ∃A (x ∈ A ∧ A ∈ A) → x ∈ B for the Axiom of Union given on p. 3 of the handout “The Axioms of Set Theory ...” and the formula ∀A ∃B ∀A ∀x (x ∈ A ∧ A ∈ A) → x ∈ B given for the Axiom of Union on p. 68 of [1] (up to the letters used for the bound variables) are provably equivalent. 8. Prove that {Pair] , Fnd} ` ∀x ¬ x ∈ x by formalizing our informal proof for this statement. 9. As in Russell’s Paradox, consider the class S of all sets A such that A ∈ / A. The L-sentence σ ≡ ¬∃s ∀x (x ∈ s ↔ ¬ x ∈ x) expresses that S is not a set. (i) Prove ` σ by formalizing our informal proof that the assumption ‘S is a set’ leads to a contradiction. (This proof will make use of the metatheorems Proof by Contradiction and Existential Initialization.) (ii) Give a proof for ` σ which uses no other metatheorems than the Generalization Theorem. (In particular, Proof by Contradiction is not used, and additional constants are not used.) Teams: A Andre Davis, Ian Gossett Problems 4, 5 B Jordan DuBeau, Michael Wheeler Problems 3, 9 C Gagan Sapkota, Lenhardt Stevens Problems 2, 7 D Athena Sparks, Trevor Jack Problems 1, 6, 8 Please • contribute to the solutions of all problems assigned to your team; • read – and if necessary, revise – the written version of your joint solution to each problem, and send the solution to me only when all team members approve it; • include the names of all team members at the top of the first page of each solution; • submit the solution to Problem M of HW assignment N to me as a PDF file called “setthNpM.pdf” (which abbreviates “set theory assignment N , problem M ”); • cc your team members on all email correspondence concerning homework problems.