Download Set Theory (MATH 6730) HOMEWORK 1 (Due on February 6, 2017

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.