Samuel Coskey / Courses / Mathematical logic 461, Fall 2008

Course information
Course title: Mathematical logic
Meeting times: MW4 (1:403:00pm)
Meeting place: SEC216 (Busch)
Office hours: M 3:304:30 & Th by appointment
Textbook: Enderton is optional 
Exam dates
First midterm: Wednesday, February 27 (exam)
Second midterm: Wednesday, April 9 (exam)
Final: Tuesday, May 13 at 12pm (exam) 
Grading
Exams: Each exam counts for about 1/4 of your grade.
Homework: Homework will count for about 1/4 of your grade. I will assign and collect homework on a weektoweek basis. I'll always discuss them in class or put them online at least a week before the due date. There should be about eleven homeworks, of which I'll drop the lowest. 
Lecture notes
(written by Simon Thomas.) 
Homework 01, due January 30
Please excuse the poor formatting. Below, N represents the natural numbers, Z the integers, and Q the rational numbers. Prove the following:
 A∩(B∩C)=(A∩B)∩C
 A∪(B∩C)=(A∪B)∩(A∪C)
 A\(B∩C)=(A\B)∪(A\C)
 Recall that Z can be constructed from N as the set of pairs (m,n) where m, n are in N. We think of (m,n) as representing the diference mn. Hence, we must specify that pairs (m,n) and (m',n') are equivalent iff m+n'=n+m'. Give an analogous construction of Q from Z. Specify the equivalence rule and give the formula for addition.
 Prove the following:

Homework 02, due February 06
Here ≤ denotes "smaller than or equal", and < denotes "strictly smaller than", since I don't know how to type the curly ones. If A is infinite (not in bijection with a finite set), then N ≤ A.
 Write an explicit bijection between NxN and N.
 For any A, A<P(A).
 Think of a category of mathematical objects such that the relevant maps don't satisfy an analog of the CantorSchroederBernstein theorem. Prove that you are correct with an example. (If you use a textbook or online resource, you still have to explain your reasoning. Some possibilities are topological spaces, groups, and linear orders.)
 Prove that there is an injection from A to B iff there is a surjection from B to A. Note explicitly where your proof uses the Axiom of Choice. (Normally in this class, you may use the Axiom of Choice without comment.) The Axiom of Choice states that for any family of nonempty sets {A_{i}}, there exists a set {a_{i}} such that for each i, a_{i}∈A_{i}.

Prove the following are equinumerous:
 R\N∼R
 R\Q∼R
 P(N)∼Sym(N), where Sym(N) denotes the set {f  f:N→N is bijective}

Homework 03, due February 13
 Describe all the equivalence relations that can be defined on {0,1,2,3}. How many are there? How many can be defined on a set of size 5 or 6 or 7? What is the growth rate of this sequence? You can use the online database of integer sequences (server is down as of this writing; wikipedia should work too).
 Prove that Z and Q are not isomorphic, as linear orders.
 Prove that any partial order on a finite set can be extended to a linear order on that set.
 Verify that ((A→(B∧C))∨(¬D)) is a wff by drawing a tree that shows its recursive construction.
 Prove that the connectives ∧, ∨, and ↔ can be defined using only ¬ and →. Refer to the truth tables in the notes as necessary.

Homework 04, due February 20
 Suppose that v,v' are truth assignments which agree on every propositional variable appearing in the wff φ. Prove that v(φ)=v'(φ). (Here the bold v means v with a bar over it.) (Argue by induction on the length of the wff φ.)
 Prove that (A→B) and ((¬B)→(¬A)) are tautologically equivalent.
 Prove that φ and ψ are tautologically equivalent iff (φ↔ψ) is a tautology.
 Let η denote a fixed negation of a tautology. Let Σ be a set of wffs. Prove that Σ tautologically implies η iff Σ is unsatisfiable.
 Let Σ={((¬A)∨B), (B→C), A}. Write down a derivation of C from Σ. (Remember that ∨ can be written using only → and ¬.)
 Prove the soundness theorem for propositional calculus: If there is a deduction of the wff φ from the set of wffs Σ, then Σ tautologically implies φ.

Review for midterm 1
Some of these will be due later, for now just solve them and use your solutions to study. Suppose that f:A→B is injective. Prove that it is leftinvertible, that is, there exists a function g:B→A such that g(f(a))=a for all a∈A.

On cardinality:
 State the CantorBernstein Theorem.
 Prove that Z[^{1}⁄_{2}]={^{m}⁄_{2^n} : m∈Z and n∈N} is equinumerous with N.
 Let Surj(N) be the be the set of surjective functions f:N→N. Prove that Surj(N) is equinumerous with P(N).

Let L be a set and R,R' binary relations on L.
 What does it mean for (L,R) to be a linear ordering?
 What does it mean for two linear orderings, (L,R) and (L',R'), to be isomorphic?

Decide whether the following pairs are isomorphic. In each
case the ordering is the "usual" ordering, <.
 Q\(0,1) and Q\[2,3]
 Q\Z and Q
 Z[^{1}⁄_{2}] and R\N

Determine whether each of the following wffs is a tautology:
 (A→(B→(A↔B)))
 ((P∧Q)→(P→Q))
 Suppose that α is a wff which only involves the connectives ∧ and ∨, and the sentence symbols A_{1},...,A_{n}. Prove that if v is a truth assignment such that v(A_{i})=T for all i then v satisfies α. (Use induction on the length of α.)

On compactness:
 State the compactness theorem for propositional logic.
 Let {S_{n}} be a collection of finite subsets of N with the following property: for each finite F_{0}⊂N there exists A_{0}⊂N such that A_{0}∩S_{n}=1 for all n∈F_{0}. Prove that there exists A⊂N such that A∩S_{n}=1 for all n.
 Give a counterexample if the S_{n} are not required to be finite.

Homework 05, due March 05
 Do problems 5,6 from the review for midterm 1.

For +4pts (out of 50) on midterm 1, redo ONE of the following
two problems. If you more or less got one of them right
already, then do the other one!
 (2c) Let C_{1/2} denote the set of power series in x with coefficients in Z which converge when x=1/2. Is C_{1/2} countable or uncountable?
 (4d) Define α≤β iff (α→β) is a tautology. What are the maximal wffs with respect to this preorder? (Here, since I don't want to rewrite the whole exam problem, let's just define that α is maximal iff α≤β implies β≤α.)

Homework 06, due March 12
 Prove that a tree is finitely splitting iff every level is finite.
 Describe a tree with elements of every possible height but no infinite branch.
 Let T be a tree. We initially defined that a subset B is a branch iff it is a maximal linearly ordered subset of T. Prove that B is an infinite branch iff B meets every level in exactly one point and B is downward closed (ie, b'<b∈B implies b'∈B).

Let A,B be a sets. For all n let A_{n}⊂A and
f_{n}:A_{n}→B. Suppose that
 (exhaustive) A=∪A_{n}, and
 (coherence) letting A_{mn}=A_{m}∩A_{n}, we have f_{n}A_{mn}=f_{m}A_{mn}.

Let S denote the space of truth assignments, ie, functions from
L={A_{n}} to {T,F}. Let [L] denote the
Lindenbaum algebra of wffs modulo tautological equivalence. For
each [φ]∈[L], let
U_{φ}={v∈S : v(φ)=T}
Verify that the map that sends [φ] to U_{φ} is a welldefined, injective homomorphism of Boolean algebras from [L] into P(S). (Wikipedia can be helpful if you missed the inclass definitions.)

Homework 07, due March 26
Since I can't write the satisfaction symbol, I'm using = for it.
Let L be a language and A be an Lstructure with domain
A. We say that B⊂A is a definable subset of A iff
there exists a wff φ with one free variable x such that
A=φ[s] iff s(x)∈B.
Prove the following two statements: The sets {0} and {1} are definable in (N;+).
 The set {2} is definable in (R;+,×).

Recall the construction (in class and the notes) of a
"nonstandard" model A of
TA={sentences σ : (N;+,×,0,1,<)=σ}
Notice that there is a copy of the ordinary natural numbers N contained in A: just identify n∈N with 1+...+1 (n times) in A. This copy of N is called the standard part of A.
Prove that the standard part of A is an initial segment of A. More precisely, prove that if n∈A is standard and a∈A is nonstandard then n<a. (Hint: first explain why A is a discrete linear order with a least element.)  Invent a language L and a set of sentences Σ such that for every natural numberr n, n is even iff there exists an Lstructure that satisfies Σ and has size n.

Let L be a language and A be an Lstructure with domain
A. We say that B⊂A is a definable subset of A iff
there exists a wff φ with one free variable x such that

Homework 08, due April 02

Let A be a nonstandard model of arithmetic, as
constructed last week.
 Prove that every definable subset of A has a minimal element. (Hint: this is obviously true in N.)
 Let N denote the standard part of A. Prove that the complement A\N has no minimal element and hence is not a definable subset of A.
 Conclude that the standard part of A is also not a definable subset of A.

Let T be a tree.
 Show that every level of the tree is a definable subset of T.
 Suppose that T is finitely splitting. Describe a sentence φ_{n,m} which says "level n has size m."
 Suppose that A is an Lstructure and that the subset P⊂A is definable. Let A' be the structure obtained from A by letting L'=L∪{Q} and letting Q^{ A'}=P. Show that any definable subset of A' is actually also definable in A. (Hint: it suffices to show that every wff in L' is equivalent to the one in L obtained by substituting occurrences of Q with the defining formula for P. Use induction on the wffs.)

Let A be a nonstandard model of arithmetic, as
constructed last week.

Review for midterm 2
 Show that + is not definable in (N;×).

Let A be a structure and let Σ=Th(A).
 Prove that if A is finite, then Σ does not have any infinite models.
 If A is infinite, can Σ have any finite models?
 Prove that wffs φ and ψ are logically equivalent iff the wff (φ↔ψ) is valid.

Let L={<,a,b,f}, where < is a binary relation symbol, a,b
are constant symbols, and f is a unary function symbol. Let
Σ denote the theory which includes the sentences:
∀x¬(x<x),
∀x∀y((x<y)∨(y<x)∨(x=y)),
∀x∀y∀z(((x<y)∧(y<z))→(x<z)),
∀x∀y((x<y)→∃z((x<z)∧(z<y))),
∀x∃y∃z((y<x)∧(x<z)),
a<b,
∀x∀y((x<y)→(fx<fy)) Generally speaking, describe the models of Σ in english.
 Let σ denote the sentence
fa<b
Prove, using counterexamples, that Σ does not logically imply σ and that Σ does not logically imply ¬σ.
 Show that the class C of 3colorable graphs with a unique vertex of degree 2 is axiomatizable in the language L={P,Q,R} with three unary predicates.
 Suppose that A is a structure and that P and Q are definable subsets of A. Let φ(x) be the defining wff for P and let ψ(x) be the defining wff for Q. If φ and ψ are logically equivalent wffs, show that P=Q.
 Let A be a nonstandard model of arithmetic. Show that every other element is even. (First, say what this means.)
 Also, study trees.

Homework 09, due April 16
 Do problems 2,5,6 of the Review for midterm 2.

Homework 10, due April 23
 Suppose that A is a nonstandard model of TA in which there is a pair p,p+2 of nonstandard prime elements. Show that the twin primes conjecture holds.

Prove that firstorder axiom groups 3 and 4 are valid, that is,
the following wffs are valid.
 (∀x(α→β))→(∀xα→∀xβ)
 α→∀xα, where x does not ocurr free in α.

Deduce the following wff only from the axioms.
(∀x(α→β))→((∀xα)→β)
At each step, state which axiom group you are using or cite modus ponens. (You should only need the first few axioms, but please reread them all again carefully from the notes.)

Homework 11, due April 30
 If z does not appear in φ(x), show that ∀xφ(x) and ∀zφ(z) are deducible from one another. Show that this need not be the case if z does appear in φ(x).
 Give an example of a structure A and a wff φ(x) (with x being its only free variable) such that ∃xφ(x) holds in A but there is no term t such that φ(t) holds in A.
 Suppose that Σ  φ and that P is a predicate symbol which appears in neither Σ nor φ. Show that there exists a deduction of φ from Σ which doesn't refer to P. (Hint: use the completeness theorem.)

Show that there exists a deduction (from Λ) of the wff:
∀x(α→β)→(∃xα→∃xβ)
This time, you may use all of the metatheorems about deduction (like generalization, deduction theorem, T, reductio ad absurdum, etc, but of course not completeness).

Review for the final exam

Sets:
 Prove that a countable union of countable sets is countable. (Hint: use the fact that N×N is countable.)
 Use this to prove that the set Z[x] of polynomials in x with integer coefficients is countable.

Relations:
 Give the definition of a linear ordering.
 Give the definition of an isomorphism of two linear orderings.

Decide whethere the following pairs are isomorphic linear
orderings.
 (Q\Z;<), (Q\{0};<)
 (Q;<), (Q\(0,1);<)

Propositional logic:
 Give the definition of a tautology of propositional logic.

Decide which of the following wffs are tautologies.
 (Q∨(¬(P→Q)))
 ((P→(Q→R))↔((P∧Q)→R))
 Suppose that α is a propositional wff involving only the connectives ∧, ∨, and ¬. Let α' be the wff obtained from α by replacing each occurence of ∧ by ∨, each occurrence of ∨ by ∧, and each occurrence of A by ¬A for A a propositional variable. Prove by induction on the length of α that α and ¬α' are tautologically equivalent.

Compactness:
 Use the compactness theorem for firstorder logic to derive the completeness theorem for firstorder logic.
 Suppose that Σ is a set of wffs which has an infinite model. Prove that Σ has an uncountable model. (Hint: although we have only established it for countable languages, the compactness theorem is valid for an uncountable language as well.)

Let L be the language {<,f}, where < is a binary relation
symbol and f is a unary function symbol. Let Σ denote the
theory consisting of the following sentences:
∀x¬(x<x),
Let σ denote the sentence:
∀x∀y((x<y)∨(y<x)∨(x=y)),
∀x∀y∀z(((x<y)∧(y<z))→(x<z)),
∀x∀y((x<y)→∃z((x<z)∧(z<y))),
∀x∃y∃z((y<x)∧(x<z)),
∀x∀y((x<y)→(fx<fy))∀y∃x(fx=y)
Prove that there is no deduction from Σ of σ and that there is no deduction from Σ of ¬σ. 
Let L be the language {<,P}, where < is a binary relation
and P is a unary predicate. Consider the following theory:
Σ=DLO∪{∀x∀y(x<y→∃z∃w((x<z<y)∧(x<w<y)∧P(z)∧¬P(w))}
 Describe the models of Σ.
 Using Vaught's test, prove that Σ is a complete theory.
 A graph is said to be locally finite if every vertex is adjacent to only finitely many other vertices. Prove that the class of locally finite graphs is not axiomatizable.

Sets: