Phil 454Advanced Symbolic Logic
Lecture Notes for weeks 1 and 2: Naive Set Theory
Dave Gilbert
Spring Semester, 2016
These notes are based on the Open Logic Text. In places the notation, and definitions, are not
identical.
1
Sets: The basics
Definition 1.

Phil 454Advanced Symbolic Logic
Midterm
Spring Semester, 2016
1. Lets add to our first-order language a new binary logical connective: . Obviously, we will
have to modify our defintions accordingly (e.g. add the clause: if and are formulas, then so is
( )

Phil 454Advanced Symbolic Logic
Incompleteness, Part 5
Spring Semester, 2016
1
Representation Theorem
Recall the axiomatizations of PA and RA.
The following are the axioms of Peano Arithmetic:
P1
P2
P3
P4
P5
P6
And
x(sx 6= 0)
xy(sx = sy x = y)
x(x + 0 = x

Phil 454Advanced Symbolic Logic
Recursively Enumerable Sets
Spring Semester, 2016
1
Recursive and Recursively Enumerable Sets
Theorem 1.1 (Kleenes Normal Form Theorem). There is a primitive recursive relation T (e, ~x, s)
and a primitive recursive functio

Phil 454|Advanced Symbolic Logic
Cantor-Schroder-Bernstein Theorem
Dave Gilbert
Spring Semester, 2016
Definition 0.1 (Equinumerous). A set X is equinumerous with a set Y if and only if there is a
total bijection f : X Y . In this case, we write |X| = |Y |

Phil 454Advanced Symbolic Logic
Mathematical Induction
Dave Gilbert
Spring Semester, 2016
1
Mathematical Induction
Mathematical induction is a deductive proof method used throughout mathematics and logic. (Do
not confuse mathematical induction with induct

Phil 454Advanced Symbolic Logic
Primitive Recursion Pt. 2
Spring Semester, 2016
1
Primitive Recursion, Continued.
Lemma 1.1. Primitive recursive relations are closed under , , and complement.
Proof. Let A and B be primitive recursive relations. That is, t

Phil 454Advanced Symbolic Logic
Primitive Recursion Pt. 3: Recursion
Spring Semester, 2016
1
Partial Recursive Functions
Recall from last time that the set of primitive recursive functions do not completely cover the set
of (intuitively) computable functi

Phil 454Advanced Symbolic Logic
Primitive Recursion
Spring Semester, 2016
1
1.1
Primitive Recursion
Functions
Consider the following definition of a function f N2 N:
f (x, 0)
f (x, y + 1)
=
=
1
f (x, y) + 1
This pretty clearly defines addition. Now consid

Phil 454Advanced Symbolic Logic
Incompleteness, Part 3
Spring Semester, 2016
1
Tarskis Truth Theorem
Tarskis undefinability theorem says that the relation arithmetical truth is not arithmetically definable.
We can begin to prove this now.
Lemma 1.1. The s

Phil 454Advanced Symbolic Logic
Homework 6
Due: Tuesday, May 3
The following are the axioms of Peano Arithmetic:
P1
P2
P3
P4
P5
P6
And
x(sx 6= 0)
xy(sx = sy x = y)
x(x + 0 = x)
xy(x + sy = s(x + y)
x(x 0 = 0)
xy(x sy = (x y) + x)
the following induction s

Phil 454Advanced Symbolic Logic
Homework 4
Solutions
1. Let L be a language consisting of a countably infinite set cfw_c1, c2, . . . of constant symbols and
the binary predicate symbol P , and also =. Let be the set of sentences
= cfw_ci 6= cj : i, j N

Phil 454Advanced Symbolic Logic
Homework 4
Due: Tuesday, April 19
1. Show that the total unary zero function Z1 : N cfw_0 is primitive recursive. (Note that the zero
function we took as basic in our definition of the primitive recursive functions is nulla

Phil 454Advanced Symbolic Logic
Homework 2 Solutions
Spring Semester, 2016
1. Show that N is non-enumerable.
Proof. First, notice that if Y is enumerable, and X Y , then X is enumerable: take any listing
of the elements of Y and just erase the elements no

Phil 454Advanced Symbolic Logic
Homework 2
Spring Semester, 2016
1. Show that N is non-enumerable.
2. Show that if X and Y are both enumerable, then so is X Y .
3. Show that the enumerable union of enumerable sets is enumerable. In other words, if X1 , X2

Phil 454Advanced Symbolic Logic
Homework 3
Spring Semester, 2016
1. Give LK derivations of the following sequents:
(a). ( ) ( )
(b). x(x) ) (y(y) )
(c). xy(x = y (x) (y)
(d). x(x) yz(y) (z) y = z) x(x) y(y) y = x)
2. Complete the proof of soundness of LK.