4. Second Order Arithmetic and Reverse Mathematics
4.1. The Language of Second Order Arithmetic.
Weve mentioned that Peano arithmetic is sucient to carry out large
portions of ordinary mathematics, but with a qualier, namely that Peano
arithmetic suces to
1. Propositional Logic
1.1. Basic Denitions.
Denition 1.1. The alphabet of propositional logic consists of
Innitely many propositional variables p0 , p1 , . . .,
The logical connectives , , , , and
Parentheses ( and ).
We usually write p, q, r, . . . f
3. Peano Arithmetic
3.1. Language and Axioms.
Denition 3.1. The language of arithmetic consists of:
A 0-ary function symbol (i.e. a constant) 0,
A unary function symbol S,
Two binary function symbols +, ,
Two binary relation symbols =, <,
i
For each n, in
On the left side we consider the situation where a sequent is underivable
and so there is a truth assignment making it falseand work towards sequent
consisting only of propositional variables, which we could easily generate
such a truth assignment from. O
1. Problems
(1) Find (by any means you like) an interpolant in Fc for the implication
[xy (P y (Qy Rx)] xRx y (P y (Qy S ) S.
You do not need to give full deductions for the implications (but you
should still check carefully that the necessary implication
1. Problems
(1) Check that being an alphabetic variant is symmetric (if is an alphabetic variant of then is an alphabetic varient of ). (Actually,
being an alphabetic variant is an equivalence relationreexivity is
obvious, and transitivity is not dicult t
Theorem 0.1. Pi
N
We show this by induction on the construction of the formula .
If is a propositional variable p then N is p, as is , so we have:
p p
p p
If is then we have
We turn to the inductive cases. If is then N is N N . We
show the two im
1. Problems
Show that Pi .
Give an entire deduction showing Pm (p p)
Show that Pi N
Prove the Inversion lemma for for Pc .
Give an entire deduction showing Pi p p (p q ) p) p
Consider a proof of (p q ) p) p obtained by applying cut to
the deduction from
1. Problems
(1) Dene [/p] recursively by:
p[/p] is ,
q [/p] is q if q is atomic and not p,
(0 1 )[/p] is 0 [/p] 1 [/p].
Give an example of , , where ([/p])[/q ] and [/q ][( [/q ])/p]
are dierent.
(2) Give deductions of the following in the sequent calc
2. First Order Logic
2.1. Expressions.
Denition 2.1. A language L consists of a set LF of function symbols, a set
LR of relation symbols disjoint from LF , and a function arity : LF LR N.
We will sometimes distinguish a special binary relation symbol =.
0
We begin with a derivation of Pierces law which contains a single cut over
of formula of rank 2.
pp
q
p , p q
p p q
pp
p p,
p , (p q ) p p
p , (p q ) p p
p , p
p p , (p q ) p p
p , p p
p p
p p (p q ) p ) p
(p q ) p ) p
There is only one cut of rank 2