1. PRIMITIVE RECURSIVE
INCOMPLETENESS
DECIDABILITY / INCOMPLETENESS
SEP
Erik A. Andrejko
University of Wisconsin - Madison
Summer 2007
ERIK A. ANDREJKO DECIDABILITY / INCOMPLETENESS

2. PRIMITIVE RECURSIVE
INCOMPLETENESS
PRIMITIVE RECURSIVE FUNCTIONS
DEFINITION
Primitive recursive functions C contains
The successor functions, λ x[x + 1] ∈ C .
1
The constant functions, λ x1 , · · · , xn [m] ∈ C are primitive
2
recursive for 0 ≤ n, m.
The identity functions, or projections, λ x1 , · · · , xn [xi ] ∈ C for
3
1 ≤ n and 1 ≤ i ≤ n.
(Composition) If g1 , g2 , · · · , gm , h ∈ C then
4
f (x) = h(g1 (x), · · · , gm (x)) ∈ C .
¯ ¯ ¯
(Primitive Recursion) If g, h ∈ C and n ≥ 1 then f ∈ C where
5
f (0, x2 , · · · , xn ) = g(x2 , · · · , xn )
f (x1 + 1, x2 , · · · , xn ) = h(x1 , f (x1 , x2 , · · · , xn ), x2 , · · · , xn )
ERIK A. ANDREJKO DECIDABILITY / INCOMPLETENESS

3. PRIMITIVE RECURSIVE
INCOMPLETENESS
PRIMITIVE RECURSION FUNCTIONS
FACT
There exists a ‘computable’ non primitive recursive function.
ERIK A. ANDREJKO DECIDABILITY / INCOMPLETENESS

4. PRIMITIVE RECURSIVE
INCOMPLETENESS
PARTIAL RECURSIVE FUNCTIONS
DEFINITION
The class of partial recursive contains C and
(Unbounded Search) If θ (x1 , · · · , xn , y ) is a partial recursive
1
function then
ψ(x1 , · · · , xn ) = µy[θ (x1 , · · · , xn , y ) ↓= 0 and
∀z ≤ y [θ (x1 , · · · , xn , z) ↓]]
ψ is partial recursive.
θ (x1 , · · · , xn , y ) ↓= 0 means that θ (x1 , · · · , xn , y ) defined and
equals 0
µyθ (x1 , · · · , xn , y ) ↓= 0 least y such that θ (x1 , · · · , xn , y ) ↓= 0
ERIK A. ANDREJKO DECIDABILITY / INCOMPLETENESS

5. PRIMITIVE RECURSIVE
INCOMPLETENESS
COMPUTABLE
DEFINITION
A function is
computable ⇐⇒ Turing computable ⇐⇒ partial recursive
ERIK A. ANDREJKO DECIDABILITY / INCOMPLETENESS

6. PRIMITIVE RECURSIVE
INCOMPLETENESS
TURING MACHINES
DEFINITION
(n)
Let ϕe be the partial function of n variables computed by the Turing
machine Pe with code e ∈ N.
DEFINITION
Write ϕe,s (x) = y if w, y , e < s and y is the output of ϕe (x) in < s
steps of the Turing machine Pe .
ϕe,s (x) converges and we write ϕe,s (x) ↓,
ϕe,s (x) diverges which is written as ϕe,s (x) ↑.
FACT
The set { e, x, s : ϕe,s (x) ↓} is recursive.
FACT
The set { e, x, y , s : ϕe,s (x) = y } is recursive.
ERIK A. ANDREJKO DECIDABILITY / INCOMPLETENESS

7. PRIMITIVE RECURSIVE
INCOMPLETENESS
RECURSIVELY ENUMERABLE SETS
DEFINITION
A set A is called recursively enumerable if A is the domain of some
partial recursive function.
We = dom ϕe = {x : ϕe (x) ↓}
and
We,s = dom ϕe,s
Let K = {x : ϕx (x) ↓} = {x : x ∈ Wx }.
FACT
K is r.e.
FACT
K is not recursive.
ERIK A. ANDREJKO DECIDABILITY / INCOMPLETENESS

8. PRIMITIVE RECURSIVE
INCOMPLETENESS
INCOMPLETENESS
N = (N, 0, S, <, +, ·, E)
THEOREM
Let A ⊆ Th(N) and suppose that the set
{ ϕ : ϕ ∈ A}
is definable. Then there is a L -sentence σ such that N |= σ and
A σ.
COROLLARY
The set { τ : N |= τ} is not definable in N.
ERIK A. ANDREJKO DECIDABILITY / INCOMPLETENESS

9. PRIMITIVE RECURSIVE
INCOMPLETENESS
PEANO ARITHMETIC
∀x [Sx = 0]
∀x∀y [Sx = Sy =⇒ x = y ]
∀x∀y [x < Sy ⇐⇒ x ≤ y ]
∀x [x < 0]
∀x [x + 0 = x]
∀x∀y [x + Sy = S(x + y )]
∀x [x · 0 = 0]
∀x∀y [x · Sy = x · y + x]
∀x [xE0 = S0]
∀x∀y [xESy = xEy · x]
induction axioms for each well formed formula ϕ
ϕ(0) ∧ ∀x [ϕ(x) =⇒ ϕ(Sx)] =⇒ ∀xϕ(x)
ERIK A. ANDREJKO DECIDABILITY / INCOMPLETENESS

10. PRIMITIVE RECURSIVE
INCOMPLETENESS
FACT
N |= PA.
ERIK A. ANDREJKO DECIDABILITY / INCOMPLETENESS

11. PRIMITIVE RECURSIVE
INCOMPLETENESS
INCOMPLETENESS
THEOREM
(Church’s Thesis) A relation is decidable iff it is definable in N.
COROLLARY
Th(N) is not recursive.
THEOREM
(Gödel Incompleteness Theorem) If A ⊆ Th(N) and A is recursive
then A is not a complete theory.
ERIK A. ANDREJKO DECIDABILITY / INCOMPLETENESS

12. PRIMITIVE RECURSIVE
INCOMPLETENESS
UNDECIDABLE THEORIES
FACT
The theory Th(N, +, ·, S, <) is undecidable.
FACT
The theory Th(N, +, ·) is undecidable.
ERIK A. ANDREJKO DECIDABILITY / INCOMPLETENESS