1. Decidability of logical theories
Giorgi Japaridze
Theory of Computability
Decidability of logical theories
Section 6.2

2. The language of arithmetic: formulas
Giorgi Japaridze Theory of Computability
Formulas --- strings produced by the following CFG:
FORMULA  ATOM |
(FORMULA) |
(FORMULA)  (FORMULA) |
(FORMULA)  (FORMULA) |
 VARIABLE (FORMULA)
ATOM  TERM = TERM
TERM  VARIABLE |
CONSTANT |
(TERM) + (TERM)
(TERM)  (TERM)
VARIABLE  v | VARIABLE’
CONSTANT  0 | 1 | 1CONSTANT
negation
conjunction
disjunction
universal quantifier

3. The language of arithmetic: sentences
6.2.b
Giorgi Japaridze Theory of Computability
An occurrence of variable x is bound in formula F, if it is in the scope of x,
i.e. F= ... x( … x …) …
Otherwise it is free.
A sentence is a formula without free occurrences of variables
Is v’ free or bound in:
v’=0
 (v’(v’=0))
 (v’’(v’=0))
v’’(v’=v’’)
v’((v’=0) (v’=v))
(v’(v’=0))(v’=v)
Is the following formula a sentence:
(0=10)
v=0
v(v=v)
v(v=v’)
v(v=1 v’(v+v’=110))

4. Truth of arithmetic sentences
Giorgi Japaridze Theory of Computability
An atomic sentence is true iff it is true under the standard
interpretation of constants and +,,=.
 A is true iff A is false
AB is true iff both A and B are true
AB is true iff either A or B (or both) are true
xA(x) is true iff for all constants c, A(c) is true
A(c) --- the result of substituting all free occurrences of x by c in A(x)
10+10=1010
v(v+v=vv)
v (v1=v)
v((v+v=v))
v(v=0(v+v=v))
vv’(v+v’=v’+v)
vv’(v(v’+1)=(v’v)+v)

5. The undecidability of truth for arithmetic sentences
Giorgi Japaridze Theory of Computability
Let Th(N,+, ) = {A | A is a true arithmetical sentence}
Theorem 6.13: Th(N,+, ) is undecidable.
Corollary: Th(N,+, ) is not Turing recognizable, either.
Proof: Suppose a TM M recognizes Th(N,+, ). Construct a TM D:
D = “On input A, an arithmetic sentence,
1. Run M on both A and A in parallel.
2. If M accepts A, accept; if M accepts A, reject”
Obviously D decides Th(N,+, ) , which is in contradiction with
Theorem 6.11.
Let Th(N,+) = {A | A is a true arithmetical sentence not containing }
Theorem 6.12: Th(N,+) is decidable.