Part 1 - University of Cambridge

7
The Power of First-Order Logic
For every finite structure A, there is a sentence ϕ A such that
B |= ϕ A if, and only if, B ∼ = A
Given a structure A with n elements, we define
ϕ A = ∃x 1 . . . ∃x n ψ ∧ ∀y ∨
1≤i≤n
y = x i
where, ψ(x 1 , . . .,x n ) is the conjunction of all atomic and negated atomic
formulas that hold in A.
For any isomorphism-closed class of finite structures, there is a first-order theory
that defines it.
Anuj Dawar February 2013