This document provides definitions and concepts related to logic, languages, structures, and the compactness theorem. It defines logical symbols and objects for propositional and first-order logic. A first-order language consists of function symbols, relation symbols, and constant symbols. A structure assigns interpretations to these symbols. The compactness theorem states that a theory is satisfiable if and only if every finite subset of the theory is satisfiable.

1. LANGUAGES
MODELS
COMPACTNESS THEOREM
COMPACTNESS
SEP
Erik A. Andrejko
University of Wisconsin - Madison
Summer 2007
ERIK A. ANDREJKO COMPACTNESS

2. LANGUAGES
MODELS
COMPACTNESS THEOREM
LANGUAGES
DEFINITION
The logical symbols of propositional logic are:
∧, ∨, =⇒ , ¬
The objects of the logic are propositional letters:
P1 , P2 , · · ·
e.g.
((P1 ∨ P2 ) =⇒ P17 ) =⇒ ¬(P2 ∧ P13 )
ERIK A. ANDREJKO COMPACTNESS

3. LANGUAGES
MODELS
COMPACTNESS THEOREM
LANGUAGES
DEFINITION
The logical symbols of first order logic are:
∧, ∨, =⇒ , ¬, =, ∀, ∃
and variables (bound or unbound)
x, y , z, · · ·
and additional symbols (constant, relational, functional)
c1 , c2 , c3 , · · · R1 , R2 , R3 , · · · f , g, h, · · ·
e.g. ∀x∃y(f (x)R1 g(y )).
ERIK A. ANDREJKO COMPACTNESS

4. LANGUAGES
MODELS
COMPACTNESS THEOREM
SPECIAL SYMBOLS
DEFINITION
1 ∀∞ means for all but finitely many.
∃∞ means there exists infinitely many.
2
∃!x means there exists a unique x.
3
DEFINITION
1 Σ : ∃xϕ(x), ϕ(x) is q.f.
1
Π1 : ∀xϕ(x), ϕ(x) is q.f.
2
Σn+1 : ∃xϕ(x), ϕ(x) is Πn .
3
Πn+1 : ∀xϕ(x), ϕ(x) is Σn .
4
e.g. ∀x∃y[x > y ].
ERIK A. ANDREJKO COMPACTNESS

5. LANGUAGES
MODELS
COMPACTNESS THEOREM
FIRST ORDER LANGUAGE
DEFINITION
A first order language L is a triple F, R, C
F is a set of function symbols,
1
R is a set of relation symbols,
2
C is a set of constant symbols.
3
ERIK A. ANDREJKO COMPACTNESS

6. LANGUAGES
MODELS
COMPACTNESS THEOREM
STRUCTURES
DEFINITION
An L -structure M is a quadruple
M = M, {f M }f ∈F , {R M }R∈R , {c M }c∈C
M is a non empty set called the universe of M,
1
{f M }f ∈F is a set of functions,
2
{R M }R∈R is a set of relations,
3
{c M }c∈C ⊆ M.
4
ERIK A. ANDREJKO COMPACTNESS

7. LANGUAGES
MODELS
COMPACTNESS THEOREM
SENTENCES
∃v∀y[xRv ∨ f (y) = v ]
DEFINITION
∀y or ∃v
BOUND
FREE otherwise
DEFINITION
An L -formula ϕ is called a sentence if all variables in ϕ are bound.
ERIK A. ANDREJKO COMPACTNESS

8. LANGUAGES
MODELS
COMPACTNESS THEOREM
EMBEDDINGS
DEFINITION
σ : M → N is an L-embedding if σ is one-to-one and
σ (f M (¯ )) = f N (σ (¯ )) for all functions f ,
a a
1
M ⇐⇒ σ (¯ ) ∈ R N for all relations R,
a∈R
¯ a
2
σ (c M ) = c N for all constants c.
3
DEFINITION
If σ is a bijective L -embedding, then σ is called an isomorphism. If
∼
there is a isomorphism between M and N then M = N.
DEFINITION
If there exists an L -embedding from M into N then M is called a
substructure of N and N is called an extension of M.
ERIK A. ANDREJKO COMPACTNESS

9. LANGUAGES
MODELS
COMPACTNESS THEOREM
|= AND
DEFINITION
First Order Propositional
∃ proof ∃ proof
|= ∃ model ∃ valuation
and |=.
The completeness theorem relates
ERIK A. ANDREJKO COMPACTNESS

10. LANGUAGES
MODELS
COMPACTNESS THEOREM
CONSISTENT
DEFINITION
A set of L -sentences is called a theory.
DEFINITION
An L -theory T is called consistent if there does not exists an
L -sentence ϕ such that
ϕ ∧ ¬ϕ
T
A consistent theory T is denoted Con(T).
ERIK A. ANDREJKO COMPACTNESS

11. LANGUAGES
MODELS
COMPACTNESS THEOREM
ELEMENTARY EQUIVALENCE
DEFINITION
M and N are called elementarily equivalent, denoted M ≡ N, if
M |= ϕ if and only if N |= ϕ
for all L -sentences ϕ.
FACT
∼
Suppose that M = N. Then M ≡ N.
ERIK A. ANDREJKO COMPACTNESS

12. LANGUAGES
MODELS
COMPACTNESS THEOREM
THEORIES
DEFINITION
An L -theory T is any set of L -sentences.
DEFINITION
A theory is called satisfiable if there is some model M |= T .
DEFINITION
Given any L -structure M
Th(M) = {ϕ : M |= ϕ}
ERIK A. ANDREJKO COMPACTNESS

13. LANGUAGES
MODELS
COMPACTNESS THEOREM
COMPACTNESS THEOREM
THEOREM
(Compactness Theorem) T is satisfiable if and only if every finite
subset of T is satisfiable.
ERIK A. ANDREJKO COMPACTNESS