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consequences

• adequacy of a syntactic calculus (our sequent calculus) for all FO-based mathematical reasoning

• finite syntactic certificates (formal proofs) for all FO truths;

recursive enumerability of all validities

example: FO group theory,

ϕ ∈ FO0({◦,e}) : {ϕG1, ϕG2, ϕG3} |= ϕ

=

ϕG1, ϕG2, ϕG3 ⊆ FO0({◦,e}) can be algorithmically generated (r.e.)

Intr.Math.Log. Winter 12/13 M Otto 33/38

model-theoretic consequences

• compactness: finiteness property for satisfiability (!)

• L¨owenheim–Skolem theorems:

(↓) countable consistent FO theories have countable models (↑) FO theories with infinite models have models of

arbitrarily large cardinalities

and further, from these:

• no infinite structure A is fixed up isomorphism by its FO theory Th(A) =

ϕ ∈ FO0: A |= ϕ

• weaknesses/strengths of first-order logic/model theory:

non-standard models, saturated models, . . . richness of classical model theory, . . .

Intr.Math.Log. Winter 12/13 M Otto 34/38

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compactness

Φ ⊆ FO satisfiable if every finite subset Φ0 ⊆ Φ is satisfiable

• a finiteness property for satisfiability

• also a topological compactness assertion

• the tool (for model construction) in classical model theory from finiteness property for consistency, via completeness

variants:

Φ unsatisfiable ⇒ some finite Φ0 ⊆ Φ unsatisfiable Φ |= ϕ ⇒ Φ0 |= ϕ for some finite Φ0 ⊆ Φ

Intr.Math.Log. Winter 12/13 M Otto 35/38

L¨ owenheim–Skolem theorems

for FO-theories Φ ⊆ FO0(σ):

(↓) Φ countable and satisfiable ⇒ Φ has a countable model

(↑) Φ has an infinite model ⇒

Φ has models in arbitrarily large cardinality

corollary: no FO-theory can determine any infinite structure up to isomorphism

Intr.Math.Log. Winter 12/13 M Otto 36/38

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non-standard models

Th(A) =

ϕ ∈ FO0(σ) : A |= ϕ

the complete FO-theory of σ-structure A

for familiar infinite standard structures A of mathematics, A |= Th(A) with A 6≃ A

is a non-standard companion of A: indistinguishable from A in FO,

but different – possibly in useful ways, especially if A ⊆ A and even A 4 A

examples: non-standard models of natural and real arithmetic with ‘infinite numbers’ and ‘infinitesimals’

Intr.Math.Log. Winter 12/13 M Otto 37/38

example: non-standard analysis

find non-standard models of (expansions of) real arithmetic R = (R,+, ·,0,1, <, . . .)

R

< R

with infinitesimals δ ∈ T

16n∈N(−1

n, 1

n) \ {0}

non-archimedean, Dedekind incomplete, real-closed field with projection map to ‘standard part’ on S

n∈N(−n,n) allows to eliminate typical limit constructions of analysis

non-standard analysis, following Abraham Robinson (1960s)

Intr.Math.Log. Winter 12/13 M Otto 38/38

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