Download We showed on Tuesday that Every relation in the arithmetical

We showed on Tuesday that
Every relation in the arithmetical hierarchy is definable in
arithmetic.
If R(~x ) is a Σn or Πn relation, then there is an expression α in
first-order arithmetic (using
0, S, +, ·, exp, =, <, ∧, ∨, ¬, →, ↔) such that for all
n1 , . . . , nk ∈ N,
R(n1 , . . . , nk ) holds ⇐⇒ α(n1 , . . . , nk ) is true.
Su Gao
Connections to Logic
Coding all expressions in arithmetic can be done in much
the same way we code all register machines, etc.
Definition
Use # to denote such a coding from all expressions in the
first-order arithmetic to N that is computable and admits
computable decoding.
Define
True = {#α : α is a true sentence in arithmetic}.
Su Gao
Connections to Logic
Proposition
If S is definable in arithmetic then S ≤m True.
Proof. The function
(n1 , . . . , nk ) 7→ #α(n1 , . . . , nk )
is computable.
Corollary
If S is a relation in the arithmetical hierarchy, then S ≤m True.
Su Gao
Connections to Logic
Tarski’s Theorem on Arithmetic Truth (1936)
The set True is not in the arithmetical hierarchy. In particular,
the set True is not decidable.
The theorem is also call Tarski’s Undecidability Theorem.
Su Gao
Connections to Logic
Gödel’s First Incompleteness Theorem essentially states that
no reasonable axiom system can “capture” all arithmetic truth
, because the set True is not semidecidable.
To illustrate the theorem we need some definitions and
observations.
Su Gao
Connections to Logic
Definition
A (first-order) proof system is a set of rules which allows
certain formulas to be derived from other formulas.
Proposition
The usual proof system (for arithmetic) is computable.
For those who worry about the deductive power of the “usual
proof system”:
Gödel’s Completeness Theorem
The usual proof system is “complete” in the sense that it can
derive all tautologies of logic.
Su Gao
Connections to Logic
Definition
A (first-order) theory in arithmetic is a set of sentences in
arithmetic that is closed under the usual proof system.
Definition
A theory in arithmetic T is axiomatizable if there is a
computable set of axioms from which every sentence in T is
derivable from the usual proof system.
Proposition
If a theory T is axiomatizable, then #T = {#α : α ∈ T } is
r.e.
Su Gao
Connections to Logic
Example T = first-order Peano Arithmetic
Peano Axioms
1. ∀x (0 6= Sx)
2. ∀x∀y (Sx = Sy → x = y )
3. ∀x (x + 0 = x)
4. ∀x∀y (x + Sy = S(x + y ))
5. ∀x (x · 0 = 0)
6. ∀x∀y (x · Sy = x · y + x)
7. ∀x (x 0 = S0)
8. ∀x∀y (x Sy = x y · x)
9. ∀~y {[α(0, ~y ) ∧ ∀x (α(x, ~y ) → α(Sx, ~y ))] → ∀x α(x, ~y )}
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Connections to Logic
Gödel’s First Incompleteness Theorem (1931)
If T ⊆ True is axiomatizable theory in arithmetic, then there is
a true sentence σ such that #σ 6∈ T .
Su Gao
Connections to Logic
Is first-order too restrictive?
The big advantage of Second Order Arithmetic is the full
induction:
∀X {[X (0) ∧ ∀x (X (x) → X (Sx))] → ∀x X (x)}
Su Gao
Connections to Logic
In fact, ZFC is a set of first-order axioms about sets that
allows the formulation of second-order arithmetic!
The set of all derivable true sentences in ar is still r.e.!
Su Gao
Connections to Logic