True but Unprovable

Goal
Logicians sometimes talk about sentences being “true but unprovable”
Our goal: Understand what they mean by this.
“Gödel’s first incompleteness theorem states that no consistent system of axioms
whose theorems can be listed by an algorithm is capable of proving all truths
about the arithmetic of natural numbers.”
“For any such consistent formal system, there will always be statements about
natural numbers that are true, but that are unprovable within the system.”

An Illustration
There’s an island on which there lives two types
of people: truthers and liars. Truthers always
speak the truth, liars always say falsehoods.
A logician that knows all of this visits the island.
He is a perfectly sound reasoner, i.e. he never
proves anything that is false.
The logician encounters an individual named Jal
living on the island.
Jal makes some statement to the logician, after
which the following two things are true:
(1) Jal’s statement logically entails that Jal is a
truther.
(2) The logician can never possibly prove that
Jal is a truther.
The question: What did Jal say?

A solution
Jal’s statement to the logician:
“You can not prove that I am a
truther.”
Only a truther could say this sentence.

A solution
Jal’s statement to the logician:
“You can not prove that I am a
truther.”
Only a truther could say this sentence.
But if Jal is a truther, then his
statement is true.
So the logician can not prove that Jal is
a truther.
Suppose Jal is a liar.
Then the sentence “You can not prove
that I am a truther” is false.
So the logician can prove that Jal is a
truther.
But the logician cannot prove a
falsehood.
Contradiction

But...
Why can’t the logician make this
exact argument, therefore proving
just as we did that Jal is a liar?
truther.
falsehood.
Contradiction

But...
Why can’t the logician make this
exact argument, therefore proving
just as we did that Jal is a liar?
Because to do so, he would have
to assert his own soundness.
Gödel’s second incompleteness
theorem tells us that no sound
reasoner can assert this of
himself.
truther.
falsehood.
Contradiction

Example 2: Busy beavers
The busy beaver numbers are a well-defined
sequence BB(n).
Run all n-state Turing machines. Some never
halt, the rest eventually do. BB(n) is the halt time
of the longest-running machine.
BB(n) is a measure of how complex a program
you can run on an n-state Turing machine.
We will now prove that no effective and
consistent proof system can enumerate all its
values.

Take any consistent formal proof system F whose
theorems can be listed by a Turing machine.
Construct an n-state Turing machine that
enumerates the theorems of F, halting only if it
reaches a contradiction.
Whether this Turing machine halts depends on if
F is consistent. BB(n) tells us how long we have
to wait to know for sure if it halts.
So if F could prove the value of BB(n), F could
prove its own consistency. Violation of Gödel’s
second!
(F can’t even upper bound BB(n)! Why?)
The busy beaver numbers are a well-defined
sequence BB(n).
Run all n-state Turing machines. Some never
halt, the rest eventually do. BB(n) is the halt time
of the longest-running machine.
BB(n) is a measure of how complex a program
you can run on an n-state Turing machine.
We will now prove that no effective and
consistent proof system can determine all its
values.

Different proof systems will have different
thresholds for unprovability.
The “standard proof system” for mathematics is
ZFC set theory.
We know ZFC can only prove at most 7,918
values of BB(n), and probably much fewer.

Final Example: Goldbach Conjecture
Every even integer greater than 2 is the sum of
two primes.
4 = 2 + 2
6 = 3 + 3
8 = 5 + 3
10 = 5 + 5
12 = 7 + 5
Etc...

two primes.
4 = 2 + 2
6 = 3 + 3
8 = 5 + 3
10 = 5 + 5
12 = 7 + 5
Etc...
Know to be true for values up to 4×1018.
Goldbach: "I regard as a completely certain
theorem, although I cannot prove it."
What if it cannot be proven true or false?
Then it must be true!

two primes.
4 = 2 + 2
6 = 3 + 3
8 = 5 + 3
10 = 5 + 5
12 = 7 + 5
Etc...
Know to be true for values up to 4×1018.
Goldbach: "I regard as a completely certain
theorem, although I cannot prove it."
What if it cannot be proven true or false?
Then it must be true!
If the conjecture were false, that would mean
that there’s a counterexample. So we could
eventually find that counterexample and prove
the conjecture false. So if it’s unprovable, then it
must be true.

So, what is it to be “true but unprovable”?
Logicians distinguish between syntax, semantics, and proof.
SYNTAX SEMANTICS PROOF
Defines an alphabet of symbols.
Defines a grammar: some subset
of the set of all finite strings of
symbols.
Defines the meaning of each symbol.
Defines a “valuation function” from
grammatical sentences to truth
values, consistent with the meanings
of the symbols.
Defines a deductive calculus.
Gives a set of axioms (sentences)
and inference rules (functions
from sentences to other
sentences) from which you can
derive theorems.

Propositional Logic
Alphabet
( ) ∧ ∨ → ¬
p1 p2 p3 …
Grammar
> pn is grammatical for each n
> If a is grammatical, so is (¬a)
> If a and b are grammatical, so
are (a∧b), (a∨b), (a→b)
We want to define a truth valuation
function V from grammatical
sentences to truth values. But not
just any function will do. The
allowed functions must respect our
intended meaning of the symbols ∧,
∨, →, and ¬.
(There’s a possible function from grammatical sentences to truth values that assigns V(p) = F, V(q) = F,
and V(p∧q) = T. We don’t want to say that this is a valid “truth valuation.” Thus our set of truth valuation
functions will be restricted to only those that align nicely with our intended semantics.)

Propositional Logic
Alphabet
( ) ∧ ∨ → ¬
p1 p2 p3 …
Grammar
are (a∧b), (a∨b), (a→b)
V: Atomic Propositions → {T,F}
V’: Grammatical Sentences → {T,F}
V’(pn) = V(pn)
V’(¬a) = T iff V’(a) = F
V’(a∧b) = T iff V’(a) = V’(b) = T
V’(a∨b) = F iff V’(a) = V’(b) = F
V’(a→b) = F iff V’(a) ≠ V’(b) = F
With our truth functions defined,
we want a purely mechanical
procedure for deriving theorems
without having to look directly at
the semantics.
We define a set of axioms and
inference rules that will blindly
churn out new sentences, and
choose them to align as best as
we can with our semantics.
Say I want to prove that a certain sentence p is a tautology. One way I could do this would be to look at all
possible valuation functions, and see that each of them assigns p the value T. But what if I want to prove it
without looking at the semantics?

Propositional Logic
Alphabet
( ) ∧ ∨ → ¬
p1 p2 p3 …
Grammar
are (a∧b), (a∨b), (a→b)
V: Atomic Propositions → {T,F}
V’: Grammatical Sentences → {T,F}
V’(pn) = V(pn)
V’(¬a) = T iff V’(a) = F
V’(a∧b) = T iff V’(a) = V’(b) = T
V’(a∨b) = F iff V’(a) = V’(b) = F
V’(a→b) = F iff V’(a) ≠ V’(b) = F
Axioms
1. a→(b→a)
2. (a→(b→c))→((a→b)→(a→c))
3. ((¬b)→(¬a))→(a→b)
Inference Rules
MP(a, a→b) = b
Notice that truth and proof are distinct!
A sentence a is a logical truth if for every function V, V’(a) = T.
A sentence is provable if it follows from the axioms via repeated use of the inference rules.

Proof of a→a
Two proofs that a→a is a logical truth:
Semantic Proof Syntactic Proof
For any V, either V’(a) = T or V’(a) = F
- If V’(a) = T, V’(a→a) = T
- If V’(a) = F, V’(a→a) = T
So for any V, V’(a→a) = T.
So a→a is a logical truth.
1. (a→((a→a)→a)) → ((a→(a→a))→(a→a)) Axiom 2
2. a→((a→a)→a) Axiom 1
3. (a→(a→a))→(a→a) MP(1,2)
4. a→(a→a) Axiom 1
5. a→a MP(3,4)
a (a→a)
T T
F T
(Reminders)
V’(a→b) = F iff V’(a) =T and V’(b) = F
Axiom 1: a→(b→a)
Axiom 2: (a→(b→c))→((a→b)→(a→c))

Semantic and Syntactic Entailment
More generally:
- A set of sentences B semantically entails (or logically entails) a sentence a (written B ⊨ a) if for every
function V such that V’ assigns T to all sentences in B, V’ also assigns T to a.
- A set of sentences B syntactically entails a sentence a (written B ⊢ a) if a follows from the axioms
AND the sentences in B via repeated use of the inference rules.
It so happens that these concepts perfectly coincide for propositional logic.
- The logical truths of propositional logic are exactly the provable sentences.
- B ⊨ a exactly when B ⊢ a
Some important logical concepts:
- Soundness: If B ⊢ a then B ⊨ a
- Completeness: If B ⊨ a then B ⊢ a
- Propositional logic is both sound and complete.

Soundness, Completeness, and Decidability
For logics with more expressive power, semantics and syntax come apart.
In particular, there is no proof system that is both sound and complete for second order logic.
Sound and Complete Proof
System
Decidable
Propositional Logic ✔ ✔
First Order Logic ✔ ✘
Second Order Logic ✘ ✘

First Order Logic
Let’s see how this plays out for first order logic! Buckle up, things are going to get a lot more complicated.
Outline
Syntax
- Terms
- Atomic Formulas
- Grammatical Formulas
Semantics
- Structures
- Variable assignments
- Valuation function

First Order Logic: Alphabet
First difference: First order logic is a collection of languages, not a single language.
All first order languages share a certain set of symbols, but also contain a set of language-specific symbols.
Shared Alphabet
( ) , ∧ ∨ → ¬ ∀ ∃ =
Variables: x1 x2 x3 …
Language-Specific Alphabet
Constant symbols: c1 c2 c3 …
Relation symbols: R1 R2 R3 …
Function symbols: f1 f2 f3 …
Arity function
Arity: (Relation symbols ∪ Function symbols) → ℕ

First Order Logic: Grammar
Grammar is a bit trickier than ‘twas for propositional logic. We do it in three steps.
Terms
All variables and constants are terms.
f(t1, t2, …, tn) is a term for any function symbol f with Arity(f) = n, and for any terms t1, t2, …, tn.
Atomic Formulas
For any terms t1, t2, (t1=t2) is an atomic formula.
R(t1, …, tn) is an atomic formula for any relation symbol R with Arity(R) = n, and for any terms t1,
…, tn.
Grammatical Formulas
Every atomic formula is grammatical.
If a is grammatical, so is (¬a).
If a and b are grammatical, so are (a∧b), (a∨b), (a→b).
If a is grammatical, x is a variable, and a doesn’t contain ∀x or ∃x, then ∀xa and ∃xa are both grammatical.

First Order Logic: Semantics
We define a structure M= <U, I> for a first order language.
U is a set: it is the set of all objects in the domain of discourse.
I is an interpretation function: it “interprets” every constant, relation, and function symbol.
Constants: I(c) = cM ∈ U
Relations: I(R) = RM: UArity(R) → {T,F}
Functions: I(f) = fM: UArity(f) → U
U
c1
M
c2
M
c3
M
R1
M
c4
M
f1
M
R2
M
R3
M

Now we define a variable assignment for our structure M = <U,I>.
s: Variables → U
We extend this to all terms as follows:
s’: Terms → U
s’(x) = s(x) for variables x
s’(c) = cM for constants c
s’(f(t1, t2,...,tn)) = fM(s’(t1), s’(t2), …, s’(tn)), for any function symbol f with Arity(f) = n.
U
c1
M
c2
M
c3
M
R1
M
c4
M
f1
M
R2
M
R3
M

With our structure M and our variable assignment s, we are finally ready to construct a valuation function!
First for atomic formulas.
VM,s: Atomic Formulas → {T,F}
VM,s(t1=t2) = T iff t1
M = t2
M
VM,s(R(t1,t2,...,tn)) = T iff RM(t1
M,t2
M,...,tn
M) = T
U
c1
M
c2
M
c3
M
R1
M
c4
M
f1
M
R2
M
R3
M

Now for the rest of the language!
V’M,s: Grammatical Formulas → {T,F}
V’M,s(a) = VM,s(a) if a is atomic
V’M,s(¬a) = T iff V’M,s(a) = F
V’(a∧b) = T iff V’M,s(a) = V’M,s(b) = T
V’M,s(a∨b) = F iff V’M,s(a) = V’M,s(b) = F
V’M,s(a→b) = F iff V’M,s(a) =T and V’M,s(b) = F
V’M,s(∀xa) = T iff V’M, s(x/d)(a) = T for all d ∈ U
V’M,s(∃xa) = T iff V’M, s(x/d)(a) = T for some d ∈ U
For the last two, we used the concept of a variant variable assignment:
sx/d(y) = s(y) if y ≠ x, otherwise d

First Order Logic: Proof System
Axioms
Every propositional tautology with atomic propositions substituted for first order sentences.
A(t) → ∃xA(x)
∀xA(x) → A(t), where t is not bound by A
(A→B) → (A→∀xB), where x is not free in A
(A→B) → (∃xA→B), where x is not free in A
x = x
(x = y) → (A → A’), where A’ is obtained by replacing any number of free occurrences of x with
y
Inference Rules
Modus Ponens: MP(A, A→B) = B

The structure M = <U,I> and the variable assignment s together uniquely determine the truth values of all
the grammatical sentences.
Notation: M,s ⊨ a iff V’M,s(a) = T
A sentence a is a logical truth if for all structures M and variable assignments s, M,s ⊨ a
Semantic entailment: B ⊨ a iff for all M and s, if M,s ⊨ b for every b in B, then it’s also the case that M,s ⊨ a.
Examples
∀x(x=x) is a logical truth
(R(x)∨(¬R(x))) is a logical truth
∀xR(x) ⊨ ∃xR(x)
∀x∀yR(x,y) ⊨ ∀y∀xR(x,y)
∀x∃yR(x,y) ⊭ ∃y∀xR(x,y)

First Order Logic: The Limitations
This proof system for first order logic is both sound and complete, but not decidable!
Decidability: there’s a decision procedure which determines whether arbitrary formulas are logical truths.
If a first-order language has at least one predicate of arity at least 2, then it is undecidable!
Expressive limitations:
- Not capable of expressing the notion that “there are finitely many things”
- Doesn’t have the expressive power to distinguish between different cardinalities of infinity
- There is no first order language in which you can uniquely pin down the natural numbers. No set of
sentences is consistent with the structure of the natural numbers and no other structures.

Second Order Logic
Second order logic fixes these problems! Second order logic is what you get when you add the ability to
quantify over predicates and functions to first order logic.
Second order logic is capable of expressing finiteness, talking about specific cardinalities, and uniquely
pinning down the natural numbers.
But as we’ll see, it has problems of its own.

Second Order Logic: Alphabet
Almost the same as with first order logic, but now we have extra variables and remove =.
Shared Alphabet
( ) , ∧ ∨ → ¬ ∀ ∃
Individual variables: x1 x2 x3 …
Relation variables: X1 X2 X3 … (actually an infinite store for each possible arity)
Function variables: F1 F2 F3 … (actually an infinite store for each possible arity)
Language-Specific Alphabet
Constant symbols: c1 c2 c3 …
Relation symbols: R1 R2 R3 …
Function symbols: f1 f2 f3 …
Arity function
Arity: (Relation symbols ∪ Function symbols) → ℕ

Second Order Logic: Grammar
Almost the same as with first order logic, but now we also allow the following grammatical constructions:
If a is grammatical and x is an individual variable, then so is ∀xa and ∃xa
If a is grammatical and X is a relation variable, then so is ∀Xa and ∃Xa
If a is grammatical and F is a function variable, then so is ∀Fa and ∃Fa

Second Order Logic: Standard Semantics
There are actually several different “second order logics” with different semantics. We’ll spend most our
time on standard semantics.
Like in first order logic, we have a structure M = <U,I>. Now, though, our variable assignment function
has to assign values to all the relation and function variables as well as the individual variables.
s1: Individual Variables → U
s2: n-ary Relation Variables → {f: Un → {T,F}}
s3: n-ary Function Variables → {f: Un → U}

Second Order Logic: Standard Semantics
Together, M, s1, s2, and s3 uniquely pin down a truth value for every sentence of second order logic. The
semantics for all the non-quantified formulas are identical to first-order logic. For quantified formulas we
have the following:
V’M,s1,s2,s3(∀xa) = T iff V’M,s1(x/d),s2,s3(a) = T for all d ∈ U
V’M,s1,s2,s3(∃xa) = T iff V’M,s1(x/d),s2,s3(a) = T for some d ∈ U
V’M,s1,s2,s3(∀Xa) = T iff V’M,s1,s2(X/D),s3(a) = T for all D ⊆ Un, for n-ary X
V’M,s1,s2,s3(∃Xa) = T iff V’M,s1,s2(X/D),s3(a) = T for some D ⊆ Un, for n-ary X
V’M,s1,s2,s3(∀Fa) = T iff V’M,s1,s2,s3(F/G)(a) = T for all G ∈ {f: Un → U}, for n-ary F
V’M,s1,s2,s3(∃Fa) = T iff V’M,s1,s2,s3(F/G)(a) = T for some G ∈ {f: Un → U}, for n-ary F

Second Order Logic: Proof System
The drawback of all this expressive power we now have…
There is no sound and complete proof system for second order logic!
For any sound proof system you choose for second order logic, and for any second-order language, there
will be logical truths in that language that the proof system will fail to prove.

Second Order Logic and Complexity Theory
- REG (the set of regular languages) is definable by monadic, second-order formulas.
- NP is the set of languages definable by existential, second-order formulas.
- co-NP is the set of languages definable by universal, second-order formulas.
- PH is the set of languages definable by second-order formulas.

Summary
Logical truth is fundamentally about semantics.
We define the truth of a sentence based off of our intended meaning for the symbols in use.
Defines an alphabet
of symbols.
Defines a grammar:
some subset of the
set of all finite
strings of symbols.
Defines the meaning of
each symbol.
Defines a “valuation
function” from
grammatical sentences
to truth values,
consistent with the
meanings of the
symbols.
Defines a deductive
calculus.
Gives a set of axioms
and inference rules
from which you can
derive more
sentences.
Sound and
Complete
Proof System
Decidable
Propositional
Logic
✔ ✔
First Order
Logic
✔ ✘
Second Order
Logic
✘ ✘

True but Unprovable

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