Incompleteness

System Relativity

Soundness and completeness are properties of particular logical systems.

There’s no sense to be made of the claim that “logic” is sound or complete. Some logics are sound, others are not, some are complete, others are not.

Incompleteness

A logic is incomplete if there are formulas that all of its models satisfy, but which is not provable in that system.

(Equivalently, if there are some consistent sets of sentences that are unsatisfiable.)

That some logic is incomplete is a boring fact.

The Boringness of Incompleteness

Here is a boringly incomplete logic.

First, it has the exact same semantics as the propositional calculus.

Second, it has one and only one proof rule, Assumption:

φ ⊢ φ

More Interesting Incompleteness

What we be more interesting would be a logic that is in principle incomplete.

The previous logic is not in principle incomplete. There are rules you could add to it that would make it complete. (These are the rules that we do in fact add to it.)

What Gödel proved was that certain logics were in principle incomplete.

Inconsistent Logics Always Complete*Well, that’s not quite true.

You see, every logic is in principle complete, in that there are always rules you can add to it to make it complete. Here’s one, the Rule of Anything Goes:

⊢ φ

Final Note on Interestingness

What would really be interesting is a logic that we can prove or at least suspect is consistent (doesn’t derive contradictions) that is in principle consistent and incomplete.

That is, there are no further rules you could add that would make the logic remain consistent and be complete.

Kurt Gödel

• 1906-1978• Austrian mathematician who

fled to America during the second world war.

• First to prove completeness of first-order predicate calculus.

• Also proved the in principle incompleteness of arithmetic

Logic and the Mind

Mental Processes are Rational

Mental processes are reason-respecting. Many of your mental states cause other mental states, and do so in a way that if the causing states represent something that is true, then the caused state represents something that is also true.

Logical Relations

From:

1. If Joe fails the final exam, he will fail the course.2. If Joe fails the course, he will not graduate.

It follows logically that:

3. If Joe fails the final exam, he will not graduate.

Logical Relations

If you believe:

1. If Joe fails the final exam, he will fail the course.2. If Joe fails the course, he will not graduate.

These beliefs can cause you to also believe:

3. If Joe fails the final exam, he will not graduate.

Functions

Examples of Functions

• f(x) = x2

• Mother of x• x’s definition in the Oxford English Dictionary• Your password for website x• It is not true that x• g(x, y) = x2 + y – 4 • y’s password for website x• x and y

Functions

A function is any relation between inputs and outputs where: for each distinct input there is only one output.

Algorithms

Algorithms

An algorithm is an effective procedure for calculating a function.

You can think of it as a list of steps where: if you follow the steps correctly, you will always get the right answer.

Change-Giving Algorithm

1. Take the largest coin of n cents where n ≤ the amount owed.2. Reduce the amount owed by n cents.3. If the amount owed is 0 cents, return all coins taken and stop.4. Go back to State (line) 1.

Computation

Computation is the concrete use of an algorithm (program) to find the output of a function given its inputs. It requires:1. A representation of the inputs.2. Basic means of manipulating its representations.3. A set of instructions that use the basic means of manipulating to

run the algorithm.

Abacus Computer

Mechanical Computers

Abacuses are nice, but they’re prone to human error. For computation to work, all the steps of the algorithm need to be followed exactly. What we want is a mechanical computer, one where physics performs the computations.

https://www.youtube.com/watch?v=GcDshWmhF4A

Logical computation

Truth Functions

A special subset of functions is the truth-functions. These are functions whose input is truth-values (true or false) and whose outputs are truth-values:• Not P• P and Q• P or Q• If P, then Q

Truth Functions

(P → Q), (Q → R)├ (P → R)

1 1. (P → Q) A2 2. (Q → R) A3 3. P A (for →I)1,3 4. Q 1,3 →E1,2,3 5. R 2,4 →E1,2 6. (P → R) 3,5 →I

Proofs

A proof is a type of program that computes conclusions from their premises:

1. A representation of the premises.2. Basic means of manipulating its representations.3. A set of instructions that leads one to a representation of the

conclusion.

Automatic Reasoner

Can we use the laws of physics to build an automatic reasoner, as we did with the marbles and addition? Yes!

Logic Gates

Mechanical and Digital Computers

In modern day digital computers, the physics isn’t gravity, but instead electromagnetism: computer chips are built with transistors.

However, the basic principle is still the same.

Automatic Reasoners

This is important!

We’ve created things that can use logic and reason on their own.

What Gödel Proved

From the SEP

First incompleteness theoremAny consistent formal system F within which a certain amount of elementary arithmetic can be carried out is incomplete; i.e., there are statements of the language of F which can neither be proved nor disproved in F.

[And, importantly, it can’t be extended to a consistent formal system which can prove or disprove any statement.]

Against Mechanism

“[G]iven any machine which is consistent and capable of doing simple arithmetic, there is a formula it is incapable of producing as being true … but which we can see to be true.”

J.R. Lucas

Gödel’s Proof: Basic Ideas

The Gödel Sentence

Suppose in the language of arithmetic (call it RA for reasons we will later explore), you could prove a sentence that said:

G: I (i.e. this very sentence) am not provable in RA.

Gödel’s Argument

Then you could argue:

RA is sound.Therefore things provable in RA are true.Therefore G is true.Therefore G can’t be proven in RA.And G can’t be disproven either (because then RA is inconsistent).So RA is incomplete.

First Theorem

IF arithmetic is consistent, THEN arithmetic is in principle incomplete.

In Principle!

Suppose we add G to RA to get RA + G. Now G is provable in RA + G. What Gödel shows, however is that even in RA + G, there is a new sentence G* that says:

G*: I (i.e. this very sentence) am not provable in RA + G.

And so on, for any formal theory that was powerful enough to do arithmetic.

The Proof: Broader Outline

Arithmetic

One standard axiomatic way of doing arithmetic is with the Peano axioms:

1. 0 is a number2. For every number n, s(n) is a number3. For all numbers n, m: if s(n) = s(m), then n = m4. 0 is not the successor of any number

Call 1-4: PA

We can extend this to addition and multiplication with the addition of several axioms:

5. x + 0 = x6. x + s(y) = s(x + y)7. x * 0 = 08. x * s(y) = (x * y) + x

Call this Robinson Arithmetic, or RA

The Claim

There exists some sentence, in the standard language of arithmetic, φ such that

RA ⊨ φ

(Every model that makes RA true, makes φ true.) but:

EITHER (a) RA ⊢ (ψ & ~ψ) OR (b) NOT: RA ⊢ φ

Some Important Notions

The Claim

There exists some sentence, in the standard language of arithmetic, φ such that

PA ⊨ φ

(Every model that makes PA true, makes φ true.) but:

EITHER (a) PA ⊢ (ψ & ~ψ) OR (b) NOT: PA ⊢ φ

Standard Language of Arithmetic

0 is written: ‘0’1 is written: ‘s0’2 is written: ‘ss0’3 is written: ‘sss0’And so on…

Names for Numerals

We will name the expression ‘ssssssssssssss0’: ‘14’ (and similarly for the other number-representations).

Don’t confuse 14 and 14. The first one designates a number, the second one designates a number representation.

Gödel-Coding Arithmetic

Arithmetic in Arithmetic

Here’s our goal: In Arithmetic, you can only talk about numbers. You can name them, or quantify over them, and say things about them. But you can’t talk about other things, like rocks, or Beethoven, or arithmetic itself.

But what if we want to talk about arithmetic in arithmetic?

What if we want to say “this arithmetical sentence is not provable in arithmetic”?

Code #1: The Primitive Symbols

‘0’ is paired with 0‘s’ is paired with 1‘=’ is paired with 2‘~’ is paired with 3‘v’ is paired with 4‘ ’ ∀ is paired with 5‘x’ is paired with 6‘(‘ is paired with 7‘)’ is paired with 8

Code for Formulas

For all numbers n, m: if s(n) = s(m), then n = m

∀x0 x∀ s0(~sx0 = sxs0 v x0 = xs0)

5 6 0 5 6 1 0 7 3 1 6 0 2 1 6 1 0 4 6 0 2 6 1 0 8

Here we’re really talking about arithmetical formulas with arithmetical formulas. We’re still talking about numbers. It’s just that each formula has a unique number that “names” it according to our coding scheme.

Representing Properties of Formulas

Now that we can write down formulas as numbers, we want to say things about them: this number is the number of a WFF, this number is the number of a conditional, this number is the number of something provable…

Strong Representation

A set S of natural numbers is strongly representable in a formal system F if there is a formula A(x) of the language of F with one free variable x such that for every natural number n:

n ∈ S ⇒ F ⊢ A(n);n ∉ S ⇒ F ¬⊢ A(n),

Strong Representation of Proof

Gödel shows that there is a way of strongly representing proof in RA:

If χ1, χ2,… , χn ⊢ φ is a proof, then RA π(⊢ χ19χ29…9χn, φ)If χ1, χ2,… , χn ⊢ φ is not a proof, then RA ~π(⊢ χ19χ29…9χn, φ)

Provability

With a notion of proof, we can define provability, Π:

Πx := y ∃ π(y, x)

Weak Representation

A set S of natural numbers is weakly representable in F if there is a formula A(x) of the language of F such that for every natural number n:

n ∈ S ⇔ F ⊢ A(n).

Representation of Provability

Gödel shows that there is a way of weakly representing provability in RA:

If φ is provable, then RA ⊢ Π(φ)

‘Π’ is thus a definable predicate in RA that “represents” provability.

Diagonalization Lemma

Finally, Gödel proves that for any predicate “Fx” of the language RA, we can construct a special formula φS where:

RA ⊢ φS ↔ F(φS)

Diagonalization Lemma

The lemma holds also for the predicate “~Πx”, so we construct the formula φG where:

RA ⊢ φG ↔ ~Π(φG)

“φG is true if and only if φG is not provable in RA”.

The Argument against Mechanism

Against Mechanism

“[G]iven any machine which is consistent and capable of doing simple arithmetic, there is a formula it is incapable of producing as being true … but which we can see to be true.”

J.R. Lucas

Form of the Argument

1. Suppose people are reasoning machines.2. Then we can write down their programs as proofs in a formal system,

call it H.3. H can clearly do arithmetic, so H can prove a Gödel sentence G.4. By Gödel’s proof, our program should not be able to arrive at (a proof

of) G.5. But we can all see that G is true.6. Hence we’re not reasoning machines,7. “the human mind (even within the realm of pure mathematics)

infinitely surpasses the power of any finite machine”

Some Replies

What Gödel actually showed was that if RA is consistent, then a G-sentence exists.

This means that we shouldn’t be able to (rationally) arrive at a G-sentence, unless we first prove our own consistency (the consistency of H).

But it’s not obvious we can do that and it’s not obvious we’re consistent in the first place.