And again, I gave a talk for the best mathematics club of the multiverse, for "younger" students, meaning that this text is not scientific. This is the script for the talk.
We want to prove the following theorem:
Theorem: There is an algorithm of which, with standard set theory, we cannot decide whether it terminates.
Firstly, we need to look at what an algorithm is. Usually, people introduce this with Turing machines. However, we will use a model which is closer to modern computers, so-called register machines. These are provably equivalent to Turing machines, except that some algorithms might take a bit longer on Turing machines – but at this point, we do not care about efficiency, we just care about whether things can be calculated at all, given enough time and space. If you care about efficiency, that would be in the realm of complexity theory, while we will work in the realm of recursion theory.
A register machine has a finite set of registers R[0]
, R[1]
, …,
which can contain arbitrarily large natural numbers. A program
consists of a sequence of instructions:
R[i]++
, which increases the
number that is contained in R[i]
.R[i]--
, which decreases the
number contained in R[i]
if it is larger than 0, and does nothing
otherwise.End
, which ends the program.if R[i]==0 then goto
n
. The instructions are numbered, and this instruction jumps to the
instruction with number n
, if R[i]
contains 0, otherwise it does
nothing.The following program checks whether R[0] >= R[1]
, and if so, it
sets R[2]
to 1:
0 if R[1]==0 then goto 6
1 if R[0]==0 then goto 5
2 R[1]--
3 R[0]--
4 if R[2]==0 then goto 0
5 End
6 R[2]++
7 End
Line 4 is only here to do an unconditional jump, therefore, we can introduce a shorthand that just does an unconditional jump:
0 if R[1]==0 then goto 6
1 if R[0]==0 then goto 5
2 R[1]--
3 R[0]--
4 goto 0
5 End
6 R[2]++
7 End
To make it a bit more readable, we can omit the line numbers, which we usually do not need, and just set labels to the lines we want to jump to:
Start: if R[1]==0 then goto Yes
if R[0]==0 then goto No
R[1]--
R[0]--
goto Start
No: End
Yes: R[2]++
End
This just increases readability, it doesn't make anything possible
that wasn't possible before. By using this algorithm, we can generally
check whether R[i] > R[j]
, and therefore, we can introduce a
shorthand notation if R[i] > R[j] then goto A
without being able to
do anything we couldn't do before.
We can do addition by
Start: if R[1]==0 then goto Done
R[0]++
R[1]--
goto Start
Done: End
and truncated subtraction () by
Start: if R[1]==0 then goto Done
R[0]--
R[1]--
goto Start
Done: End
Therefore, we can add the instructions R[i]+=R[j]
which adds R[j]
to R[i]
, and R[i]-=R[j]
, without being able to do anything more
than before. Having these instructions, we can define multiplication
by repeated addition, and division and modulo by repeated
subtraction. This is left to the reader.
Now, we cannot know for sure whether everything that is computable at all is computable by register machines. However, we do not know any computable function that cannot be computed by a register machine. Hence, it is generally believed that there is none. This is called the Church-Turing thesis.
These programs operate on numbers only. Real computers work with
images and text. However, this is, computationally, no difference, and
there are several injections between these kinds of data. Especially,
programs themselves can be represented as numbers, which is called
Gödelization. For this, we use the Cantor Pairing, which
gives a bijective function
by
. This function
enumerates the backward diagonals on the grid of natural numbers, as
this graphic shows. The pairing itself can obviously be
calculated with the above functions by a register machine. Inverting
the function is also easy, and left to the reader.
Now we can represent every program we have in the following way: We first map numbers to the single instructions:
R[i]++
is mapped to R[i]--
is mapped to if R[i]==0 then goto j
is mapped to End
is mapped to Therefore, every instruction has its own code. A program can be
encoded as a sequence of these codes; the program with the
instructions can be encoded by
.
Therefore, it is well-defined to talk about "programs getting other programs as parameters". And having seen this, it is easy to write an universal register machine, which evaluates such a program, given a sequence of register values:
The state of a program is entirely determined by its registers and the
number of the current instruction. It has a finite sequence of
registers , and an instruction line
,
and therefore, we can encode the state by
.
Now, let contain the program code, and
contain
the current state. Let
be the first
element of the program state, which tells us, at which position of
we are. Let
be the current
instruction given by
.
R[k-1]++
.R[k-1]--
.While it is really intricate, we can see that it is possible to
"simulate" a register program inside a register program. A natural
question which arises is: Is there an algorithm such that, given a
program (or its Gödelization), and an input state
,
the algorithm determines whether
terminates. This problem
is called the Halting problem. We now show that it cannot be
solved. Formally, we show that there is no program
that,
given the Gödelization of a program
in
and an
input state in
, leaves
being 0 if and only if
with the given input state terminates. We do this by
contradiction: Assume such an M exists. Then we could, from this M,
generate the following program:
"execute M"
if R[2]==0 then goto Loop
End
Loop: goto Loop
This program terminates if and only if the given program with
the given state does not terminate. We modify this program once again
by one line:
"set R[1] to <R[0]>"
"execute M"
if R[3]==0 then goto Loop
End
Loop: goto Loop
We call this program .
only takes one argument. It
terminates if and only if the given program, given its own
Gödelization, does not terminate. Such programs are called
self-accepting.
Now, as is itself a program, we can Gödelize it, so let
be the Gödelization. By setting
, we can
calculate
.
Now assume terminates. This means that in
after
executing M there will be
R[2]==0
. Therefore
would not terminate. Contradiction.
But assuming would not terminate would mean, by the same
argument, that
R[2]
is not 0 after M. Therefore, would
terminate. Also a contradiction.
Such a program cannot exist. Therefore,
cannot exist.
This proves that we cannot generally decide whether an algorithm terminates. However, it is not yet what we want: We want an algorithm, of which we cannot decide whether it terminates, at all. To get it, we need to do a bit of logic. We will mainly focus on Zermelo-Fraenkel set theory here, as it is the foundation of mathematics.
We first define what a mathematical formula is, which is essentially a string that encodes a mathematical proposition.
Now, the set of formulae is the smallest set, such that
We now give the axioms of set theory:
NUL: There is an empty set:
We can introduce a common shorthand notation for by
, and rewrite this axiom as
If we want to talk about the empty set now, we need to introduce some
variable , and add
to the
formula. Therefore, our system doesn't get stronger if we introduce a
symbol
for the empty set, instead of always adding
this formula, and it increases readability, which is why we do that.
We furthermore define the shorthand notation
by
, and
by
.
EXT: The axiom of extensionality says that sets that contain the same elements are also contained in the same sets:
FUN: The axiom of foundation says that every set contains a set
that is disjoint to it. From this axiom follows that there are no
infinite -chains.
or, with additional obvious shorthand notation
PAR: The axiom of pairing says that there is a set that contains
at least two given elements, meaning, for all , there
exists a superset of
:
UN: The axiom of union says that the superset of the union of all sets in a set exists:
POW: The axiom of the powerset: A superset of the powerset of every set exists:
.
INF: The axiom of infinity says that a superset of the set of
natural numbers exists. Natural numbers are encoded as ordinals:
, and
. Writing it out
as formula is left as an exercise.
The other two sets of formulae we need are given by axiom schemes: They are infinitely many axioms, but they can be expressed by a simple, finite rule:
SEP: The axiom scheme of separation says that, for every formula
and every set
, the set
exists:
Let a formula be given with free variables among
, and
not occur freely. Then the formula
.
is an axiom of set theory.
RPL: The axiom scheme of replacement is a bit more complicated.
A formula is called a functor on a set
(which is not the same as a functor in category theory), if for all
there is a unique
such that
holds. Therefore, in some sense,
defines something
similar to a function on
, and we write
for this unique
. Then the set
, the "image" of
,
exists. Formalizing this scheme is left as an exercise.
AC: It should be noted that usually the axiom of choice is added. However, we do not need to care whether it is added or not, so we omit it here.
We already talked about embedding natural numbers into this set theory. We can also define general arithmetic inside this set theory. Most of mathematics can be formalized inside Zermelo-Fraenkel set theory.
Now, we can formalize propositions. Now we want to formalize proofs. Normally, I would introduce the calculus of natural deduction here, because it corresponds to the dependently typed lambda calculus, so every proof is a term. However, for the specific purpose we need, namely, formalizing proof theory in arithmetic, the equivalent Hilbert calculus is the better choice. It corresponds to the SKI calculus for proof terms.
Firstly, we further reduce our formulae: We can express
as
, and
as
. Furthermore,
can be expressed by
. Hence, we only need
,
and
to express all formulae. We now
define additional logical axiom schemes, where
range over all formulae. (Notice:
is right-associative.)
A proof of a formula is a finite sequence of formulae
, such that
and for
all
, either
is an axiom of set
theory, or a logical axiom, or there exist
such that
. Essentially this means that
everything in the formula is either an axiom or follows from former
formulae applying modus ponens.
Completeness Theorem: If a formula is true in set
theory, then there exists a proof of it.
To prove this, we would need model theory, which would lead too far, so we leave out the proof.
Now, as we did for programs before, we can gödelize formulae and
proofs. Let us denote by the gödelization of
.
Diagonalization Lemma: For every formula with one
free variable
, there exists a formula
, such
that
holds.
Proof: First, we notice that, given the formula , we
can express the substitution of another variable
for
, therefore, we can give a function that satisfies
. Now we can
define
. Now,
define
. Then we have
. This concludes the proof.
Notice that the definition of is computational: It can
be done effectively by a computer. As we can find such a formula for
every
, we denote it by
.
Now, we can also gödelize proofs and their correctness
criterion. Therefore, we can give a formula meaning
"
is the gödelization of a correct proof of the gödelized
formula
". Therefore,
says
that the gödelized formula
is provable.
By the diagonalization lemma, there is a formula
such that
. Now,
assume that
does not hold. Then
also cannot hold,
therefore, it would be provable, which is a contradiction. Hence,
must hold. But then, it cannot be
provable. This is a (sketch of a) proof of
Gödel's first incompleteness theorem: In Zermelo-Fraenkel set theory, there are propositions that can neither be proved nor disproved.
More generally, this theorem holds for all axiom systems that are capable of basic arithmetic, because this is all we used. Specifically for Zermelo-Fraenkel set theory, there are other examples of such propositions, namely the continuum hypothesis, and the existence of large cardinals.
Now, something we always implicitly assumed is that set theory is
consistent: If is provable, then
cannot be provable. This is, however, unknown, which follows from:
Gödel's second incompleteness theorem: Set theory cannot prove its own consistency.
Proof: We use our from the proof of the
first incompleteness theorem. Furthermore, we can define
such that
. Now, we can
define what it means to be consistent, namely:
. Now, we know that
, and therefore, since
false propositions imply anything,
for all formulae
,
and obviously this implies
. Therefore,
. But this contradicts what
we proved in the first completeness theorem. Hence,
cannot be provable.
Let . Obviously,
if and only if set theory is inconsistent (since it is
wrong). Now consider the following algorithm:
Retry: if ν(R[0], a) then goto Found
R[0]++
goto Retry
Found: end
Does this algorithm terminate?
If it terminates, it has found an inconsistency in set theory. Assuming that set theory is consistent, it would not terminate. But if we could prove that it does not terminate, we would be able to prove that set theory is consistent, and this contradicts the second incompleteness theorem.
Hence, we have an algorithm of which we cannot decide whether it terminates.