Transcript Incompleteness Suppose L is a logic and H(T,x) is a statement in L expressing that Turing machine T halts on input x. 

Incompleteness
Suppose L is a logic and H(T,x) is a
statement in L expressing that Turing
machine T halts on input x.
 Thus H(T,x) is true if and only if T halts on
input x.
 Recall -- L is sound and effective. So:

–
–
If H(T,x) is provable in L then it is true so T
halts on input x.
If H(T,x) is provable in L then H(T,x) is false
so T does not halt on input x.

There is a Turing machine M such that M
halts on input i if and only if H(Ti,i) is
provable in L.
–
M can be constructed from a theorem prover
for L
Let j be such that M is Tj.
 Claim: Tj loops on input j but H(Tj,j) is
not provable in L. (Incompleteness.)

Proof: If Tj halts on input j then by
definition of M, H(Tj,j) is provable in L.
Thus Tj loops on input j. Contradiction.
 Thus Tj loops on input j. By definition of M
this implies that H(Tj,j) is not provable in
L.
 Thus H(Tj,j) is true but not provable in L.

The formula H(Tj,j) asserts its own
unprovability
 If H(Tj,j) is true then Tj does not halt on
input j.
 This means that H(Tj,j) is not provable in
L.
 So if H(Tj,j) is true, it is unprovable.
 If H(Tj,j) is provable, L is unsound.

Let S be the sentence “This sentence is
false.”
 This is self-referential like H(Tj,j)
 If S is true it is false.
 If S is false it is true.
 Either way, HL is contradictory unless this
is not considered a valid sentence of HL.
 But which sentences then does HL allow?

Similar to Russell’s paradox:
 Let X be {S : S  S}
 Then is X  X?
 If so X  X.
 If not X  X.
 Either way set theory is contradictory.
 So set theory was modified to disallow this
set.

Constructing H(T,x)

Legend:
–
–
–

tape(i,t) Tape symbol i at time t.
state(t) State at time t
scan(t) Symbol scanned at time t
Transition function of T for (state,symbol):
–
–
–
ns(state,symbol) New state
wr(state,symbol) Symbol written
dir(state,symbol) Direction moved, 0,-1,or +1

Let A be the conjunction of these formulae:
–
–
–
–
–
–
–
–
(0 < i < |x|)  tape(i,0) = xi.
(0  i  i  |x|)  tape(i,0) = blank
state(0)=s (the start state of T)
scan(0)=0
ti (scan(t)  i  tape(i,t)=tape(i,t+1))
t (state(t+1)=ns(state(t),tape(scan(t),t)))
t (scan(t+1)=scan(t)+
dir(state(t),tape(scan(t),t)))
t (tape(scan(t),t+1)=
wr(state(t),tape(scan(t),t)))
Assume ns, wr, and dir are defined in A too.
 Let Hn(T,x) be A  (state(n) F)

–
–
–

F is the set of halting states of T
N is the natural numbers {0,1,2,3,…}
Means that T halts in n steps on input x
Let H(T,x) be n N Hn(T,x)
–
Means that T halts on input x
H(T,x) is true if T halts on input x
 H(T,x) is false if T loops on input x

Note that H(T,x) can be constructed from T
 Thus if j is known, H(Tj,j) can be
constructed. This formula is true but not
provable in L.
 j can be obtained from a Turing machine M
that on input i tests if H(Ti,i) is provable in
L.
 H(Ti,i) can be constructed as above
 From L a theorem prover for L can be
constructed and M can be found
 Thus one can easily construct H(Tj,j)


But H(Tj,j) is true but not provable in L
Constructing H(T,x) for a particular T and x
 Consider this TM:

–
–
–
–
–

pass, 1, pass, 1, R
pass, 0, pass, 1, R
pass, B, del1, B, L
del1, 1, del2, B, L
del2, 1, stop, B, R
Tape: B111011B Start state: pass

Let A be the conjunction of these formulae:
–
–
–
–
–
–
–
tape(0,0)=B tape(1,0)=1 tape(2,0)=1
tape(3,0)=1 tape(4,0)=0 tape(5,0)=1
tape(6,0)=1 (because x = B111011)
(0  i  i  7)  tape(i,0) = B
state(0)=pass
scan(0)=0
ti (scan(t)  i  tape(i,t)=tape(i,t+1))
t (state(t+1)=ns(state(t),tape(scan(t),t)))
t (scan(t+1)=scan(t)+
dir(state(t),tape(scan(t),t)))
–
–
–
–
–
–
t (tape(scan(t),t+1)=
wr(state(t),tape(scan(t),t)))
ns(pass,1)=pass wr(pass,1)=1 dir(pass,1)=1
ns(pass,0)=pass wr(pass,0)=1 dir(pass,0)=1
ns(pass,B)=del1 wr(pass,B)=B dir(pass,B)=-1
ns(del1, 1)=del2 wr(del1, 1)=B dir(del1, 1)=-1
ns(del2, 1)=stop wr(del2, 1)=B dir(del2, 1)=1
Let Hn(T,x) be A  (state(n) =stop)
 For this T and x, H(T,x) is true and T halts
on input x.






In general H(Tj,j) is true but not provable in L
Paradox: How do we know that this formula is
true when L does not know it?
What logical system are we in when we think this
way?
We seem to have the ability to get outside of any
logical system. Thus we must not reason using a
fixed logical system.
Lucas argued on this basis that human thought
processes cannot be simulated by Turing
machines.
Let HL be “human logic”
 If we know HL then we can construct a
formula X that is true but not provable in
HL
 But we just proved X in HL, contradiction.
 Possible solutions:

–
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–
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We can never know HL
HL is not sound.
HL is not effective. (We excel TM in thought.)
We can never know that HL is sound.

All of these seem strange
–
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–
–
Don’t we even know how we think?
Can’t we think logically?
Can we really exceed Turing machines in
reasoning power?
Is our thought process really so subtle that we
cannot know it is sound?
In fact HL does not need to contain all of
human logic.
 HL just needs enough logic to be able to
show that H(Tj,j) is true but not provable
in L.
 This “restricted HL” is known to us and
very likely free from error.
 Either this restricted HL goes beyond TM or
cannot be shown (in HL) to be sound for the
formula H(Tj,j).
 Pick Peano arithmetic as “restricted HL.”


What does “restricted HL” need to derive to
obtain the contradiction?
–
–
It needs to show that Tj halts on input j if and
only if Tj constructs a proof that Tj does not halt
on input j.
It needs to show that if there is a proof that Tj
does not halt on input j then in fact Tj does not
halt on input j.
The first part is straightforward.
 The second part is the problem.

Thus no sufficiently powerful logic L can
derive the statement “(X is provable in L)
implies X.”
 In fact this only has to be derivable for X of
the form H(T,x)
 Any sufficiently powerful finite logic that
can do this, is inconsistent (contradictory).
 But humans use the fact “If X is provable in
L then X” all the time.
 Does this mean our thought processes
cannot be captured by any logical system?

If L is a logic such that L can derive the
statement “(X is provable in L) implies X”
then either L is inconsistent (can derive a
contradiction) or L cannot be represented by
a Turing machine (L is not effective).
 We can try to strengthen L.
 Let L’ be L with the statement
“Tx[(H(T,x) is provable in L) implies
H(T,x)] ” added.
 In L’ one cannot derive the statement
“Tx[(H(T,x) is provable in L’) implies
H(T,x)] ” !

We can also add the statement H(Tj,j) to
L.
 But then for the new L, j will be different
and H(Tj,j) will not be derivable for the
new j!
 For a logic L let G(L) be a statement which
permits H(Tj,j) to be derived. G(L) may
be the statement H(Tj,j) itself or G(L) may
be the statement “Tx[(H(T,x) is
provable in L) implies H(T,x)].”

Then in L  G(L) one can derive H(Tj,j).
But one cannot derive G(L  G(L))!
 Define Li by L0 = L and Li+1 = Li  G(Li).
 Let be the limit (union) of all these logics.
 In L one still cannot derive G(L )!
 Then one can construct L  G(L ) et
cetera. Related to ordinals.
 The process never stops.
 A staircase to infinity.
 An attempt to reach the inexpressible.

We can advance farther and farther but there
is always an infinite distance between us
and ultimate understanding.
 Can a Turing machine construct the same
sequence of logics starting with some sound
logic such as Peano arithmetic?
 What would it mean if it could?
 There is a shortcut to obtain a logic to solve
the halting problem, but it has a drawback:

Let L* be L with all true statements of the
form H(T,x) added.
 Then L* can solve the halting problem.
 If L is Peano arithmetic (assuming it is
consistent) L* is consistent too.
 But L* is not effective. L* does not have a
theorem prover.
 No finite logic can solve the halting
problem.
 The question remains: Do humans excel
Turing machines?


The process of strengthening Peano
arithmetic using Godel’s theorem can be
made more formal:
Let PA be Peano arithmetic.
 Let HL’ be the logic permitting us to apply
Godel’s theorem repeatedly to PA and
obtain PA, PA  G(PA) and other stronger
logics from PA.
 Write HL’  L to mean that logic L can be
obtained from PA by such strengthening.
 Thus HL’  PA.
 Also if HL’  L then HL’  L  G(L)
 Also if HL’  Li for all i and the sequence
Li is effectively computable then HL’ 
iLi

Make HL’ into a logic -- say one can prove
formula H(T,x) if for some logic L, HL’ 
L and H(T,x) can be proved in L.
 Suppose HL’  HL’. Then HL’  HL’ 
G(HL’). Thus G(HL’) is provable in HL’.
But this is a contradiction.
 Suppose it is not true that HL’  HL’. This
seems strange -- if HL’ has a finite
axiomatization then we should be able to
write it down and reason about it.

Because PA is (very likely) sound, HL’ is
also very likely sound.
 If HL’  HL’ then HL’ does not have a
finite axiom system and we excel Turing
machines.
 This is a more concrete version of Lucas’
paradox.

The essence of consciousness -- awareness
of ourselves. The ability to view our own
thought processes as objects “from the
outside.” To infer HL’  HL’.
 The ability to formalize any logical system
we use.
 Can a computer with sensory input be selfaware in this sense?
 Could it be that a computer with sensory
input and the ability to see and think about
itself, excels Turing machines?


Note that such sensing devices would be
able not only to observe the machine M but
also the sensing devices themselves.
Lucas’ view

Goedel's theorem must apply to cybernetical
machines, because it is of the essence of being a
machine, that it should be a concrete instantiation
of a formal system. It follows that given any
machine which is consistent and capable of doing
simple arithmetic, there is a formula which it is
incapable of producing as being true---i.e., the
formula is unprovable-in-the-system-but which we
can see to be true. It follows that no machine can
be a complete or adequate model of the mind, that
minds are essentially different from machines.
Lucas’view (cont’d)

Goedel's theorem seems to me to prove that
Mechanism is false, that is, that minds cannot be
explained as machines. So also has it seemed to
many other people: almost every mathematical
logician I have put the matter to has confessed to
similar thoughts, but has felt reluctant to commit
himself definitely until he could see the whole
argument set out, with all objections fully stated
and properly met.1 This I attempt to do.

For every machine there is a truth which it
cannot produce as being true, but which a
mind can. This shows that a machine cannot
be a complete and adequate model of the
mind. It cannot do everything that a mind
can do, since however much it can do, there
is always something which it cannot do, and
a mind can.

The paradoxes of consciousness arise
because a conscious being can be aware of
itself, as well as of other things, and yet
cannot really be construed as being divisible
into parts. It means that a conscious being
can deal with Goedelian questions in a way
in which a machine cannot, because a
conscious being can both consider itself and
its performance and yet not be other than
that which did the performance.

A machine can be made in a manner of
speaking to "consider" its own performance,
but it cannot take this "into account"
without thereby becoming a different
machine, namely the old machine with a
"new part" added. But it is inherent in our
idea of a conscious mind that it can reflect
upon itself and criticize its own
performances, and no extra part is required
to do this: it is already complete, and has no
Achilles' heel.

We can even begin to see how there could
be room for morality, without its being
necessary to abolish or even to circumscribe
the province of science. Our argument has
set no limits to scientific enquiry: it will still
be possible to investigate the working of the
brain. It will still be possible to produce
mechanical models of the mind.

Only, now we can see that no mechanical
model will be completely adequate, nor any
explanations in purely mechanist terms. We
can produce models and explanations, and
they will be illuminating: but, however far
they go, there will always remain more to
be said. There is no arbitrary bound to
scientific enquiry: but no scientific enquiry
can ever exhaust the infinite variety of the
human mind.12
Lucas’ view

The arguments I put forward in "Minds,
Machines and Goedel" and then in Freedom
of the Will have been much attacked.
Although I put them forward with what I
hope was becoming modesty and a certain
degree of tentativeness, many of the replies
have been lacking in either courtesy or
caution. I must have touched a raw nerve.

In recent years I have been less zealous to
defend myself, and often miss articles
altogether. There may be some new decisive
objection I have altogether overlooked. But
the objections I have come across so far
seem far from decisive.
Encyclopedia article

Roger Penrose claims that this (alleged) difference
between "what can be mechanically proven" and
"what can be seen to be true by humans" shows
that human intelligence is not mechanical in
nature. This view is not widely accepted, because
as stated by Marvin Minsky, human intelligence is
capable of error and of understanding statements
which are in fact inconsistent or false.

However, Marvin Minsky has reported that
Kurt Gödel told him personally that he
believed that human beings had an intuitive,
not just computational, way of arriving at
truth and that therefore his theorem did not
limit what can be known to be true by
humans.
Penrose’s view

The famous "incompleteness" theorem of Gödel
(illustrated by a particularly striking but elementary
example of it known as Goodstein's theorem) tells us
that human mathematical understanding cannot be
encapsulated in any (knowably sound) computational
procedure. This has the implication that there is
something involved in human understanding that lies
beyond the actions of any computer. Understanding is a
particular manifestation of human consciousness, so it
appears that it is conscious mentality that lies outside
computational activity.

Why do I believe that consciousness involves
noncomputable ingredients? The reason is Gödel's
theorem. I sat in on a course when I was a research
student at Cambridge, given by a logician who
made the point about Gödel's theorem that the
very way in which you show the formal
unprovability of a certain proposition also exhibits
the fact that it's true. I'd vaguely heard about
Gödel's theorem — that you can produce
statements that you can't prove using any system
of rules you've laid down ahead of time.

But what was now being made clear to me
was that as long as you believe in the rules
you're using in the first place, then you must
also believe in the truth of this proposition
whose truth lies beyond those rules. This
makes it clear that mathematical
understanding is something you can't
formulate in terms of rules. That's the view
which, much later, I strongly put forward in
my book The Emperor's New Mind.
Criticism of Penrose
In The Emperor's New Mind [Penrose, 1989] and especially
in Shadows of the Mind [Penrose, 1994], Roger Penrose
argues against the “strong artificial intelligence thesis,"
contending that human reasoning cannot be captured by
an artificial intellect because humans detect nontermination
of programs in cases where digital machines do not.
Penrose thus adapts the similar argumentation of Lucas
[1961] which was based on Goedel's incompleteness
results to one based instead on the undecidability of the
halting problem, as shown by Turing [1936]. Penrose's
conclusions have been roundly critiqued, for example, in
[Avron, 1998; Chalmers, 1995a; LaForte et al., 1998;
Putnam, 1995].
In a nutshell, Penrose's argument runs as follows:
1. Collect all current sound human knowledge about nontermination.
2. Reduce said knowledge to a computer program.
3. Create a self-referential version of said program.
4. Derive a contradiction.
The conclusion (by reductio ad absurdum) is that the second
step is invalid: A program cannot incorporate everything
humans know!
(The reasoning is that humans can know that a self-referential
version of this program does not halt, but the computer program
cannot know this.)
Human mathematical intuition
Ramanujan had several extraordinary
characteristics which set him apart from the
majority of mathematicians. One was his lack of
rigor. Very often he would simply state a result
which, he would insist, had just come to him from
a vague, intuitive source, far out of the realm of
conscious probing. In fact, he often said that the
goddess Namagiri inspired him in his dreams.
This happened time and again, and what made it
all the more mystifying -- perhaps even imbuing it
with a certain mystical quality -- was the fact that
many of his “intuition theorems” were wrong.
Mahalanobis: Now here is a problem for you.
Ramanujan: What problem, tell me?
I read out the question from the Strand Magazine.
Ramanujan: Please take down the solution. (He dictated a
continued fraction.)
The first term was the solution I had obtained. Each
successive term represented successive solutions for the same
type of [problem] as the number of houses in the street would
increase indefinitely. I was amazed.
Mahalanobis: Did you get the solution in a flash?
Ramanujan: Immediately I heard the problem, it was clear that
that solution was obviously a continued fraction. I then
thought, “Which continued fraction?” and the answer came to
my mind. It was as simple as this.
Johann Martin Zacharias Dase, who lived from 1824 to 1861 and
was employed by various European governments to perform
computations, is an outstanding example. He not only could
multiply two number each of 100 digits in his head; he also had an
uncanny sense of quantity. That is, he could just “tell”, without
counting, how many sheep were in a field, or words in a sentence,
and so forth, up to about 30 … . Incidentally, Dase was not an
idiot.
The statement H(Tj,j) is true but not
provable in L.
 This means that neither H(Tj,j) nor H(Tj,j)
can be proven in L.
 If A is a statement and neither A nor A can
be proven in L, then A is said to be
undecidable in L.
 Goedel’s theorem gives an undecidable
statement but other such statements are
known.

Other undecidable statements

The combined work of Gödel and Paul Cohen has
given concrete examples of undecidable
statements (statements which can be neither
proven nor disproven): both the axiom of choice
and the continuum hypothesis are undecidable in
the standard axiomatization of set theory. These
results do not require the incompleteness theorem.

In 1973, the Whitehead problem in group theory
was shown to be undecidable in standard set
theory. In 1977, Kirby, Paris and Harrington
proved that a statement in combinatorics, a version
of the Ramsey theorem, is undecidable in the
axiomatization of arithmetic given by the Peano
axioms but can be proven to be true in the larger
system of set theory. Kruskal's tree theorem,
which has applications in computer science, is also
undecidable from the Peano axioms but provable
in set theory. Goodstein's theorem is a relatively
simple statement about natural numbers that is
undecidable in Peano arithmetic.

Gregory Chaitin produced undecidable statements
in algorithmic information theory and in fact
proved his own incompleteness theorem in that
setting.

The complexity (or Kolmogorov complexity) of a
string is the length of the shortest program which,
when run without any input, outputs that string.

Chaitin's incompleteness result: though we know
that most strings are complex in the above sense,
the fact that a specific string is complex can never
be proven (if the string's length is above a certain
threshold). The precise formalization is as follows.

Suppose we fix a particular consistent axiomatic
system for the natural numbers, say Peano's
axioms. Then there exists a constant L (which only
depends on the particular axiomatic system and
the choice of definition of complexity) such that
there is no string s for which the statement
I(s)  L

can be proven within the axiomatic system (even
though, as we know, the vast majority of those
statements must be true).

Thus there are very many true, unprovable
statements in any sufficiently strong axiom
system.
Nonstandard Models
H(Tj,j) is n N Hn(Tj,j)
 This formula is true but not provable in L
 It must be false in a nonstandard model
 Thus there must be a nonstandard integer n
such that Hn(Tj,j) is true.
 This must be so in any sound, effective
logic that can express the integers and
Turing computations.

There is no formula A such that A(x) is true
if and only if x is a standard integer.
 If there were then the formula n N (A(n)
 Hn(T,x)) would hold in all models if and
only if T did not halt on input x
 Thus this formula would be provable if and
only if T does not halt on input x
 This would provide a solution to the halting
problem.


We can’t formally describe the integers.
–
–
–

Any attempt to do so will also have
nonstandard models that describe “infinite”
integers
Then how do we know what the “standard”
integers are?
Is this evidence of intuition that goes beyond
Turing machines?
No formal system can even formally
describe what it means for something to be
finite.





         
      

       
 


 { nonstandard integers … }
{standard integers}
1
2
3
4 5 6 7 8 9 10
In fact in any formal system there are
models of the real numbers that have the
same size as the integers
 But we know there are more reals than
integers (a bigger infinity)
 So no formal system can fully capture the
idea of an infinity larger than the integers
 Then what makes us think we know what
such infinities are if we can’t fully describe
them? Do we excel Turing machines again?


Gödel's theorem thus shows that there must
always exist such unusual, unintended
interpretations of the system; as Henkin
says, quoted in [Turquette 50]:
We tend to reinterpret Gödel's incompleteness result as
asserting not primarily a limitation on our ability to
prove but rather on our ability to specify what we mean
... when we use a symbolic system in accordance with
recursive rules [Gödel & the synthetic a priori].
Damjan Bojadziev, Mind versus Gödel, in M. Gams, M.
Paprzycki and X. Wu (eds.), Mind Versus Computer,
IOS Press 1997, pp. 202-210.
–


Similarly, Polanyi says, though only in
connection with the second theorem:
–
we never know altogether what our axioms
mean [Personal Knowledge, p. 259]. We must
commit ourselves to the risk of talking
complete nonsense if we are to say anything at
all within any such system [p. 94].

(from Mind versus Gödel)

Bojadziev concludes that our self awareness is
limited (we can never know HL):


Applied to minds, it would translate to some
principled limitation of the reflexive cognitive
abilities of the subject: certain truths about oneself
must remain unrecognized if the self-image is to
remain consistent [Hofstadter 79, p. 696].
Conclusion: It is not possible to see oneself
completely, in the literal, metaphorical
("see=understand"), formal and computational
sense of the word. Gödel's theorems do not
prevent the construction of formal models of the
mind, but support the conception of mind (self,
consciousness) as something which has a special
relation to itself, marked by specific limitations.
How many true but unprovable statements
are there?
 Are they rare or common?
 Chaitin showed that they are very common
in any formal system.
 Why then does this not cause more of a
problem for mathematics research?
