Incompleteness in the natural world


Gödel Incompleteness Theorem
Gödel Incompleteness Theorem (Photo credit: janoma.cl)

A post inspired by Godel, Escher, Bach, Complexity: A Guided Tour, an article in this week’s New Scientist about the clash between general relativity and quantum mechanics and personal humiliation.

The everyday incompleteness: This is the personal humiliation bit. For the first time ever I went on a “Parkrun” today – the 5km Finsbury Park run, but I dropped out after 2.5km 2km – at the top of a hill and about 250 metres from my front door – I simply thought this is meant to be a leisure activity and I am not enjoying it one little bit. I can offer some excuses – it was really the first time ever I had run outdoors and so it was a bit silly to try a semi-competitive environment for that, I had not warmed up properly and so the first 500 metres were about simply getting breathing and limbs in co-ordination – mais qui s’excuse, s’accuse.

But the sense of incompleteness I want to write about here is not that everyday incompleteness, but a more fundamental one – our inability to fully describe the universe, or rather, a necessary fuzziness in our description.

Let’s begin with three great mathematical or scientific discoveries:

The diagonalisation method and the “incompleteness” of the real numbers: In 1891 Georg Cantor published one of the most beautiful, important and accessible arguments in number theory – through his diagonalisation argument, that proved that the infinity of the real numbers was qualitatively different from and greater than the infinity of the counting numbers.

The infinity of the counting numbers is just what it sounds like – start at one and keep going and you go on infinitely. This is the smallest infinity – called aleph null (\aleph_0 ).

Real numbers include the irrationals – those which cannot be expressed as fractions of counting numbers (Pythagoras shocked himself by discovering that \sqrt 2 was such a number). So the reals are all the numbers along a counting line – every single infinitesimal point along that line.

Few would disagree that there are, say, an infinite number of points between 0 and 1 on such a line. But Cantor showed that the number was uncountably infinite – i.e., we cannot just start counting from the first point and keep going. Here’s a brief proof…

Imagine we start to list all the points between 0 and 1 (in binary) – and we number each point, so…

1 is 0.00000000…..
2 is 0.100000000…..
3 is 0.010000000……
4 is 0.0010000000….
n is 0.{n – 2 0s}1{000……}

You can see this can go on for an infinitely countable number of times….

and so on. Now we decide to ‘flip’ the o or 1 at the index number, so we get:

1 is 0.1000000….
2 is 0.1100000….
3 is 0.0110000….
4 is 0.00110000….

And so on. But although we have already used up all the counting numbers we are now generating new numbers which we have not been able to count – this means we have more than \aleph_0 numbers in the reals, surely? But you argue, let’s just interleave these new numbers into our list like so….

1 is 0.0000000….
2 is 0.1000000…..
3 is 0.0100000….
4 is 0.1100000….
5 is 0.0010000….
6 is 0.0110000….

And so on. This is just another countably infinite set you argue. But, Cantor responds, do the ‘diagonalisation’ trick again and you get…

1 is 0.100000…..
2 is 0.110000….
3 is 0.0110000….
4 is 0.1101000…
5 is 0.00101000…
6 is 0.0110010….

And again we have new numbers, busting the countability of the set. And the point is this: no matter how many times you add the new numbers produced by diagonalisation into your counting list, diagonalisation will produce numbers you have not yet accounted for. From set theory you can show that while the counting numbers are of order (analogous to size) \aleph_0 , the reals are of order 2^{\aleph_0} , a far far bigger number – literally an uncountably bigger number.

Gödel’s Incompleteness Theorems: These are not amenable to a blog post length demonstration, but amount to this – we can state mathematical statements we know to be true but we cannot design a complete proof system that incorporates them – or we can state mathematical truths but we cannot build a self-contained system that proves they are true. The analogy with diagonalisation is that we know how to write out any real number between 0 and 1, but we cannot design a system (such as a computer program) that will write them all out – we have to keep ‘breaking’ the system by diagonalising it to find the missing numbers our rules will not generate for us. Gödel’s demonstration of this in 1931 was profoundly shocking to mathematicians as it appeared to many of them to completely undermine the very purpose of maths.

Turing’s Halting Problem: Very closely related to both Gödel’s incompleteness theorems and Cantor’s diagonalisation proof is Alan Turing’s formulation of the ‘halting problem’. Turing proposed a basic model of a computer – what we now refer to as a Turing machine – as an infinite paper tape and a reader (of the tape) and writer (to the tape). The tape’s contents can be interpreted as instructions to move, to write to the tape or to change the machine’s internal state (and that state can determine how the instructions are interpreted).

Now such a machine can easily be made of go into an infinite loop e.g.,:

  • The machine begins in the ‘start’ state and reads the tape.  If it reads a 0 or 1 it moves to the right and changes its state to ‘even’.
  • If the machine is in the state ‘even’ it reads the tape. If it reads a 0 or 1 it moves to the left and changes its state to ‘start’

You can see that if the tape is marked with two 0s or two 1s or any combination of 0 or 1 in the first two places the machine will loop for ever.

The halting problem is this – can we design a Turing machine that will tell us if a given machine and its instructions will fall into an infinite loop? Turing proved  we cannot without having to discuss any particular methodology … here’s my attempt to recreate his proof:

We can model any other Turing machine though a set of instructions on the tape, so if we have machine T we can have have it model machine M with instructions I : i.e., T(M, I)

Let us say T can tell whether M will halt or loop forever with instructions I – we don’t need to understand how it does it, just suppose that it does. So if (M, I) will halt T writes ‘yes’, otherwise it writes ‘no’.

Now let us design another machine T^\prime that takes T(M,I) its input but here T^\prime loops forever if T writes ‘yes’ and halts if T writes ‘no’.

Then we have:

M(I) halts or loops – T(M, I) halts – T^\prime loops forever.

But what if we feed T^\prime the input of T^\prime(T(M, I)?

M(I) halts or loops – T(M, I) halts – T^\prime(T(M,I)) loops forever – T^\prime(T^\prime(T(M,I))) – ??

Because if the second T^\prime(T^\prime(T(M,I))) halted then that would imply that the first had halted – but it is meant to loop forever, and so on…

As with Gödel we have reached a contradiction and so we cannot go further and must conclude that we cannot build a Turing machine (computer) that can solve the halting problem.

Quantum mechanics: The classic, Copenhagen, formulation of quantum mechanics states that the uncertainty of the theory collapses when we observe the world, but the “quantum worlds” theory suggests that actually the various outcomes do take place and we are just experiencing one of them at any given time. The experimental backup for the many worlds theory comes from quantum ‘double-slit’ experiments which suggest particles leave traces of their multiple states in every ‘world’.

What intrigues me: What if our limiting theories – the halting problem, Gödel’s incompleteness theorem, the uncountable infinite, were actually the equivalents of the Copenhagen formulation and, in fact, maths was also a “many world” domain where the incompleteness of the theories was actually the deeper reality – in other words the Turing machine can both loop forever and halt? This is probably, almost certainly, a very naïve analogy between the different theories but, lying in the bath and contemplating my humiliation via incompleteness this morning, it struck me as worth exploring at least.

A (partial) answer to my Goedelian conundrum?


Douglas Hofstadter in Bologna, Italy
Douglas Hofstadter in Bologna, Italy (Photo credit: Wikipedia)

Last week I puzzled over what seemed to me to be the hand waiving dismissal, by both Alan Turing and Douglas Hofstadter of what I saw as the problem of humans being able to write true statements that the formal systems employed by computers could not determine – the problem thrown up by Goedel’s Incompleteness Theorems.

Well, Douglas Hofstadter has now come to his own (partial) rescue as I continue to read on through Godel, Escher, Bach – as he describes Tarski’s Theorem, which essentially states that humans cannot determine all such statements either (unless we posit the Church-Turing thesis is wrong, of course and there is some inner human computational magic we have yet to describe).

I am now going to quickly run through Hofstadter’s exposition – it might not mean too much to those of you not familiar with GEB, but if so and if you are interested in computation (and genetics and music) and you want to improve your mind this summer you could always think about buying the book. I don’t promise it’s an easy read, the style can vary from the nerdy to the deeply frustrating, but it is still a rewarding read.

So here goes:

We imagine we have a procedure that can determine the truth of a number theoretical statement, i.e.:

TRUE\{a\} can tell us whether the number theoretical statement with Goedel number a is true.

So now we posit, T with Goedel number t :

\exists a:<\sim TRUE\{a\} \wedge ARITHMOQUINE\{a^{\prime\prime},a\}>

Now, of you have read GEB you will know that to “arithmoquine” a number theoretical statement is to replace the free variable – in this case a^{\prime\prime}, with the Goedel number for the statement itself…

\exists a:<\sim TRUE\{a\} \wedge ARITHMOQUINE\{SSS \ldots SSS0,a\}>

Which we can state as “the arithmoquinification of t is the Goedel number of a false statement”.

But the arithmoquinification of t is T ‘s own number, so this statement is the equivalent of saying “this statement is false”: just another version of the famous Epimenides Paradox, but one that is decidedly not hand waving in form: it’s about natural numbers.

The outcome is that TRUE\{a\} cannot exist without our whole idea of natural numbers collapsing and we are forced to conclude there is no formal way of deciding what is a true statement of theoretical number theory using only theoretical number theory – and so humans are no better off than computers in this regard: we use concepts from without the formal theory to establish truth and we could surely program our computers to do the same. Turing’s “imitation game” conception of intelligent machines thus survives.

At least I think so!

“Ultimate L” explained


Hugh Woodin, American mathematician, at Berkeley
Image via Wikipedia

Well, that is what I wanted to write about – Hugh Woodin‘s theories of infinity, and indeed I found a blog that discussed a lot of this – http://caicedoteaching.wordpress.com/2010/10/19/luminy-hugh-woodin-ultimate-l-i/

But that is somewhat beyond me! And there does not appear to be a wikipedia article on this, so you may have to wait for some time before you see me try to take it on.

Interestingly, Woodin’s wikipedia entry, accurately, suggests that his earlier work was sceptical as to the truth of Cantor’s continuum hypothesis  – something that his latest work turns on its head.

Goedel’s Incompeteness theorem surpassed?


Portrait of Kurt Gödel, one of the most signif...
Image via Wikipedia

Gödel’s Incompleteness Theorems are one of the cornerstone’s of modern mathematical thought but it is also a major blot on the mathematical landscape – as it establishes an inherent limit on the ability of mathematicians to describe the mathematical world: the first theorem (often thought of as the theorem) states that no consistent (ie self-contained) axiomatic system is capable to describing all the facts about natural numbers.

To today’s physical scientists – used to concepts such as relativity and quantum uncertainty – the broad idea that there could be an uncertainty at the heart of mathematics is maybe not so difficult to take, but it is fair to say it broke a lot of mathematical hearts in the 1930s when first promulgated. (This book – Godel’s Proof – offers an excellent introduction for the non-mathematician who is mathematically competent – ie like me!).

Gödel thought at the time that this kink in mathematical reality could be smoothed out by a better understanding of infinities in mathematics – and, according to the cover article in this week’s New Scientist (seemingly only available online to subscribers), by Richard Elwes, it is now being claimed by Hugh Woodin of UC Berkeley that just that has been shown.

Along the way, this new hypothesis of “Ultimate L” also demonstrates that Cantor’s continuum hypothesis is correct. I do not claim to understand “Ultimate L”, and in any case, as is their style, the New Scientist don’t print the proof, they just describe it in layman’s terms. I do have a basic understanding of the continuum hypothesis, though, and so can show the essential points that “Ultimate L” claims to have found.

Georg Cantor showed that there were multiple infinities, the first of which, so-called \aleph_0 (aleph null) is the infinity of the countable numbers – eg 1, 2, 3… and so on. Any infinite set that can be paired to a countable number in this way has a cardinality of \aleph_0 . (And, as the New Scientist point out, this is the smallest infinity – eg if you thought that, say, there must be half as many even numbers as there are natural numbers, you are wrong – the set of both is of cardinality \aleph_0 – 2 is element 1 of the set, 4, is element 2 and so on: ie a natural number can be assigned to every member of the set of even numbers and so the set is of cardinality \aleph_0 .)

The continuum hypothesis concerns what might be \aleph_1 – the next biggest infinity. Cantor’s hypothesis is that \aleph_1 is the real numbers (the continuum): I discuss why this is infinite and a different infinity from the natural numbers here.

We can show that this set has cardinality 2^{\aleph_0} – a number very much bigger than \aleph_0 . But is there another infinity in between?

Mathematicians have concentrated on looking at whether any projections (a word familiar to me now from relational algebra) of the set of reals has a cardinality between \aleph_0 and 2^{\aleph_0} – if they did then it would be clear the reals could not have the cardinality of \aleph_1 – but some other, higher, \aleph .

No projections with a different cardinality have been found, but that is not the same as a proof they do not exist. But if Woodin’s theory is correct then none exist.

(Just one more chance to plug the brilliant Annotated Turing: if you are interested in computer science you should really read it! This is the book that first got me interested in all this.)

Gödel, Turing and decidability


Portrait of Kurt Gödel, one of the most signif...
Image via Wikipedia

I am still wandering around in the world opened to me by The Annotated Turing
– perhaps a little lost, but I have been reading some guide books.

The most recent of these has been Gödel’s Proof which gives a gentle(ish) introduction to Gödel’s incompleteness theorem.

This is a short, but fascinating book that I repeatedly made the mistake of trying to read when tired or not able to give it my full attention (eg on a flight from London to Budapest last week when three days of hard work had really taken it out of me). But I finally managed to finish it last night.

I think the really startling point it makes – which admittedly Charles Petzold’s book also makes but I didn’t fully grasp at the time – is about the nature of the human mind.

Gödel’s theorem states that no consistent system of axioms whose theorems can be listed by an “effective procedure” (such as a computer program) is capable of proving all facts about the natural numbers.

Before I read the book I had not really thought about what this meant: after all Turing and Church had shown that most numbers/mathematical problems were not computable and so this seemed of a part with that conclusion.

But the key thing here is that metamathematics can show the correctness of theorems that axiomatic proofs cannot. In other words – I think – that computer programs are actually a poor model of our ability to solve mathematical problems.