You are currently browsing the category archive for the ‘TCS’ category.

[Cross-posted from goodreads]

If Godel proved that no sufficiently complex system, i.e one that is capable of arithmetic, can prove its own consistency or if you assume the system is consistent there will always exist (infinitely many) true statements that cannot be deduced from its axioms, in what system did he prove it in? Is that system consistent? In what sense is the Godel statement true if not by proof? You’ll have hundreds of questions popping in your mind every few minutes, and this short book does a very good job of tackling most of them.

Godel numbering is a way to map all the expressions generated by the successive application of axioms back onto numbers, which are themselves instantiated as a “model” of the axioms. The hard part of it is to do this by avoiding the “circular hell”. Russell in Principia Mathematica tried hard to avoid the kind of paradoxes like “Set of all elements which do not belong to the set”. Godel’s proof tries hard to avoid more complicated paradoxes like this :

Let p = “Is a sum of two primes” be a property some numbers might possess. This property can be stated precisely using axioms, and symbols can be mapped to numbers. ( for e.g open a text file, write down the statement and look at its ASCII representation ). The let n(p) be the number corresponding to p. If n(p) satisfies p, then we say n(p) is Richardian, else not. Being Richardian itself is a meta-mathematical property r = “A number which satisfies the property described by its reverse ASCII representation”. Note that it is a proper statement represented by the symbols that make up your axioms. Now, you ask if n(r) is Richardian, and the usual problem emerges : n(r) is Richardian iff it is not Richardian. This apparent conundrum, as the authors say, is a hoax. We wanted to represent arithmetical statements as numbers, but switched over to representing meta-mathematical statements as numbers. Godel’s proof avoid cheating like this by carefully mirroring all meta-mathematical statements within the arithmetic, and not just conflating the two. Four parts to it.

1. Construct a meta-mathemtical formula G that represents “The formula G is not demonstratable”. ( Like Richardian )

2. G is demonstrable if and only if ~G is demonstrable ( Like Richardian)

3. Though G is not demonstrable, G is true in the sense that it asserts a certain arithmetical property which can be exactly defined. ( Unlike Richardian ).

4. Finally, Godel showed that the meta-mathematical statement “if ‘Arithmetic is consistent’ then G follows” is demonstrable. Then he showed that “Arithmetic is consistent” is not demonstrable.

It took me a while to pour over the details, back and forth between pages. I’m still not at the level where I can explain the proof to anyone clearly, but I intend to get there eventually. Iterating is the key.

When I first came across Godel’s theorem, I was horrified, dismayed, disillusioned and above all confounded – how can successively applying axioms over and over not fill up the space of all theorems? Now, I’m slowly recuperating. One non-mathematical, intuitive, consoling thought that keeps popping into my mind is : If the axioms to describe arithmetic ( or something of a higher, but finite complexity ) were consistent and complete, then why those axioms? Who ordained them? Why not something else? If it turned out that way, then the question of which is more fundamental : physics or logic would be resolved. I would be shocked if it were possible to decouple the two and rank them – one as more fundamental than the other. I’m very slowly beginning to understand why Godel’s discovery was a shock to me.

You see, I’m good at rolling with it while I’m working away, but deep down, I don’t believe in Mathematical platonism, or logicism, or formalism or any philosophical ideal that tries to universally quantify.