Diagonalisation & +1 Hypothesis

Note...
The proof that the power series of S is always greater than S means that the set of all sets is a contradiction. This is analogous to the SRH that there is no absolute theory

Diagonalisation

Turning to Hofstadter he gives an excellent account of diagonalisation in GEB and offers his proof using diagonalisation that some functions taking 1 input cannot be computed.

If a table is made of all functions against their output for all the natural numbers (computers can only work in natural numbers - the decimals they process are really just natural numbers and an index) then the output of a function not in the list can be created very easily. Take the output of program 1 for input 1 and make the new non-computable function output some other value (say +1). Then for program 2 take its output for an of 2 and make the new function output something else (again say +1). Do this for all functions in the list. The new function is then not in the list because it differs from all the functions in the list at, at least, the value we have defined it to be different. This is diagonalisation and means that there is always at least 1 function that is not in the list.

So the +1 hypothesis was that any system would need to be larger than itself to model itself and the notion of itself hence +1. The basic idea that the meta level must be larger than the level, if only because it needs to accomodate (encode) the new idea of the self. If we make the level self-referential is this true... need think...

Is it not true that we can construct a Turing machine not in the list of all Turing machines using the Diagonalisation argument? Take the output of each Turing machine for each input, and +1 to the output ... what if it doesn't output ok the halting problem... enter Turing himself... what if we had a special code to say it didn't output (if we had an infinite amount of time) so it did effectively output? say {0,0} if it didn't output and {1,output} if it did. Creating a pair is like creating a meta-level interesting...

So a three level system would be {1,2,3} where 2 could refer to system 3 and 1 refer to system 2... whatever

still to read... got lunch
http://plus.maths.org/issue5/turing/