Doesn't this mean that 1) There exist mathematics (systems of logic) for which there is no way to reach logically from what we know now to how this system work

Doesn't this mean that 1) There exist mathematics (systems of logic) for which there is no way to reach logically from what we know now to how this system works, it must be intuited 2) But does this mean that reality is embedded in one of these mathematics? This is not necessary as far as I can see, so this still leaves the door open for a theory of everything in the physical world (string theory or some such). 3) But it does seem to imply that no infinite system can be described by a finite system ( a restatement of Godel ). So we can't set up a computer program to run the universe - no real surprise there. or are the implications bigger? from http://plus.maths.org/issue37/features/omega/feat.pdf http://plus.maths.org/issue37/features/omega/feat.pdf "To put it bluntly, if the incompleteness phenomenon discovered by Gödel in 1931 is really serious and I believe that Turing's work and my own work suggest that incompleteness is much more serious than people think then perhaps mathematics should be pursued somewhat more in the spirit of experimental science rather than always demanding proofs for everything. Maybe, rather than attempting to prove results such as the celebrated Riemann hypothesis, mathematicians should accept that they may not be provable and simply accept them as an axiom" So, for example, if we have the axioms of set theory, then for any theorem, it may not be possible to prove this theorem from the axioms as a set of linear deductions, somewhere along the line we may find a new theorem that requires to be stated as axiomatic, i.e. we have axioms A,B we have theorems C,D,E provable from A,B then we find F which seems to be true, but we can't prove F from A,B, F must be stated as an axiom, probably not that surprising really. Again just because systems like this exist, doesn't mean we live in one.