I'm not so sure that I do perceive positive integers directly. But regardless of that, I remain convinced that all properties of them that I can perceive can be written as a piece of ASCII text.
The description doesn't need to be axiomatic, mind you. As I have mentioned, the Schmidhuber ensemble of descriptions is larger than the Tegmark ensemble of axiomatic systems. Cheers Hal Finney wrote: > > But as an example, how about "the positive integers"? That's a pretty > simple description. Just start with 0 and keep adding 1. > > >From what we understand of Godel's theorem, no axiom system can capture > all the properties of this mathematical structure. Yet we have an > intuitive understanding of the integers, which is where we came up with > the axioms in the first place. Hence our understanding precedes and is > more fundamental than the axioms. The axioms are the map; the integers > are the territory. We shouldn't confuse them. > > We have a direct perception of this mathematical structure, which is > why I am able to point to it for you without giving you an axiomatic > description. > > Hal Finney > ---------------------------------------------------------------------------- A/Prof Russell Standish Director High Performance Computing Support Unit, Phone 9385 6967, 8308 3119 (mobile) UNSW SYDNEY 2052 Fax 9385 6965, 0425 253119 (") Australia [EMAIL PROTECTED] Room 2075, Red Centre http://parallel.hpc.unsw.edu.au/rks International prefix +612, Interstate prefix 02 ----------------------------------------------------------------------------