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.

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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
>
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