[time 147] Re: [time 81] Discreteness and p-adics

Matti Pitkanen (matpitka@pcu.helsinki.fi)
Sun, 4 Apr 1999 19:54:53 +0300 (EET DST)

On Sun, 4 Apr 1999, Ben Goertzel wrote:

> >p-Adic numbers with norm bounded by some upper bound
> >are of from x= p^rk , where k is possibly *infinite* integer
> >having pinary expansion k= SUM(n>=0) k_np^n. These numbers form a
> >discrete set! The real counterparts of these numbers in canonical
> >identification SUM x_np^n --> SUM x_np^(-n) however form a continuum!
> I don't understand this. It sounds like you are saying that a discrete set
> (cardinality aleph null) is isomorphic to a continuum (cardinality aleph one),
> which is impossible, and so obviously is not what you're really saying!!!

The point is that k can be also *infinite* as integer: these have finite
p-adic norm also. For instance 1/1-p =1+p+p^2+.... is completely well
defined p-adic number with p-adic norm one.

One could argue that discreteness is only apparent but the fact p-adic
Fourier basis obeys Kronecker delta normalization suggests that this is
not the case.


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