Comment by FilosofumRex
3 days ago
One, in theory, can construct number sets (fields) with holes in them - that's truly discrete numbers. such number sets are at most countably infinite, but need not be. One useful such set might have Planck's (length) constant as its smallest number beyond which there is a hole. The problem with using such number sets is that ordinary rules of arithmetic breakdown, ie division has to be defined as modulus Planck constant
Such a thing would not be a field.
You can define an additive group $\frac{1}{n}\mathbb{Z}$ if you like. However (for $n > 1$) it would not even be a ring, because it would not be closed under multiplication. (It's closure under multiplication would be $\mathbb{Z}[\frac{1}{n}]$, which would not have a smallest positive element, contrary to your design criterion.)
(Of course, you could define a partial multiplication on it. I don't think there's a good name for such a thing. I guess you could just call it "a subgroup of the rational numbers under addition, equipped with a partial multiplication operation that is defined and agrees with the usual multiplication on rational numbers when the result would still be in the subgroup")
The field Qp of p-adic numbers is complete with respect to the p-adic norm, but is not ordered in the same sense as field of real numbers. It's still uncountable infinite. If there is a sense in which "gaps" or Holes can be introduced without breaking its completeness, that would make it very useful for modeling reality
p-adics may be useful, yes, as may other fields.
They do not constitute a field with a smallest non-zero element.