In mathematics, the supernatural numbers, sometimes called generalized natural numbers or Steinitz numbers, are a generalization of the natural numbers. They were used by Ernst Steinitz in 1910 as a part of his work on field theory.
A supernatural number formula_1 is a formal product:
 formula_2
where formula_3 runs over all prime numbers, and each formula_4 is zero, a natural number or infinity. Sometimes formula_5 is used instead of formula_4. If no formula_7 and there are only a finite number of non-zero formula_4 then we recover the positive integers. Slightly less intuitively, if all formula_4 are formula_10, we get zero. Supernatural numbers extend beyond natural numbers by allowing the possibility of infinitely many prime factors, and by allowing any given prime to divide formula_1 "infinitely often," by taking that prime's corresponding exponent to be the symbol formula_10.
There is no natural way to add supernatural numbers, but they can be multiplied, with formula_13. Similarly, the notion of divisibility extends to the supernaturals with formula_14 if formula_15 for all formula_3. The notion of the least common multiple and greatest common divisor can also be generalized for supernatural numbers, by defining
 formula_17
and
 formula_18.
With these definitions, the gcd or lcm of infinitely many natural numbers (or supernatural numbers) is a supernatural number.
We can also extend the usual formula_3-adic order functions to supernatural numbers by defining formula_20 for each formula_3.
Supernatural numbers are used to define orders and indices of profinite groups and subgroups, in which case many of the theorems from finite group theory carry over exactly. They are used to encode the algebraic extensions of a finite field.
Supernatural numbers also arise in the classification of uniformly hyperfinite algebras.
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