Supernatural numbers
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In mathematics, the supernatural numbers (sometimes called generalized natural numbers or Steinitz numbers) are a generalization of the natural numbers.
A supernatural number ω is a formal product:
where p runs over all prime numbers, and each np is either a natural number or infinity. Sometimes we write vp(ω) for np. If no 
and there are only a finite number of non-zero np then we recover the natural numbers. Supernatural numbers extend beyond natural numbers by allowing the possibility of infinitely many prime factors, and by allowing any given prime to divide ω "infinitely often," by taking that prime's corresponding exponent to be the symbol 
.
There is no natural way to add supernatural numbers, but they can be multiplied, with 
. Similarly, the notion of divisibility extends to the supernaturals with 
if 
for all p. We can also generalize the notion of the least common multiple and greatest common divisor for supernatural numbers, by defining
With these definitions, we can now take the gcd or lcm of infinitely many natural numbers to get a supernatural number. We can also extend the usual p-adic order functions to supernatural numbers by defining vp(ω) = np for each p
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.
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