Supernatural numbers

In mathematics, supernatural numbers are a set of numbers, which together with the set of natural numbers, forms the generalized natural numbers. Supernatural numbers are closely tied to Gödel's incompleteness theorems, and are useful for representing different kinds of infinitely large numbers.

In particular, a generalized natural number $omega$ is a formal product:

: $omega\; =\; prod\_p\; p^\{n\_p\},$

where $p$ runs over all prime numbers, and each $n\_p$ is either a natural number or infinity. If no $n\_p\; =\; infty$ and there are only a finite number of non-zero $n\_p$ then we recover the natural numbers. If one or more $n\_p\; =\; infty$ or there are an infinite number of non-zero $n\_p$ then the resulting number is a supernatural number. Supernatural numbers extend beyond natural numbers by allowing the possibility of infinitely many prime factors, and by allowing any given prime to divide $omega$ "infinitely often," by taking that prime's corresponding exponent to be the symbol $infty$.

We can extend the usual $p$-adic order functions to supernatural numbers by defining $v\_p(omega)=n\_p$ for each $p$. and extend the notion of divisibility by declaring $omega\_1midomega\_2$ if $v\_p(omega\_1)leq\; v\_p(omega\_2)$ for all $p$. Finally, we can also generalize the notion of the least common multiple and greatest common divisor for supernatural numbers, by defining

: $displaystyle\; operatorname\{lcm\}(\{omega\_i\})\; displaystyle\; =prod\_p\; p^\{sup(v\_p(omega\_i))\}$

: $displaystyle\; operatorname\{gcd\}(\{omega\_i\})\; displaystyle\; =prod\_p\; p^\{inf(v\_p(omega\_i))\}$

With these definitions, we can now take the gcd or lcm of infinitely many natural numbers to get a supernatural number.

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 and can be further extended for use in analysis through superreal numbers

References

* [http://planetmath.org/encyclopedia/LcmOfSupernaturalNumbers.html Planet Math: Supernatural number]
*Douglas Hofstadter, 1979. "". Vintage Books. ISBN 0465026850. 1999 reprint: ISBN 0465026567. MathSciNet|80j:03009