Formalizing the Fundamental Theorem of Arithmetic
- From: mmweiss@xxxxxx
- Date: 23 May 2007 14:59:51 -0700
I was wondering how to state the Fundamental Theorem of Arithmetic in
First-order Logic.
The Theorem says: ``Every positive integer can be uniquely written as
a product of powers of distinct primes (up to ordering)."
With more gobbledigook: Let P map the nonnegative integers into the
primes in strictly increasing order. Forall x, there's a unique
mapping F from N to N and a positive integer n such that x is the
product of all the P_{i}^{F(i)} with 0\leq i \leq n, i.e.,
x = prod_{0\leq i \leq x} P_{i}^{F(i)}
or in English, such that
This is not first-order logic.
Of course, we only *need* to invoke the existence of an F whose domain
is an initial segment of N. Such F with finite domain can be "coded"
as finite sequences hence as numbers. (E.g., as 2^{F(0)} * 3^{F(1)}
*... * P_n^{F(n)}.)
But, is it possible to state the theorem *without* coding?
What I'm looking for is a formula in the language of first-order
arithmetic that neatly expresses the mathematical content of the
theorem.
Thanks,
Max
.
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