## Journal of Symbolic Logic

### Prime Numbers and Factorization in $\mathrm{IE}_1$ and Weaker Systems

Stuart T. Smith

#### Abstract

We show that $\mathrm{IE}_1$ proves that every element greater than 1 has a unique factorization into prime powers, although we have no way of recovering the exponents from the prime powers which appear. The situation is radically different in Bezout models of open induction. To facilitate the construction of counterexamples, we describe a method of changing irreducibles into powers of irreducibles, and we define the notion of a frugal homomorphism into $\hat\mathbb{Z} = \Pi_p\mathbb{Z}_p$, the product of the $p$-adic integers for each prime $p$.

#### Article information

Source
J. Symbolic Logic, Volume 57, Issue 3 (1992), 1057-1085.

Dates
First available in Project Euclid: 6 July 2007

https://projecteuclid.org/euclid.jsl/1183744058

Mathematical Reviews number (MathSciNet)
MR1187466

Zentralblatt MATH identifier
0768.03022

JSTOR
Smith, Stuart T. Prime Numbers and Factorization in $\mathrm{IE}_1$ and Weaker Systems. J. Symbolic Logic 57 (1992), no. 3, 1057--1085. https://projecteuclid.org/euclid.jsl/1183744058