By the fundamental theorem of arithmetic, we know that any positive integer can be uniquely defined by its prime factors. Now, suppose $S_{\infty}$ is the set of all primes, and let $s_i \in S$ such that $s_0=2$, $s_1=3$, $\ldots$. Further, let $S_i = \{s_0,s_1,\ldots s_i\}$. Let $A$ be the set of nonnegative integers, and $a_i \in A$ where $a_m$ and $a_n$ are not necessarily distinct. Consider the set $$U_n = \{\Pi_{0\leq i \leq n} {s_i}^{a_i}| a_i \in \mathbb{Z}^+ \cup \{0\}, s_i \in S_i\}.$$ The definition of $U_{\infty}$ naturally follows. Now, my question is: without calculations, is it possible for us to sort any $U_n$? i.e., given some $u \in U_n$, can I determine the smallest $v \in U_n$ with $v>u$ without doing any direct calculations?
This question arose because of my intuitive feeling that if primes are building blocks of the integers, then we should be able to sort the integers solely using their primitive representations: the prime factorizations. Thanks in advance for any help.
