Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Integer Complexity and Well-Ordering   Define $\|n\|$ to be the complexity of $n$, the smallest number of ones
needed to write $n$ using an arbitrary combination of addition and
multiplication. John Selfridge showed that $\|n\| \ge 3\log_3 n$ for all $n$.
Define the defect of $n$, denoted $\delta(n)$, to be $\|n\| - 3\log_3 n$. In
this paper, we consider the set $\mathscr{D} := \{\delta(n): n \ge 1 \}$ of all
defects. We show that as a subset of the real numbers, the set $\mathscr{D}$ is
well-ordered, of order type $\omega^\omega$. More specifically, for $k\ge 1$ an
integer, $\mathscr{D}\cap[0,k)$ has order type $\omega^k$. We also consider
some other sets related to $\mathscr{D}$, and show that these too are
well-ordered and have order type $\omega^\omega$.

24
The categories of this paper are: dg-ga
0
The categories of this paper are: hep-ph
4
The categories of this paper are: cond-mat
6
The categories of this paper are: astro-ph
32
The categories of this paper are: atom-ph
8
The categories of this paper are: hep-ex
20
The categories of this paper are: acc-phys
30
The categories of this paper are: q-alg
26
The categories of this paper are: bayes-an
18
The categories of this paper are: econ
35
The categories of this paper are: mtrl-th
1
The categories of this paper are: cs
15
The categories of this paper are: nucl-ex
12
The categories of this paper are: nucl-th
7
The categories of this paper are: hep-th
9
The categories of this paper are: nlin