# Well-ordered reference

I'm wondering if there is a standard reference for the following straightforward facts about well-orderings. (They are quite easy to prove, I just need a handy/standard reference.)

1. Let $(S,<)$ be a well-ordered set. The set $T=\{(s_1,s_2,\ldots, s_k)\ : \ k\in \mathbb{N},s_i\in S\}$, of all ordered finite tuples, is well-ordered in the degree-lexicographical ordering $\prec$. In other words, given different elements $\vec{a}=(a_1,a_2,\ldots, a_m),\vec{b}=(b_1,b_2,\ldots, b_n)\in T$ then we have $\vec{a}\prec \vec{b}$ exactly when either (1) $m<n$ or (2) $m=n$, and if $i$ is the first index where $a_i\neq b_i$, then $a_i<b_i$.

2. Let $(S,<)$ be a well-ordered set. The set $U$ of all finite subsets of $S$ is well-ordered by the following relation: given different finite subsets $S_1,S_2\subset S$ we say $S_1\prec S_2$ if it happens that the maximal element of (the finite set) $(S_1\cup S_2)\setminus (S_1\cap S_2)$ belongs to $S_2$.

• Not a set theorist, but I think these are both sufficiently standard that no reference would be needed. By the way, in #2 you can identify finite subsets of $S$ with functions from $S$ into $\{0,1\}$, and what you have described is the ordinal exponent $\{0,1\}^S$. – Nik Weaver Jul 7 '15 at 18:37
• The referee on a paper I wrote asked me to provide references if they are readily available. I think you are correct that these are both very standard. I did find that in Sierpinski's book "Cardinal and ordinal numbers" he essentially proves 2 on page 309 when he defines ordinal exponenentiation (so I guess I'll go with that). – Pace Nielsen Jul 7 '15 at 18:41
• Yes, that sounds reasonable. – Nik Weaver Jul 7 '15 at 23:16
• I think these can be seen as special cases of Kruskal's Tree Theorem. – Christoph-Simon Senjak Jul 8 '15 at 3:38
• I would not be surprised if these facts turned out to originate with Georg Cantor. – Johan Wästlund Jul 8 '15 at 6:51