2
v. Every infinite subset of A has an infinite chain of type
w
.
vi. Every subset of A has a finite subset such that every
element of A is
≥
some element of that finite subset.
Proof: i
Æ
ii. Color (i,j), i < j, according to whether xi
£
xj or not. By Ramsey’s theorem for pairs, there is an
infinite subsequence which is either increasing (
£
), or with
y
i
£
y
j
, i < j. Only the former is possible.
ii
Æ
iii. Suppose not. Define an infinite subsequence
y1,y2,.
.. so that y1 = x1, and each yi+1 is not
≥
y1,.
..,yi.
This contradicts wqo.
iii
Æ
iv. Enumerate the set without repetition.
iv
Æ
v. Enumerate the set without repetition, and argue as
in i
Æ
ii.
v
Æ
vi. Enumerate the set without repetition, and argue as
in ii
Æ
iii.
vi
Æ
i. Let x
1
,x
2
,... A. Let {x
1
,x
2
,...,x
n
} include the
finite set. QED
Mainly application of Ramsey’s theorem for pairs. Also
simple inductive constructions. QED
J.B. Kruskal treats finite trees as finite posets, where
there is a root, and the set of strict predecessors of any
vertex is linearly ordered, and studies the qo
there exists an inf preserving embedding from T
1
into T
2
.
I.e., h:T
1
Æ
T
2
, where h is one-one, preserves
£
, and h(x
inf y) = h(x) inf h(y). (Every finite tree has an obvious
inf operation on the vertices).
THEOREM. (J.B. Kruskal). The above qo of finite trees as
posets is a wqo.
Let’s see what is set theoretic about the proof. Nash
Williams’s introduction of the minimal bad sequence
technique seriously simplified the original Kruskal proof,
and is also used extensively in wqo theory. See [NW65].