Download ordinals proof theory

Ordinals from a Proof Theoretic point of
view
Rodrigo Hernandez-Gutierrez
In [1] there is an explanation of the construction of the ordinals less than
ǫ0 . I liked it and it seems it’s important because of its relation with Gentzen’s
proof of the consistency of Peano arithmetic.
The point is that we can consider ǫ0 as a set of formal symbols defined
recursively. The definition is as follows:
(a) 0 is an element of ǫ0 ,
(b) if a, b are elements of ǫ0 , then a + b is also an element of ǫ0 ,
(c) if a is an element of ǫ0 , then ω a is also an element of ǫ0 , and
(d) an element of ǫ0 can only be obtained by successive applications of (a),
(b) and (c).
These definitions consider the elements of ǫ0 as strings of symbols. So + is
not formally an operation but just a symbol that denotes some contatenation.
So there are no parenthesis to worry about.
We already have 0 defined and we denote 1 = ω 0 . We will say that an element
of ǫ0 is a natural number if it is contained in the smallest class containing 0, 1
and closed under (b). So formally, natural numbers are expressions of the form
0, 1, 1 + 1, 1 + 1 + 1, 1 + 1 + 1 + 1 and so on. We also define ω = ω 1 .
But then there are questions we must address. First, from arithmetic we
would like 0 to be a “neutral element for addition” although + does not represent
an operation. Then we must address order, for example: is any natural number
strictly smaller than ω? Finally we also need to consider that some expressions
may be equivalent to others, for example: in the context of ordered sets, ω + ω 2
is isomorphic to ω 2 .
These questions are solved by defining an order < in ǫ0 and at the same time
a simplification rule as follows.
(0) for all a in ǫ0 , a + 0 = 0 + a = a,
(1) for all a in ǫ0 , 0 < a,
(2) the relation < behaves as a transitive tricotomy,
(3) if a < b, then ω a + ω b = ω b ,
1
2
(4) let k be a natural number (in the metatheory) and b > ak ≥ ak−1 ≥
ak−2 ≥ . . . ≥ a2 ≥ a1 ≥ a0 be in ǫ0 , then
ωb
>
>
>
ω ak + ω ak−1 + ω ak−2 + . . . + ω a2 + ω a1 + ω a0
ω ak + ω ak−1 + ω ak−2 + . . . + ω a2 + ω a1
ω ak + ω ak−1 + ω ak−2 + . . . + ω a2
>
...
ω ak + ω ak−1 + ω ak−2
>
>
ω ak + ω ak−1
ω ak
(1)
Exercise 1. Show that if k is a natural number, then 0 < k < k + 1 and k < ω.
If k is a natural number greater than 0, a is an element of ǫ0 , then k · ω a
denotes ω a + ω a + . . . + ω a , where there are k summands.
Exercise 2. Show that if k is a natural number and a < b are elements of ǫ0 ,
then k · ω a < (k + 1) · ω a and k · ω a < ω b .
The use of natural numbers in this sense is only an abbreviation. It is
indeed possible to not use the notation k · ω a and instead write this as a finite
summation. Of course, that makes no difference to us humans that have learned
how to use natural numbers in an intuitive way without having to write the
complete sum.
The following can be proved from the definitions.
Theorem 1. If b is an element of ǫ0 , then there is a minimal expression for b:
b = ck · ω ak + ck−1 · ω ak−1 + . . . + c1 · ω a1 + c0 · ω a0 ,
where k is a non-zero natural number, ci is a non-zero natural number for each
i ≤ k and ak > ak−1 > . . . > a1 > a0 are elements of ǫ0 . This expression is
unique.
The form in Theorem 1 is called “Cantor normal form”. The definition of
the sum of ordinals is straightforward from the definition of ǫ0 itself. To define
the product we follow the following rules.
(i) If a is in ǫ0 , then 0 · a = a · 0 = 0.
(ii) Let b = ck · ω ak + ck−1 · ω ak−1 + . . . + c1 · ω a1 + c0 · ω a0 be expressed in
normal form and c > 0 be both elements of ǫ0 . Then b · ω c = ω ak +c .
(ii) Let b > 0 and c = ck · ω ak + ck−1 · ω ak−1 + . . . + c1 · ω a1 + c0 · ω a0 be
expressed in normal form, both elements of ǫ0 . Then b · c = ck · (b · ω ak ) +
ck−1 · (b · ω ak−1 ) + . . . + c1 · (b · ω a1 ) + c0 · (b · ω a0 ).
3
Now we finally define ω ↑ n by recursion. Let ω ↑ 0 = 1 and if k is a natural
number, let ω ↑ (k + 1) = ω ω↑k .
Theorem 2. If α is an element of ǫ0 , there exists a natural number k such that
α < ω ↑ k.
In [1], it is shown how to prove that ǫ0 is a well-order. This is obvious from
the usual von Newman definition of ordinals. However, the proof in [1] is taken
from the finitist point of view.
Theorem 3. Every sequence a0 > a1 > a2 > . . . of elements of ǫ0 is finite.
Proof. Sketch of proof I will not give any details. The main idea is to use the
normal form and using induction. We know that any decreasing chain of natural
numbers is finite just by the definition of natural numbers. Then using this, we
may reduce the chain a0 > a1 > a2 > . . . to another chain b0 > b1 > b2 > . . .
where bi is some part of ai . If the bi are in some way simpler than the ai we
have succeded. It’s just a matter of defining what simple is by looking at the
normal form. All details are in [1].
As I understand (although I have not read the details) the proof of the
consistency of Peano arithmetic is to use ǫ0 as a numbering device for all proofs
in the language of PA. Every time you use a logical rule, the complexity of the
formula rises. It seems that that proof relies on Theorem 3 to simplify formulas.
This reminds me of the famous hydra game (Google it). In fact, the interpretation of ǫ0 can be given in terms of Hydras. We define a Hydra to be a set
hT, p, <i where T is a finite tree (with some given order), p ∈ T is the minimal
vertex in T and < is a lexicographic order in T that extends the tree order.
Then we can define the ordinals as Hydras in the following fashion.
(a) 0 is the Hydra h{p}, p, ∅i,
(b) if a = hT0 , p0 , <0 i and b = hT1 , p1 , <1 i are Hydras, then a + b is the Hydra
that results in pasting T0 and T1 by identifying p0 with p1 and placing all
elements in T0 \ {p0 } lexicographically before all elements of T1 \ {p1 },
(c) if a = hT, p, <i, then ω a is the Hydra such that one adds q ∈
/ T exactly
below p.
Then the reduction algorithm can be given in terms of Hydras. Maybe I
need to write this better some other time.
References
[1] G. Takeuti; “Proof Theory” Second edition. With an appendix containing
contributions by Georg Kreisel, Wolfram Pohlers, Stephen G. Simpson and
Solomon Feferman. Studies in Logic and the Foundations of Mathematics,
81. North-Holland Publishing Co., Amsterdam, 1987. x+490 pp. ISBN:
0-444-87943-9