A constructive theory of ordinals 1
**
Thierry Coquand , Henri Lombardi
* Computer
**
, and Stefan Neuwirth
Science and Engineering Department, University of Gothenburg, Fax: +46-31-772-3663,
** Laboratoire
Tel.: +46-31-772-1030,
[email protected].
de mathématiques de Besançon, Université de Franche-Comté, Fax: +33-3-8166-6623,
[email protected], Tel.: +33-3-8166-6330,
[email protected], Tel.: +33-3-8166-6351.
September 2016
Abstract Martin-Löf [1970] describes recursively constructed ordinals. acceptable version of Kleene's computable ordinals.
He gives a constructively
In fact, the Turing de nition of com-
putable functions is not needed from a constructive point of view. We give in this paper a constructive theory of ordinals that is similar to Martin-Löf 's theory but more detailed. In our setting, the operation upper bound of two ordinals plays an important rôle through its interactions with the two relations x approach as much as we may the notion of
linear order
6 y
and x
< y .
when the property α
This allows us to
6β
or
β 6 α
is
provable only within classical logic. Our problem is to give a formal de nition corresponding to intuition, and to prove that our constructive ordinals satisfy constructively all desirable properties.
Contents 1 Introduction
2
2 Linear orders associated to a set of indexors
3
2.1
Indexors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3
2.2
Axioms
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4
2.3
Some properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5
3 Inductive construction of ordinals
6
3.1
Subordinals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7
3.2
De nition of the sup law
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7
3.3
De nition of
0 } ∪ {N} with convenient operations for disjoint unions. Any other indexor set
will contain
F-indexed family of elements F-indexed families of elements
An set of
F
F2 .
of E of
E
is a family
(xi )i∈I where I ∈ F Fam(F, E).
and the
xi 's ∈ E .
The
is denoted by
We shall restrict the use of subscripts for ordinal variables to this meaning, and use superscripts for all other uses.
2 In
the category of sets.
4
A constructive theory of ordinals
2.2 Axioms A structure of
•
β i > αi for each i ∈ J , and by the characteristic property of succ we get γ > α. k k It remains to show that α > γ , i.e., that α > β for each k ∈ J , but β = succ(αk ), and since α > αk , we have α > succ(αk ). Let
Remark law
α = succi∈J (αi ), β i = succ(αi )
.
2.4
The previous result shows that we could de ne the structure using only the in nitary
sup : Fam(F, E ? ) → E ?
Axiom
for
succ : E → E ? . In this case we succ(α) 6 β if and only if α < β .
and the unary law
12 by stating only its unary version:
have to modify
6
A constructive theory of ordinals
Fact 2.5. We have sup(α, β) < succ(α, β) and more precisely succ(α, β) = succ(sup(α, β)). More generally, if αk < γ for k ∈ J1..rK, then sup(α1 , . . . , αr ) < γ , and succ(α1 , . . . , αr ) = succ(sup(α1 , . . . , αr ))
Proof.
.
and γ > β , so that γ > sup(α, β) by Axiom 9. By γ > succ(sup(α, β)). It remains to show that γ 6 succ(sup(α, β)). In view of the de nition of γ , we need to show that α < succ(sup(α, β)) and β < succ(sup(α, β)). By transitivity this follows from succ(sup(α, β)) > sup(α, β), sup(α, β) > α and sup(α, β) > β . Let
γ = succ(α, β).
We have
succ
the characteristic property of
γ >α
we get
Fact 2.6. Let α, β be elements of E . 1. succ(α) < succ(β) if and only if α < β . 2. succ(α) 6 succ(β) if and only if α 6 β . Proof.
Use the characteristic property of
We write
F ⊆f I
succ(α),
in order to express that
F
transitivities and Axiom 8. is a nite, possibly empty list, of elements of
I.
Fact 2.7. Let α, β 1 , . . . , β m ∈ E . 1. Assume that α = succi∈J αi with (αi )i∈J ∈ Fam(F, E) and that αi < sup(β 1 , . . . , β m ) for all i ∈ J . Then α 6 sup(β 1 , . . . , β m ). 2. Assume that β k = succi∈Jk (β k )i with ((β k )i )i∈Jk ∈ Fam(F, E) for k ∈ J1..mK. Let F1 ⊆f J1 , . . . , Fm ⊆f Jm not all be empty. If α6
(β k )j ,
sup k∈J1..mK, j∈Fk
then α < sup(β 1 , . . . , β m ). Proof. 1. This is the de nition of succ. 2. Suppose, e.g., that F1 is nonempty. Then α 6 supj∈F1 (β 1 )j < β 1 6 sup(β 1 , . . . , β m ). inequality comes from Fact 2.5 because
3
β1
is
>
than all
The strict
(β 1 )j 's.
Inductive construction of ordinals In Sections 3 and 4, the indexor set
Ord F-orders.
We shall de ne a set of ordinals initial object in the category of
ord
F
is xed but seldom explicited.
(more precisely
OrdF )
and we shall prove that it is an
F-indexed ordinals by an inductive de nition. The N: it is given an element 0 and a successor map x 7→ s(x) : N → N. The inductive de nition of ord is very similar to that of N. In N each element is either 0 or an s(x) for an x ∈ N. Similarly, in ord, each element is either 0 or the succ ? of an F-indexed family in ord; we denote by ord the set of elements of this second type. First we de ne a set
of names for
simplest inductive de nition of an in nite set is that of
De nition 3.1.
The set
a distinguished element
0
ord
(more precisely
ordF )
is de ned in an inductive way: it is to admit
and a map
succ : Fam(F, ord) → ord N.b.: the only constraint in this inductive de nition is that to
succ be indeed a map from Fam(F, ord)
ord. An element of When
F = F2
ord
will be called an
ordinal
in the sequel.
we get the set of names of countable ordinals, denoted by
ord2 .
7
Coquand, Lombardi, Neuwirth
Remark
.
3.2
Each element
α ∈ ord?
is given with two data:
1. the indexor used in the de nition of 2. the element
χord (α, i)i∈Inα
α:
it will be denoted by
Fam(F, ord)
of
such that
Inα ;
α = succi∈Inα χord (α, i).
ord implies the existence of a map α 7→ Inα : ord? → F and ? family (α, i) 7→ χord (α, i) which is de ned for α ∈ ord and i ∈ Inα .
Thus the inductive de nition of the existence of a dependent
In order to make the text more readable we will perform a slight abuse of notation: we shall not mention the construction of the dependent family for
χord , and the notation αi
will be an abbreviation
χord (α, i). α = succi∈Inα αi
With these conventions we may write For
α1 , . . . , αr ∈ ord
we de ne
.
succ(α1 , . . . , αr ) = succi∈J1..rK (αi )
.
α ∈ ord, its immediate successor succ(α) is the element β = succi∈Inβ βi Inβ = N1 = {0} and β0 = α. The sequence (m)m∈N in ord is de ned inductively by m + 1 = succ(m). Then we can de ne ω = succn∈N n . In order to prove a property of α = succi∈Inα αi it is su cient to prove the property for each αi . In a similar way we can construct inductively a map whose domain is ord, or de ne inductively a predicate on ord. This is given precisely in Fact 3.4. In particular, if
where
3.1 Subordinals Here is a correct inductive de nition.
De nition 3.3. Let α = succi∈In αi ∈ subordinal of α if β = αi for an i ∈ Inα . α
α
if it is a de nitional subordinal of
this
α
ord? .
β of ord is said to be a de nitional β l1 α. The element γ is a subordinal of of a de nitional subordinal of α. We write
An element
We write this
or a subordinal
γ l α. 0
Thus
is the only element of
ord
which has no subordinal.
Let us remark that the notion of subordinal is de ned within the set be possible to descend this notion to the quotient
ord
and that it will not
Ord.
Fact 3.4. Relations l1 and l on ord are well-founded. Consequently there is no in nite branch in the tree of subordinals of an element of
ord.
Precisely:
Fact 3.5. A sequence (αj )j=1,2,... in ord, where each αj+1 is a subordinal of αj , reaches in a nite number of steps αr = 0. Remark that in order to perform a construction (or a proof ) by the case
0
l1 -induction or by l-induction,
has to be dealt with separately since it has no subordinal. Nevertheless, until ordinal
arithmetic page 15, we shall be able to avoid this case distinction.
3.2 De nition of the sup law De nition 3.6. 1. The law Let If
j
sup : Fam(F, ord? ) → ord?
(α )j∈J
be a family in
j
j
α = succi∈Ij (α )i , • K
then
ord
?
with
j
J ∈ F.
supj∈J (α )
is the disjoint union of the
• (εk )k∈K
is de ned in the following way.
is the element
ε = succk∈K εk
Inαj 's;
is the family de ned by
εk = (αj )i
if
ιj (i) = k
where
8
A constructive theory of ordinals
ιj : Inαj → K is the injective map from Inαj supj∈J1..rK (αj ) = sup(α1 , . . . , αr ).
(Here
to the disjoint union of the
Inαj 's).
We shall write 2. Finite sups in
ord
are de ned in the following way.
( 1
r
def
sup(α , . . . , α ) =
0
if
sup
α1 = · · · = αr = 0 of the
αk ∈ ord?
otherwise.
In0 = N0 . ord.
We note that Item 2 is formally included in Item 1 if we adopt the convention
F-indexed
However, this convention would not allow us to de ne arbitrary
sups in
3.3 De nition of 6 and of < The main job remains to be done, i.e., to de ne two binary relations
6
and
1)
of elements of
ord:
.
α 6 sup(β 1 , . . . , β m )
et
α
1, α, β 1 , . . . , β m ∈ ord, and γ ∈ ord? . We have 1. 0 6 β 1 , . . . , β m ; 2. 0 < γ, β 2 , . . . , β m ; 3. α < 0, . . . , 0 is impossible. | {z } m times
Proof.
Straightforward from the de nitions.
Remark
.
3.8
Axiom
as an element of
15 will be valid in Ord because every element of ord is given either as 0 or always > 0 by Item 2 of 3.7.
ord? ,
Fact 3.9. Let m, n ∈ N. Then 1. m 6 n if and only if m 6 n; 2. m < n if and only if m < n; 3. m 6 n and n < m are incompatible. Proof.
Concerning the direct implications in 1 and 2, we write n = m + r and we do an induction r. For the reverse implications, cases m = 0 and n = 0 are already known. Next we see that m + 1 6 n + 1 implies m 6 n, and that m + 1 < n + 1 implies m < n. This allows us to conclude by induction on m. on
Item
3
follows from items
An element an
m ∈ N.
α ∈ ord
1
and
2.
is said to be
nite
if
α =Ord m
for an
m ∈ N,
bounded
if
α 6 m
for
Bounded ordinals are much more complicate than nite ordinals (see examples 3.20
and 3.21).
3.5 In classical mathematics Proposition 3.10 shows that the law of excluded middle (LEM) dramatically simpli es and/or obscures what is the set
ordF .
Proposition 3.10. Assume LEM. Then for α, β β < α, there exists an i ∈ Inα such that β 6 αi . Proof.
we have α 6 β or β < α. Moreover, if
We prove by simultaneous induction the two following properties. α
6β
or
β < α
By induction hypothesis, we have for all β
∈ ord,
6 αi
or
and
i ∈ Inα
β
6α
and all
or
α < β .
j ∈ Inβ , α 6 βj
or
βj < α,
and also
αi < β .
βj < α for all j ∈ Inβ or there is j ∈ Inβ α 6 βj . In the rst case, we have β 6 α by de nition of · 6 · · · . In the second case, we α < β by de nition of · < · · · , with for F ⊆f Inβ the list [ j ]. The rst disjunction implies by LEM that either
such
that
have
Symmetric reasoning for the second disjunction.
10
A constructive theory of ordinals
N.b.: for countable ordinals, the limited principle of omniscience (
LPO) is su cient for proving
the proposition.
Corollary 3.11. Assume LEM. Any ordinal α > 0 is either an immediate successor, or the upper bound of ordinals γ < α. Proof.
Consider
α = succi∈Inα αi
and compare
α
with
supi∈Inα αi .
The details are left to the
reader.
Corollary 3.12. Assume LEM. Any bounded ordinal is nite. Proof.
Left to the reader: use Fact 3.15.
3.6 First consequences The following fact shows that the
succ
law will satisfy the characteristic property given in Axiom
12 when we shall know that it descends to the quotient Ord.
Fact 3.13. Proof.
succdef.
We have α 6 β if and only if αi < β for all i ∈ Inα .
This property is tautological: this is the de nition of
Similarly, the following fact shows that the in Axiom
α 6 β.
sup law will satisfy the characteristic property given
13 when we shall know that it descends to the quotient Ord.
Fact 3.14. supdef. Let (αj )j∈J be a family in ord? with J ∈ F, γ = supj∈J (αj ), and β ∈ ord. We have γ 6 β if and only if αj 6 β for all j ∈ J . In particular, if sup(α, β) 6 β , then α 6 β . N.b.: the result is equally true for nite sups in ord. Proof.
Another linguistic tautology. We have
and of
6,
the inequality
that for each
j∈J
γ 6 β means αj 6 β .
that
αj = succi∈Ij (αj )i for an Ij ∈ F. By de nitions of γ j for each j ∈ J and each i ∈ Ij we have (α )i < β , i.e.,
we have
The following fact shows that Axiom
8
will be valid when we shall descend to the quotient
Ord.
Fact 3.15. Proof. α
