Σ1 and Π1 Transfinite Induction

is isomorphic

is isomorphic to a proper ioitial segment of -
p

'I'*(n)

3n
be the Kleene/Brouwer ordering of the tree of unsecured seThus by

is a well ordering. because

1

be a

ATR

This works

proves comparability of well orderings.

Lemma.

Let

t(m)

and

formulas.


Then

induction (on the natural numbers) proves Vm[t(m) 7 3n[t(n)&
Assume

we may also assume 8(k,s) and

say that

Vm[t(m) 73n[t(n)&q,(m,n)]]. Vm[v(m) 71!n
s

3s8(k,s)

1

By

~~-ACO

is

TIl

so we can use

statement holds for all

k.

Thus we have

consequence of f(O) = m and

1

and the previous lemma By ATR O m such that t(m) holds. Let

encodes a finite sequence of length

Vi
statement

Fix

1

TIl

k+l

such that

s(O)

(a consequence of ATR the O)' induction to prove that this

VkJ!sO(k,s).

Hence, by

1

6 (a l-CAO Vk9(k,f[k+l]), i.e.

Ll-AC there exists f such that O), Vi[t(f(i»&
m

n: transfinite induction

~: and

2.5

Theorem.

The following

plus []l ATR O 1 Zl plus ATR O 1

1.

2. 3.

ATR

4.

Zl-TI O'

induction (along the natural numbers) ; 1

[]l-CA O'

1

Proof.

pairwise equivalent:

induction (along the natural numbers) ;

plus finite

O

~

245

The pairwise equivalence of

1, 2,

and

3

is by Lemma 2.1.

1

be a linear ordering of the natural numbers on'which

Zl-TI

a []l

-+

1

formula

'!rem)

such that

we obtain a function

f

3IIDjr(m)

such that

nite descending sequence through Obviously mains only to prove that

Zl-TI

1

Zl-AC O' Let -< arithmetical formula 8(y,X).

we have weak

asserts that

In order to prove we may assume exists

Ifm<

Ifm<

-+

i.e.

Let

~(n,X)

= {(y,m):m~n

n

is an infi-

be the arithmetical formula which 8

along

~

n~(m'(Y)m).

up to

n,

i.e.

& 8(y,{(x,k):k'
there is at most one

Hence

n & 8(y'(Y)m)}.

Let

X such that

~(n,X).

n

be fixed. By Zi-TIO 1 Ifm<: n3!X~(m,X) so by weak Zl-AC there O Then clearly ~(n,X) i f we put \fn3X~(n,X).

This completes the proof.

(Friedman [3], Steel [13]). The systems

Corollary.

f

4.

induction on the natural numbers so it re-

we must prove

O

fails, i.e. we have O 3n
1

n3X~(m,X).

Y such that

= {(y,m):m~

2.6

ATR

1

<

By Lemma 2.2 Assume Zl-TI implies ATR O O' O' be a well ordering and suppose we are given an

It is easy to see that for each

X

This proves

X is the result of iterating X

Ifm['!r(m)

\fk ['!r (k ) &f (k+l ) < f(k)],

~

includes 1

and

Let

ATR

and

(both with full induction on the natural numbers) are equivalent. We shall now show that

1

is properly stronger than ATR This result O O' contradicts a claim which was made in Theorem 8 of [4] and on page 22 of [13].

2.7

Lennna.

Zl-TI

Over

the following are equivalent:

1.

Zl

induction (on the natural numbers) ;

2.

[]l

soundness of

able in

1

3

ACA

Proof.

O 2

ACAO'

is true. -+

1:

and

\fk(~(k) -+ ~(k+l»

let

M be a model of

i.e. the assertion that any

and O

~(n)

plus

3

sentence prov-

Zl induction, i.e. ~(O) 1 for some fixed n. By []l soundness of ACA O 3 plus ~(n). plus Ifk(~(k) -+ ~(k+l» ~(O)

Suppose that we have a failure of ACA

[]l

246

S.G. SIMPSON

The standard integers are canonically identified with an initial segment of the integers of n

M.

Z = [m ; M f= "",,(m)}.

Let

yet has no least element.

1

1

Zl

~ 2:

contains the standard integer

Z

This is absurd.

Z~-ACO'

Reasoning in

Then

a

let

induction to prove consistency of

ACA

be a true

O

plus

a.

z;

sentence.

Let

L

C i E w. i' is arithmetical and let ~(U,X,Y)

second order arithmetic augmented by set constants

a

= 3UVX3Y8(U,X,Y)

where

e

We shall use

be the language of Write be the arithme-

tical formula

~ denotes the i t h set recursively enumerable in X.

where

L-theory consisting of

RCA

O hension plus quantifier free induction) together with axioms

We shall prove consistency of

Let

T

be the

ordered semiring axioms plus recursive compre-

(=

T.

~(CO,Ci,Ci+l)'

Let

T be the restriction of T to k VX3ye(UO'X'Y)' By zi-ACo we have

i ~ k. Fix a set U such that o VX3Y~(UO,X,Y). Hence by zi induction we have

i E co,

C i'

It follows by cut elimination that ness theorem a model of

T

is consistent.

Vk(T is consistent). Hence by the compactk But from any model of T we can easily extract

ACA

plus a by throwing away all sets except those which are reO for some i E w. Thus ACA C 6 plus a is consistent. This comi pletes the proof. cursive in

2.8

=r==

consistency of Proof. 1

ATR

Let

a

in

n; ATR

soundness of O

is true.

be a true

1

Zl-TI

O'

ATR i.e. the assertion O' 1 In particular Zl-TI proves O

By Theorem 2.5 we have

Z~ sentence.

We know that

Now apply Lemma 2.7 to conclude that 0

Corollary.

Proof. theorem.

proves

O'

We reason in

ATR plus O 2.9

O

ATR

ATR

O

O

and hence

consists of

n2l sentence so we may as well assume that a includes this n~

sentence. i.e.

Zl-TI

n3 sentence prova bl e

t h at any

Zl-AC O' plus a

1

Theorem.

is

ACA

O

plus

a

is consistent,

cons~stent.

ATR

O

ACA

does not prove

Immediate from Theorem 2.8 plus Godel's second incompleteness

O

L: and IT: transfinite induction 2.10 finite

Corollary.

does not prove

247

induction,

induction, or

IIi-CAo'

Proof. §3.

Immediate from the previous Corollary plus Theorem 2.5.

Partition calculus in

1 Zl-TI

O'

In this section we use the notation of §3 of [6].

[00]00 of infinite sets of natural numbers.

the space

We study closed sets in It was shown in [6] that

ATR is equivalent to ACA plus the Galvin!Prikry theorem for closed sets, O O i.e. the assertion that for every closed set C ~ [00]00 there exists A E [00]00

[A]w ~ C or

such that either

[A]oo

nC=

¢.

The purpose of this section is to 1

prove a similar result in which A set some

U

C

00

ATR is replaced by the stronger theory Zl-TI O O·' is said to be hyperarithmetic if U is recursive in ~ for

O.

b E

3.1 ~ (ATR Let C i
~

Proof.

then there exists

= ¢

A E [00]00

such that

Vi < n[A]oo c C.•

-

~

The proof of Theorem 3.8 of [6] actually establishes this stronger

result. 3.2

Theorem.

Over

ACA

the following are equivalent:

O

L

2. A E

For any finite sequence of closed sets

[w]oo

such that for each

Proof.

1

~

2:

i < n

either

C i

[A]

00

~

~

C i

[00] 00 , i < n,

there exists

££

By relativization we may safely assume that the given se-

quence of closed sets

C i < n, is recursively coded. i' We claim that there exists a hyperarithmetic set U E [w]oo

each

i < neither

[U]oo

n

C.

n

~

=¢

or there is no hyperarithmetic

C = ¢. Suppose not. Let t(k) i a hyperarithmetic V E [00]00 and a finite set that

[V]oo

Vi E s (i
n Ci

=

¢).

such that for

Clearly

t(O)

V E [U]oo

such

be the assertion that there exists s and

of cardinality Vk(t(k)

k

~ t(ktl».

such that By

Z~-ACO

(a consequence of ATR the formula V(k) is equivalent to a n~ formula. O) l Hence by n induction we have t(ntl) which is absurd. This proves the claim. 1

Let

1 nl-GAO let The claim tells us that there is no hyperarithmetic By finite

U be as in the above claim.

X = {i
¢}.

3iEX {V]OO ViEX [A]

co

n Ci = ¢.

:=.

Ci •

Hence by LeDDlla 3.1 there exists

Hence for each

i
either

[A]oo c C -

i

248

(if

S.G. SIMPSON

i

E X)

2

~

or

1:

[AjUl

n c.

= 1>

~

(if

i;' X).

We already know (by Theorem 3.2 of [6j) that the partition theorem

3.2.2 implies

ATR By Theorem 2.5 it remains to show that the partition theorem O' 1 1

T For any X E [UljUl let rr E UlUl X i• ates the elements of X in increasing order. Put

be the function which enumer-

X E C if and only if TI X i & f[jjET or equivalently i) by K6nig's lemma Yk3t(tET & th(t)=k & Yj
majorizes a path through

T i,

i.e.

3fYj(f(j)~rrX(j)

[AjUl £ C or [AjUl n C = 1>. Then i i for i < n we have ~(i) if and only if ~[AjUl ~ C The latter formula is i• we have 3XYi
A E [UljUl

such that for each

i < neither

The following corollary establishes a conjecture which was stated after Theorem 3.9 in [6j. 3.3

Corollary.

Proof.

The partition theorem 3.2.2 is not provable in

ATR

O'

Immediate from Theorems 3.2 and 2.9.

An argument similar to the above proof of Theorem 3.2 establishes the follow-

ing result which was discovered jointly by S. Shelah and the author, long before the author's discovery of Theorem 3.2. 3.4

~.

Over

ACA

O

the following are equivalent:

1.

2.

For any infinite sequence of closed sets

A E [UljUl [A/{k}jUl

such that for each

n Ci

§4.

=

1>.

(~

k E A and

o

there exists

[A/{k}jUl c C. -

1

or

-

transfinite induction.

= U nl-TI nEUl n 0

Un EUl6;-RFN

~

A/{k} = {nEA:n>k}.)

Friedman [3) has shown that over TIl-TI

i < k

C.1C- [UljUl, i E Ul,

o.

Here

ACA

is equivalent to the

L~-RFNO

the transfinite induction scheme O l_RFN w-model reflection scheme 6 = 0

asserts that for any

z;

sentence

~(Xl"",Xm)

with set parameters

Xl"",X there exists a countable w-model M of ACA O m Xl'''' ,Xm E M and M 1= co ~ (Xl'" • ,Xm). It is natural to ask how much transfinite induction is equivalent to how much w-model reflection. As a rule,

such that

special cases of this question appear to be difficult.

However, one special case

n: transfinite induction

~: and

249

is answered by the following theorem. 4.1

Theorem.

Over

ACA

the following are pairwise equivalent:

O

1

L

ITl-TI O•

2.

Zl-DC O•

3.

Z3- RFN O·

1

1

Remark. lence of

1

2

The equivalence of and

2

3

and

is due to Friedman [1].

The equiva-

may be derived from the appendix of Howard [8] together with

BI to BI in Howard/Kreisel [9]. The equivalence of 1 and I O subsumes several results which have been stated by Friedman in Theorem 4.2 of

the reduction of

2

[3] and Theorem 8 of [4]. Proof of Theorem 4.1. 1

that

Zl-DC

says

1

VX3Ycp(X,Y)

-+

Similar to the proof of Lemma 2.2.

2:

-+

3ZVkcp«Z)k,(Z)k+l)

where

O there exist Skolem functions for the arithmetical matrix of 1

Zl-DC

is cp,

Recall

1

Zl. so to prove

it suffices to prove

O

Vf3gVn6(f[n],g[n]) where

f,g,h

are function variables,

( f(O), ••• ,f(n-l),

and

-+

6

3hVkVm6(h

k[n],hk+l[n])

is arithmetical,

hk(m) = h«m,k)).

If the conclusion fails then is a well ordering.

Say that

t E T

ness is

zi

T

be

if and only if

is good if h[eh(t)]=t).

Clearly the empty sequence is good, and the hypothesis t

t E T

T has no path, i.e. the Kleene/Brouwer ordering of

3h(Vk
implies that each good

f[n]

Assume the hypothesis and let

the tree of unsecured sequences for the conclusion, i.e.

T

cp

Vf3gVn6(f[n],g[n])

has a good immediate extension.

so we have a failure of

ITi-TIO

The property of good-

along the Kleene/Brouwer ordering of

T. 2

-+

3:

parameter

Similar to Lemma 2.7. UO.

U such that l and

where exists

~ Z

countable

Write VX3Y6(U

is the

i th

such that (i)-model of

cp(U

Let

cp(U

O) =3VVX3Y6(U O'V,X,Y) O)

O'Ul,X,Y).

Let

cp(X,Y)

Z~ sentence with a set

be a true 6

where

say that

is arithmetical.

(Y)O = Uo and

Vi3 j3k[6(U O,Ul,(X)i'(Y)j) & ~ = (Y)k] set recursively enumerable in Vkcp«Z)k,(Z)k+l). ACA

O

plus

cp(U

O).

Clearly

By

X.

M = {«Z\) i

This proves

1

1

Zl-DC

Z3- RFNO·

(Y)l

Fix =

there O kE(i)&iE(i)} is a

Ul

S.G. SIMPSON

250

3

-+

1:

..<.

Let

be a linear ordering of the natural numbers and assume that 1

we have a failure of I1 - TI on <:, i.e. 'in ('im ~ n (m) -+


O

let

Z

-<

ment under

=

-<

in: M ~oo "
-<

Hence

ntp(m)

-+
"
Z is nonempty and has no least ele-

Thus

is not a well ordering.

This completes the proof of

Theorem 4.1. 4.2

of

Corollary.

l:i~TI.

(ii)

Proof. consists of

1 I1 - TI plus ATR proves the existence of an O O l proves the existence of an oo-model of l1i-TI.

(i)

ATR O

The first part is immediate from Theorems 4.1 and 2.5 since ACA

plus a

11

1

sentence.

O 2 the proof of Theorem 3.7 of [6] gives

c E 0+ \ 0

Ml

ive in

H It is not hard to see that a)}. 1 and hence of IIl-TIo Corollary.

oo~odels

Proof. form of

Neither of

l:i-TI

O ATR

and a countable oo-model

satisfying "c E 0 and H exists." Since O c be such that 'ib:':O C<'ia
4.3

ATR

For the second part, reasoning in

ACA

exist

oo-model

c;' 0

let

{X: 3aEA(M

is a countable

and

O

M

O

O' of

A 5. {a: a <0 c }

F00

X

is recurs-

oo-model of

1

l:l-DC

O

implies the other, and there

for the independence.

Both directions are immediate from Corollary 4.2 plus the

G~del's

oo-model

second incompleteness theorem (for which see Friedman [5] or

Steel [12]). Remark.

The two previous corollaries are not really new since it is well

known [7] that the hyperarithmetic sets satisfy not provable in §5.

ATR

O'

1

l:l-DC

although it is provable in

O'

However, this fact is

1

l:l-TI

O'

Remark on a system considered by Jager.

After the main part of this paper was written, Harvey Friedman pointed out that the idea of the proofs of Theorems 2.8 and 4.1 above can be used to settle the relationship between ATR and a related system ATRJ considered by Jager [10].

With Friedman's permission we include this result here.

Let ATR~ be just like ATR except that arithmetical transfinite recursion O J is only assumed to hold for well orderings which are primitive recursive. ATR is

ATR; plus full induction on the integers. Say that

X

C 00

X

is low if

numbers which is recursive in

001

CK

001'

i.e. any well ordering of the natural

X is isomorphic to a primitive recursive well

ordering of the natural numbers.

~: and 5.1 than

Lemma.

X and

Let

Y.

Proof.

transfinite induction

ATRJ proves O

s

3Y~(X,Y) -+

3

low

Y~(X,Y).

We use the notation of §3 of [6]. In ACA we can prove that for O l 1 in X and hence not Zl in X. Write 1

X, OX

is complete n

{2n:nEX}U{2n+l:nEY}.

X~

3Y~(X,Y). By ATRJo we have that for each such that ~(X,Y) and ~~ exists. Since OX is not e

Assume now that there exists

X is low and

Y

X it follows that there exist

H(X~Y,e,Z),

Y,e,Z

Z is a pseudo-HX~.

i.e.

As in §4 of [6] write

Lemma.

Proof.

X«

J

proves

O

and This

X ~ Y.

Y to mean that there exists

i, X and the Turing jump of ATR

oX,

X ~ Y is low.

We claim now that

e

such that for all

~(X,Y), e E O~ \

such that

follows from Theorem 4 of [5] relativized to

5.2

251

be arithmetical with no free set variables other

~(X,Y)

Then

X low

all

n:

(Z)i+l

Z recursive in

are recursive in

Y

(Z)i'

X3Y (X« Y).

V low

This a straightforward combination of the proofs of Lemma 5.1 above

and Lemma 4.6 of [6].

~'

5.3

For every model of Proof.

a

Write

ATR

Let X

o

J

ATR

and

ATR

J

there is a model of

be a model of

= 3X~(X)

a low set

(Friedman).

~

where

such that

ATR

J

holds.

a

plus

J

Z EM

By ACA

O

we can write our

zi

e

says that be good i f

is arithmetical. f

O

= X

o

and

f

a

J

is a

Zl

1

sentence.

M

apply Lemma 5.1 to get

Zl

assertion

1

(Z)k+l]' J

M•

Unfortunately we can-

Z which is first order definable over

k[m],fk+l[m])

Disregarding Skolem functions, k«

f

k+1•

Within

(3low f) [Vk
J

M

Vm8(f k[m],fk+l[m]) define a finite sequence t to

k[m],fk+l[m])

& f[lh(t)]=t].

Clearly the empty sequence is good, and by Lemmas 5.1 and 5.2 each good sequence has a good immediate extension.

J

M•

assertion in the form

3fVkVm8(f where

where

such that this holds in

not do this, but we shall find such a

sentences.

Within

3ZVk[(Z)O = Xo & (Z)k« We would like to find

ni

Consider the

is arithmetical.

~(XO)

prove the same

ATR with the same integers •.

S.G. SIMPSON

252

By induction on

n

we can prove that there exists a lexicographically left-

most good sequence of length

n.

Let

f

be the leftmost "path" through the

"tree" of good sequences.

(We use quotation marks to indicate that the objects in J J.) question are not elements of M but merely first order definable over M By

Lemma 4.6 of [6] the "sets" which are recursive in

for Some k

M of ATRO with the same integers as

M is first order defin-

M.

f k This model

form a model

~ and therefore satisfies full induction since MJ does.

able over

is a model of

ATR.

M

Also

contains

X

o

the theorem.

5.4

Corollary.

Proof. ordinal as

and hence satisfies

The proof theoretic ordinal of

ATR

is

f

GO

Thus

M

This proves

~.

•

From Theorem 5.3 it follows that ATR has the same proof theoretic J• J!lger [10] has shown that the proof theoretic ordinal of ATRJ

ATR

By a similar but easier argument one has:

5.5

Theorem

ATR

ATRO' Proof. Write

~

Let

=' 3X(jl(X)

M0J

k

be a model of

where

a sequence of sets Here

(jl

Zk,k E CD,

ATR

submodel of

Zk J

for some

k

J

plus

0

1 sentence. 1 By Lemmas 5.1 and 5.2 we can find where

o

o

is a

2:

is arithmetical. J such that M satisfies

ranges over standard integers.

recursive in

1

J

and ATR prove the same TIl sentences. O O ATR~ has a submodel with the same integers which is a model of

Every model of

(Friedman).


form a model

M of O

ATRO'

This model

M is a O

MO'

The next corollary was proved earlier by Friedman [4]. [6J. 5.6

Corollary.

Proof.

The proof theoretic ordinal of

From Theorem 5.5 it follows that

proof theoretic ordinal. of

ATR~

is

ATR

O

ATR and

O

is

fO'

ATR~ have the same

J!lger [10] has shown that the proof theoretic ordinal

f '

O

We do not know the proof theoretic ordinal of 1 1 1 TIl-TI O or of 2: - TI + TIl-TI. l

dr of

It is fairly clear that the proofs of Theorems 5.3 and 5.5 can be made to yield general· results in the style of Theorems 2.8 and 4.1. general formulations to the reader.

We leave these

l:: and n: transfinite induction

253

Bibliography [1]

H. M. Friedman, Subsystems of set theory and analysis, Ph.D. thesis, M. 1. T. (1967), 83 pp.

[2 ]

H. Friedman, Bar induction and

[3]

H. Friedman, Some systems of second order arithmetic and their use, Proceedings of the International Congress of Mathematici~, Vancouver 1974, Vol. 1, Canadian Math. Congress (1975), pp. 235-242.

[4]

H. Friedman, Systems of second order arithmetic with restricted induction I, II (abstracts), J. Symb. Logic 41 (1976), pp. 557-559.

[5]

H. Friedman, Uniformly defined descending sequences of degrees, J. Symb. Logic 41 (1976), pp. 363-367.

[6]

H. Friedman, K. McAloon and S. G. Simpson, A finite combinatorial principle which is equivalent to the I-consistency of predicative analysis, preprint, 1981, to appear in the Proceedings of Logic Symposion I, Patras, Greece, 1980, edited by G. Metakides, North-Holland Publishing Company.

[7]

J. Harrison, Recursive pseudowellorderings, Trans. Amer. Math. Soc. 131 (1968), pp. 526-543.

[8]

W. A. Howard, Functional interpretation of bar induction by bar recursion, Compo Math. 20 (1968), pp. 107-124.

[9]

W. A. Howard and G. Kreisel, Transfinite induction and bar induction of types zero and one and the role of continuity in intuitionistic mathematics, J. Symb. Logic 31 (1966), pp. 325-358.

1

TII-CA, J. Symb. Logic 34 (1969), pp. 353-362.

[10]

G. Jager, Theories for iterated jumps, handwritten notes, Oxford, January 1980, 9 pp.

[11]

G. Kreisel, A survey of proof theory, J. Symb. Logic 33 (1968), pp. 321-388.

[12]

J. Steel, Descending sequences of degrees, J. Symb. Logic 40 (1975), pp. 59-61.

[13]

J. Steel, Determinateness and subsystems of analysis, Ph.D. Thesis, Berkeley (1976), 107 pp.

[14]

R. van Wesep, Ph.D. Thesis, Berkeley (1977), 112 pp.