Binumerability in a sequence of theories

R ^ENDICONTI

del

S ^EMINARIO M ^ATEMATICO

della

U NIVERSITÀ DI P ^ADOVA

F RANCO P ARLAMENTO

Binumerability in a sequence of theories

Rendiconti del Seminario Matematico della Università di Padova, tome 65 (1981), p. 9-12

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Binumerability Sequence

FRANCO PARLAMENTO

(*)

SUMMARY - In this note we answer a

question

^raised

by

A. Ursini in [2].

In that work he defines a denumerable sequence of arithmetic theories

Qn ,

whose union is

complete,

^{and asks a}

question concerning

the binumer-

ability

^{of the}

relations

ⁱⁿ

^{Qn .}

We show that a relation is in if and

only

^{if it is} binumerable in

Qn.

We are

going

^{to follow the notations and}

terminology

^of

[1]

^and

[2].

In

particular Ko

^{is the}

language

of first order

arithmetic,

^{and a}

K-system

^{is a set} of sentences in the

language

^.K.

y)

^{is the}

relation y is ^a

proof

^{of x} from axioms in T ». Let’s recall

that, given

^a

K-system,

^where

Ko

^c

_K,

^a numerical relation con is called nu-

merable in T is there is a formula _{... ,} in .g such that

In that case we say

that q

numerates R in T.

A relation R is binumerable in T if there exists a

formula p

^{in K}

such

that p

^{numerates numerates con - .R.}

The result

expressed

^{in the}

following proposition

^{is obtained as a}

straigtforward application

^{of the «} Rosser trick ^)}.

PROPOSITION 1. Let T be a consistent

K-system,

^where

KoÇ .K,

such that ⁼

0vs =

^{== 9 and}

Let the relation

y)

be binumerable in T and wn.

(*) Indirizzo dell’A.:

University

^of California,

Berkeley,

^{Cal. and Uni-} versith di Torino.

10

If both R and ^are numerable in T then R is binumerable in T.

PROOF. For notational convenience let’s suppose R C ^(o. Let

q(s) numerate R and y(x)

^{numerate w -}

R,

^{and let}

Pr f (x, ^{y )}

^binumerate

y )

^{in T.}

Consider then the

following

^formula

We claim that

z(z)

binumerates R in T.

Since T is consistent it is

clearly enough

^{to show that}

and

(i)

^{E ..K then T I-}

since 99

numerates P in

T,

^therefore

for some

h)

holds and therefore

On the other

hand, since

numerates o) ^-

R,

^{we have}

and thus

From

(1)

^and

(2)

^{we have}

(ii)

^{Let’s assume}

that k 0

^R.

Since y

^{numerates m - R in}

T,

^{we have T -}

y(k)

^and

therefore,

as

above,

y for some r E o

On the other hand for

i)

^doesn’t

^{hold, hence,}

^as

above

Therefore

From

(3)

^and

(4)

^{we have}

namely

In

[2]

two sequences and with the

following properties

^{are defined.}

1) Q n

⁼⁼

2) ^Rn

^{is a valid}

Ko-system containing

Robinson’s Arith- metic

Q .

3)

^{if and}

only

if it is numerable in

Rn .

4) ^.Rn

is binumerable in

Q n

^{via a formula ocn}

(Proposition

^{8 in}

[2]).

Applying Proposition

^{1 we have the}

following result,

PROPOSITION if and

only

^{if R is} binumerable in

Qn .

PROOF. Let’s first notice that

by 1) numerability (and

^binumer-

ability)

^{of a relation in}

Rn

^{or in}

Qn

^are

equivalent. Hence, by 4)

^we

have that

Z~n

^is binumerable in

Rn

^{via ocn . Thus the relation}

y)

is binumerable in

Rn, via,

say

y),

^{since it is}

just

^a

primitive

recursive combination of

..Rn

^{and P.R}

relations,

^{which are} binumerable in

I~n by 2).

^{Also from}

2)

^{we have that}

^Rn

is consistent

and,

If we have that both Rand c~ -1~ are numerable in We can thus

apply Proposition

^{1 to}

get

^{that R} is binumerable in

Rn . Conversely

^if .R is binumerable in it follows

immediately

^from

3)

that both R and its

complement

^{are in}

namely

^R

12

REFERENCES

[1] ^S. FEFERMAN, Arithmetization

of

metamathematics in a

general setting,

Fundamenta Matematicae, 49 (1960), pp. 35-92.

[2] ^A. URSINI, A sequences

of

theories whose union is

complete,

^{Rend. Sem.}

Mat. Univ. Padova, 57 (1977), pp. 75-92.

Manoscritto

pervenuto

ⁱⁿ redazione il 28

gennaio

^1980.