SET THEORY CA N PROVIDE FOR ALL
ACCESSIBLE ORDINALS
Rolf SCHOCK
A (stro n g ly) inaccessible o rd in a l is a n uncountable o rd in a l
x such th a t the cardinals o f the p o we r set and u n io n o f a n y
less p o we rfu l subset o f x are also sma lle r than x. A n o rd in a l
is accessible if it is sma lle r than a n y inaccessible ordinal. Let
ZF and ZFP b e th e class theories o f Zermelo, Fraenkel, a n d
Skolem a n d o f v o n Neumann, Bernays, a n d Gö d e l re sp e ctive ly ('). Fro m a n in t u it ive p o in t o f vie w, i t seems th a t th e
axioms o f ZF and ZFP p ro vid e f o r the existence o f a ll accessible ord inals. Th is in t u it io n is n o t e n t ire ly clear, b u t, sin ce
the set o f ordinals o f any standard complete model o f ZF and
ZFP is a n o rd in a l o f the metalanguage, i t ca n be understood
to mean the conjunction o f the fo llo win g t wo model-theoretic
statements.
A. Th e set o f ordinals o f a n y standard complete mo d e l o f
ZF is an inaccessible ordinal.
B. Th e set o f o rd in a ls o f a n y standard complete mo d e l o f
ZFP is th e successor o f a n inaccessible o rd in a l.
In Montague and Vaught [2], it was shown that the existence
of an inaccessible o rd in a l imp lie s the existence o f a stro n g ly
standard complete model o f ZF whose se t o f ordinals is less
than inaccessible and so th e f a lsit y o f A . Th e co n d itio n ca n
be reduced t o t h a t o f th e existence o f a standard mo d e l o f
ZF by means of the strong Lbwenheim-Skolem Theorem. Cohen
and Mo sto wski applied th is procedure in [1] and [3] t o p ro ve
(') Fo r a des c ription o f t he ax ioms o f ZF and ZI P and definitions o f t he
metamathematic al t erms o f t h e pres ent s t udy , t h e re a d e r i s re f e rre d t o
Schock [ 4] .
596
R
.
S CHO CK
that the existence o f a standard model o f ZF imp lie s the existence o f a denumerable min ima l mo d e l o f ZF, a n d so o f a
standard complete model o f ZF whose set o f ordinals is denumerable. Wit h the aid o f the same procedure, i t was shown in
Schock 141 th a t the assumption that ZF is founded imp lie s the
existence o f denumerable min ima l models o f both ZF and ZFP
and so the fa lsity o f both A and B. On e is therefore tempted
to conclude that intuition has once more gone astray and doubt
both A and B. (
5
proofs
of the fa lsity of A and B referred to above, the assumption
made
) B u t is essetial to the proof of the existence of a standard
complete
in wh ich not a ll accessible ordinals are attains u c model
h
ed.
In
fact,
a
cTheorem
o n 1.c Th e fo llo win g conditions are equivalent.
l u s i o
ZF is founded
(
n
1
i
(2) ZFP is founded.
)
s
A is false.
(
p
r
e
3
m (4)
a Bt is false.
)
u
r
e
. If (1) holds, (2) does b y Theorem 2 of Schock [4]. I f (2) holds,
I(1) does and A is false b y Theorem 5 of Schock [41. B y Lemma
n3 o f Schock 14], (3 ) imp lie s (4). Fin a lly, i f (4) holds, ZFP a n d
tso ZF is founded b y Theorems I and 2 o f Schock 141.
h By means o f Theorem I , a clo se ly related equiconsistency
theorem
can be proved. Let ZF a be the result of adding the
e
axiom a to ZF, and simila rly fo r Z.FP. Also, le t en and inc be the
formulas o f ZFP about sets wh ic h assert t h e co n ju n ctio n o f
A and B and the existence of an inaccessible ordinal respectively, let Id and Id ° be such formulas wh ich assert the foundedness
of Z F and ZFP respectively, a n d e t mo d a n d modo b e su ch
formulas wh ich assert the existence o f standard complete mo dels o f ZF and ZFP respectively. Fin a lly, assume that N is 'not'.
Then,
(1 I n 121, Mont ague and V a u g h t i n f ac t doubt ed wh a t appears t o b e A
weakened t o s t ro n g ly s t andard c o mp le t e models , b u t n o t B wes t ened
s imilarly since it is prov able.
SET THEORY CA N PROVIDE FOR A L L ACCESSIBLE O RDI NALS 5 9 7
Theorem 2. I f T is ZF o r ZFP, then the fo llo win g theories a re
equiconsistant.
(1) T
(2) T + NirIC
(3) T + Nmod + NmodP
(4) T + Nid + NM"
(5) T + en.
Assume f irst th a t T is Z R I t wa s sh o wn in Sheperdson [5]
that (1) a n d (2) a re equiconsistant since th e existence o f a n
inaccessible o rd in a l imp lie s th e existence o f a standard co mplete mo d e l o f ZFP and so o f ZF b y Lemma 2 o f Schock [4].
Consequently, i f (3) is consistent, so is (2). On the other hand,
if (3) is inconsistent, ZF implies that ZF has a model and so its
own consistency. B u t t h e n Z F i s in co n siste n t b y GOdel's
theorem on the u n p ro va b ility of consistency and (2) is as we ll.
That is, (2) and (3) a re equiconsistent. Also , b y Theorem 1 o f
Schock [4], the foundedness o f ZF o r ZFP is equivalent to the
existence o f a standard complete model o f either. Since these
proofs and the proof o f Theorem 1 can be g ive n wit h in ZF, i t
is clear from Theorem 1 that (3) through (5) are equiconsistent.
if T is ZF. Fro m Theorem 2 o f Schock [4], i t fo llo ws th a t (1)
through (5) a re also equiconsistent if T is ZFP.
Consequently, i t is possible to assume with o u t contradiction
in ZF o r ZFP th at a ll accessible ordinals a re p ro vid e d f o r b y
either th e o ry, b u t o n ly a t th e co st o f d e n yin g th a t th e re i s
an inaccessible ordinal, that either theory has a standard complete model, and th a t e ith e r th e o ry is founded. Thus, th e assumption does n o t seem t o b e v e ry plausible. O n th e o th e r
hand, notice that a ll these implausible assumptions hold in the
min ima l models o f ZF and ZFP.
598
R. S CHO CK
REFERENCES
COHEN, P., A min ima l model f o r s et t heory , B ull. A mer. Mat h. Soc. 69,
1963.
[2] MONTAGUE R. and VAUGHT R., Nat ural models of set theories, Fund. Math.
47, 1959.
MOSTOWSKI, A., O n models of Zermelo-Fraenk el s et t heory s atis fy ing t he
axiom of c ons truc tibility , Studiae L o g i c o
Hels
ink i, 1965.
Mathernatic a
e
t
[4] SCHOCK, R., O n s tandard models o f s et theories , Logique et Analy s e 63,
P h i l o s o p h i c a ,
1973.
E
5
I
SHEPERDSON, J., I nner models f o r set t heory I I I , J . Sy mb. Logic 18, 1953.