Table Of ContentLecture Notes ni
Mathematics
Edited by .A Dold and .B Eckmann
619
Set Theory dna yhcrareiH V Theory
Bierutowice, Poland 1976
Edited yb
.A Lachlan, .M Srebrny, dna .A Zarach
galreV-regnirpS
Berlin Heidelberg NewYork 1977
Editors
Alistair Lachlan
Department of Mathematics
Simon Fraser University
Burnaby 2, B.C.
V5A 1S6/Canada
Marian Srebrny
Mathematical Institute
Polish Academy of Sciences
P.O. Box 137
00-950 Warszawa/Poland
Andrzej Zarach
Institute of Mathematics
Technical University
ul. VVybrze~'e Wyspiar~skiego 27
50-3?0 Wroclaw/Poland
AMS Subject Classifications (1970): 02 20, B 02 25, B 02 29, F 02 H ,02
02 J05, 02K 05, 02 K10, 01 N15, 54J05
ISBN 3-540-08521-1 Springer-Verlag Berlin Heidelberg NewYork
ISBN 0-387-08521-1 Springer-Verlag NewYork Heidelberg Berlin
This work to is copyright. subject llA rights era ,devreser whether eht whole
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FOREWORD
This volume represents the proceedings of the Third
Conference on Set Theory and Hierarchy Theory held at
Bierutowice a mountain resort south-west of Poland, September
17-24, 1976.
It was organized by the Mathematical Institute of the Wroc~aw
Technical University. There were 62 registered participants
representing 8 countries.
The programme committee was: Alistair Lachlan /Canada/,
Wiktor Marek /Warsaw/, Janusz Onyszkiewicz /Warsaw/, Leszek
Pacholski /Wroclaw/, Marian Srebrn~ /Warsaw/, Bogdan W~glorz
/Wroc~aw/, J~drzej Wierzejewski /Wroc~aw/- secretary, Andrzej
Zarach /wroc~aw/ - chairman.
We would like to express our appreciation to all those
whose help contributed to the success of the conference.
We are also grateful to all those who assisted us in giving
these proceedings their present form. In particular: Gert
H.~ller, Wiktor Marek, John Truss, Peter Hinman, Petr H~jek
and many colleagues from Warsow and Wroc~aw who helped us
considerably in the editorial task.
A listair Lachlan
April 1977
Marian Srebrn~
Andrze j Zarach
CONTRIBUTED PAPERS :
Z~Adamowicz, Constructible semi - lattices of degrees
of constructibility . . . . . . . . . . . . . . . q
B.Balcar and P.Vojta~s, Refining systems on Boolean algebras .45
A. Baudisch and H. Weese, The Lindenbaum algebras of the
theories of well-orderings and Abelian
groups with the quantifier Q~ ....... 59
K. Bieli~ski, Extendability of structures as an infinitary
property ......... • . . ........ 75
M.Boffa, A reduction of the theory of types ......... 95
L.Bukovsky', Random forcing . . . . . . . . . . . . . . . . . . 101
J . C- nuaacek, ..v Topological problems in Alternative Set Theory..119
J.Cicho~, On the Baire propert-y of Boolean algebras ..... 135
K.~uda, The relation between ~ I procedures and the
infinitely small in nonstandard methoda ....... 143
M.Diekman, Structures ~-satur@s ............. 153
R.Frankiewicz, On the inhomogenity of the set of
P(m)- points of oc .......... . .169
W.Guzicki, On the projective class of the continuum
hypothesis . . . . . . . . . . . . . . . . . . . . 181
P.Hinman, A survey of finite-type recursion ....... . .187
L.Kirby and J.Paris, Initial segments of models of Peano's
axioms . . . . . . . . . . . . . . . . . . . . . . 211
A.Krawczyk and W.~arek, On the rules of proof generated
by hierarchies . . . . . . . . . . . . . . . . . .227
A.Petry, On cardinal numbers in Qu~ne's New Foundations...2~1
S.Roguski, The theory of the class HOD ............ 251
D.Seese and P.Tuschik, Construction ~f nice trees ...... 257
A.Sochor, Differential calculus in the Alternative
Set Theory . . . . . . . . . . . . . . . . . . . . . 273
Z.Szozepaniak, The consistency of the theory
ZF 1 + L ~ HOD . . . . . . . . . . . . . . . . 285
P.Tuschik, On the decidability of the theory of linear
orderimgs in the Language L(Qq) .......... 291
A.Wilkie, On the theories of end-extensions of models
of arithmetic . . . . . . . . . . . . . . . . . . . 305
B.W@glorz, Some properties of filters ............ 311
A. Zarach, Constructibility done more constructively . . • • 329
LIST OF REGISTERED PARTICIPANTS
Zofia Adamowicz Warszawa
Bohuslav Balcar Praha
Andreas Baudisch Berlin
Konrad Bieli~ski Kielce
Haurioe Boffa Mons
Jan Brzuchowski Wroclaw
Lev Bukovsky l Kosxce
Jaroslav Chuda~ek Praha
Jacek Cicho~ Wroclaw
Marcel Crabbe Bruxelles
Karel ~uda Praha
Bern~ Dahn Berlin
Max Dickmau Paris
Na~gorzata Dubiel Warszawa
Arleta Dylu~ Wroe ]:aw
Anna Fedyszak Warszawa
Ulrich Fe igner T~bimgen
Adam Figura Wroc law
R~szard Frankiewicz Katowice
Andrzej Gutek Ka~owioe
WoJciech Guzicki Warszawa
Pet~ Ha~ek t . l~aha
Peter ~H nman Ann Arbor
Czeslawa Jakubowicz Wroclaw
Andrzej Jankowski Warszawa
Laurence Kirby Nanohester
Stanislaw Krajewski Warszawa
Adam Krawczyk Warszawa
M~chal Krynick~ Warszawa
Wieslaw Ku~aga Wroc taw
Anna Lema~ska W~szawa
~raucis Lowenthal Brussel
Wiktor Marek Warszawa
Roman Murawski Pozna~
Gert .H M~ller Heidelberg
Jeff Paris Manchester
Bogdan Pawlik Wroc;~aw
Andre Perry Lxege
vlll
Leszek Paoholski rW oo~.aw
Zygmunt Ratajczyk Warszawa
Stanislaw Roguski Wroolaw
Detlef Seese Berlin
El~bieta Sepko Warszawa
Kostas Skandalis Wroclaw
Anion Sochor ~Taha
Marian Srebrny awa Warsz
Yoshindo Suzuki
Zbigniew Szczepaniak Wroc~aw
Leslaw Szczerba Warszawa
Rastislav Telgars~ Wroc~aw
Jerzy Tomasik Wroclaw
Peter Tuschik Berlin
Anita Wasilewska Warszawa
Jan Waszkiewicz Wr oc law
Martin Weese Berlin
Philip Welch Oxford
Bogdan W~glorz Wroc~aw
J~drzej Wierzejeweki Wr oc law
Alex Wilkie London
George Wilmers Manchester
Wlodzimierz Zadroiny Wroolaw
Pawel Zbierski Warszawa
Andrzej Zarach Wr oc law
CONSTRUCTIBLE SEi~-LATTICES OF DEGREES OF CONSTRUCTIBILITY
by
ZOFIA ADA~OWICZ
We shall prove owt theorems :
T ~ Iv~E .1 If [~ is a countable standamd model of ZFC + V = L
and ~ is an upper semi-lattice in i, countable and well-
founded in ,~i and ~ sa~l a greatest element ~ , then
there is a model of ZFC, N, such that ~I~ N, On~ = On N and
the ordering of the degrees of constructibili%y of sets of
ordinals of N is isomorphic in N with ~ enlarged by a lowest
element 0 •
TH~ORE~I 2. If ~l is a countable standard model of ZFC + V = L
and L is an upper semi-lattice in ,M well-founded in I~ and
such that every initial segment of ~ of the form
{ ~I~I : ~ <~ ~ ) for a given ~ in I(cid:127) ~ is countable
in M, then there is a model of ZF, N such that MC_ N,
On M = On N and the ordering of the degress of constructibili%y
of sets of ordinals of N is isomorphic in N with ~ enlarged
by a lowest element 0.
Evidently Theorem 2 implies Theorem .I Theorem I is
formulated separately because the methods of its proof are slightly
different from the derivation of Theorem 2 from Theorem .I The paper
has two parts : the first devoted to the fi2st pumpose and the
second ot the second.
Introducti on.
Let us describe briefly the idea of the proofs. The basic
technique is forcing and taking symmetric submodels of generic
models. In the case of Theorem 1 the top degree is represented in N
by a real.
Let us first describe the methods of ]lf[ used ot obtain such
a real. These m~thods are refined here. In the case ef a countable
lattice ~ (not necessarily finite) the refinement is inessential.
In what follows we discuss not the present paper but [I] and a
straightforward generalization of ]I[ for a countable lattice°
The top real can eb regarded as a generic of certain perfect
,secart i.e. trees of finite sequence ramifying infinitely many
times (see [I]) - this idea goes back ot Sacks [5]. Still further,
one can refer ot Spector's work on a minimal Turing degree. A dif-
ferent development of Spector's idea- in recursion theory- is
Lachlan, Lerman [4], Lachlan- Lebeuf [3].
Other degrees are supposed ot eb represented in N by "sections"
of the real representing the top degree, and thus it is more
convenient ot take trees included in
( U [~I) n ( ) m instead of trees included in n m (as in
Sacks [5]).
The sections of a generic real C~ G 6 (m,~i) are then defined
as tG~ (n) = tG(n)(~) for ~ ~ I~ • The real tG~ (by ~ we
denote the ~o being the ~ ~ of ~ has the same degree as ~G"
For the sake of the case where ~ has no ~ it is simpler ot
dis~e~rd the function G t and ot speak only about reals t%6 m • m
as they are supposed ot represent the degrees we want the
forcing conditions to ensure the following
(q) is ensured by the fact that the trees which we use have the
following property :
(~) any two branches with the same ~'th section have the
same ~'th section ; thus knowing any tree in G and
tG~ we knc~ G t
(~) requires that in every ramification of a tree p the
values of elements of p lying in ~ 'l~fI ~ should have the
erty prop
(i) s(~) = t(~) > s(~) = (t 0
(2) is ensured by the fact that our trees have the following
property :
(m~) if we fix the ~° th section of a branch, there si still
freedom for the ~'th section at infin~y many arguments.
Then (2) is shown by an analogue of the usual forcing argument
showing that a generic real is not constructible.
(~) requires than in infinitely many ramifications of a tree
p there are values of elements of p lying in ~'~|~ I with the
pro perry
(ii) s(~) = t(~) & s(~) ~ t (~)
Similarly we ensure that any G t is non-constructible.
Properties (i) and (ii) lead to the idea of allowing the trees
ot have, in consecutive ramifications, increasing finite subsets of
~I containing more and more pairs with the property (ii) and
all having the property (i). This idea is due to Lerman (L~]).
Also other properties of these finite sets are required ot
ensure the being no more degrees in ~ than those of the reals t@~.
They lead to a definition of an increasing sequence of finite sets
(Ui)i E ~ which is to be put in ramilications of trees.
Such a sequence was called by Lerman a sequential representation
of~ . Lerman's notion was modified in [q] for the sake of forcing
and to permit considerations of other degrees than these of reals.
Hg~e we simplify Lerman's definition by requiring the sequence
(Ui)iE ~ ot have a universality (embedding) property (Def.4, point
(3)) which generalizes all previously required detailed properties°
In addition to simplifying the definition the embedding
property (3) of Def. 4 allows us to combine representations of
different parts of ~ together.
This idea is due to Lachlan-Lebeuf ([3]), who introduced eht
notion of an embedding property. However, their formulation of an
embedding property is different from ours and does net generalize
otha~ properties of the sequence (Ui)iE ~ . Our reguirem~nts
imposed on a sequential representation seem ot be the weakest
possible.
Our modification of the Lachlan-Lebeuf definition of a
sequential re~esentation aims solving two new difficulties: one
concerns representing a countable ~ instead of l inite and the other
consists in combining together uncountably many representatiomsof
p~rts of ~ in Part 2, instead of countably many representations as
in Lachlan-Lebeuf [3].
~ortunately ~e can ensure all the required properties of
the sets i U in the case oi a countable ~ in conntably many steps
and thus we can define a Sequence (Ui)iE~ - there is on deep
dll ference from the case of a finite,C.
We have thus closed the discussion fo sequential re~esentations.
The trees that we have bee~ men~io~ing are aezined in this
paper as "pre-conditions" Des. 5 and in [I] as conditions.
Row forget that they are not conditions aere a~d regard them
as conditions. Also regard ~/ as .)G(~:
tsoi~i of the properties of pre-conditions that are needed are
i~idden in the properties of the representation of ~ •
(i) follows i rom (2) 2 Def. and also froi~ (3) Dez. .5 In contra-
distinction ~o Sacks [5] the trees ramify at even levels (property
(2) De2.5) for the sake o± more symmetry and thus less complication.
]~zoperty (4) ~el.5 is formulated for a "iusion iemma" formulat-
ed as Lemma .4
The cumbersome technique o± the paper si bound with the proper-
ties of pre-conditions.
What we want o~ show is the following : ii ~ x ~/ is a set
of ordinals and ~ is the least element of~ suah that x6L[t G ]
then tG~ ~ L[x]; moreover ilcus a least ~ exists.
%
The existence of such a least ~ follows from the fact that
is well-founded, from the fact ~hat theze is a ~op degree whence
6 x L[tGI ] and from the following facts:
fi C_on 6 [t r]oL[tG ],
~en (~) x~ T[t~ ~'] i~ ~ ~ ^ ~" exi~s or x e,[~£~,]
(see Def. b, 11,18) ~^
otherwise and
(~x) ifx@ L[G ~,]_ then there is an ~ in ~ such
The well-foundedness of~ is absolute with respect to the
standard models ,~ .~d However, we use in the proof only the
wali-f~undedness of~ in lVi ot make the argument easy ot translate
ot the case of non-standard models if anyone wa~ted ot consider
non-standard models.
The fact (~) easily folzows from an appropriate property of
the representation of~ (Lemma 15). The idea o± this property is
