Pseudo-Hierarchies in Admissible Set Theory without Foundation

176 Pages·2012·0.93 MB·English

Pseudo-Hierarchies in Admissible Set Theory
without Foundation and Explicit Mathematics
Inauguraldissertation
der Philosophisch-naturwissenschaftlichen Fakulta(cid:127)t
der Universita(cid:127)t Bern
vorgelegt von
Dieter Probst
von Langnau i. E.
Leiter der Arbeit:
Prof. Dr. G. Ja(cid:127)ger
Institut fu(cid:127)r Informatik und angewandte Mathematik

Contents
Introduction 1
Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
I Languages, theories and provable ordinals 9
I.1 General conventions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
I.1.1 Mathematical reasoning and formal proofs . . . . . . . . . . . 9
I.1.2 Notational conventions . . . . . . . . . . . . . . . . . . . . . . 10
I.1.3 Sets, functions and relations . . . . . . . . . . . . . . . . . . . 10
I.1.4 Orderings, trees and ordinals . . . . . . . . . . . . . . . . . . . 11
I.1.5 Recursive and primitive recursive functions and relations . . . 13
I.1.6 Some primitive recursive functions and relations . . . . . . . . 14
I.1.7 Indices for the [primitive] recursive functions and relations . . 15
I.2 Languages, theories and structures . . . . . . . . . . . . . . . . . . . 16
I.2.1 Languages, theories and structures . . . . . . . . . . . . . . . 16
I.2.2 Theories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
I.2.3 The languages L and L of (cid:12)rst and second order arithmetic . 20
1 2
I.2.4 The language L(cid:3) of Kripke-Platek set theory . . . . . . . . . . 21
I.2.5 The language L for explicit mathematics . . . . . . . . . . . . 22
I.2.6 First and second order predicate logic and the logic of partial
terms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
I.2.7 Peano Arithmetic . . . . . . . . . . . . . . . . . . . . . . . . . 24
I.2.8 The semi-formal systems PA(cid:3) . . . . . . . . . . . . . . . . . . 25
I.2.9 The theories BS0 and KPu0 . . . . . . . . . . . . . . . . . . . 26
I.2.10 ACA and ACA : Second order theories with arithmetical com-
0
prehension . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
I.2.11 The theory EETJ . . . . . . . . . . . . . . . . . . . . . . . . . 29
0
I.2.12 Translations and embeddings . . . . . . . . . . . . . . . . . . 32
I.2.13 On the dispensability of primitive recursive function symbols . 34
I.2.14 Syntactical extensions of L . . . . . . . . . . . . . . . . . . . 35
2
I.3 Proof-theoretic basics . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
I.3.1 A notation system based on the ternary Veblen function . . . 36
iii
iv Contents
I.3.2 Cut-Elimination . . . . . . . . . . . . . . . . . . . . . . . . . . 37
I.3.3 The proof-theoretic ordinal jTj of a theory T . . . . . . . . . . 40
II Pseudo-hierarchies in second order arithmetic 43
II.1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
II.1.1 Universal formulas . . . . . . . . . . . . . . . . . . . . . . . . 44
II.1.2 Trees and normal forms of (cid:5)1 and (cid:6)1 formulas of L . . . . . . 46
1 1 2
II.1.3 Hierarchies and choice sequences . . . . . . . . . . . . . . . . 52
II.1.4 N-models of (cid:6)1-AC and (cid:6)1-DC . . . . . . . . . . . . . . . . . . 54
1 1
II.1.5 The jump-hierarchy . . . . . . . . . . . . . . . . . . . . . . . . 58
II.1.6 The hyper-arithmetical sets HYP . . . . . . . . . . . . . . . . 62
II.2 Pseudo-hierarchy arguments . . . . . . . . . . . . . . . . . . . . . . . 66
II.2.1 On HYP . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66
II.2.2 The theory ACA +((cid:1)-TR) . . . . . . . . . . . . . . . . . . . 76
0
II.2.3 Fixed points of monotone and non-monotone operators . . . . 78
II.2.4 Fixed points and hyperarithmetical sets . . . . . . . . . . . . 81
II.2.5 The proof-theoretic analysis of FP(cid:0) . . . . . . . . . . . . . . . 87
0
II.2.6 Additional results on the class FixA . . . . . . . . . . . . . . . 90
IIIPseudo-hierarchy arguments in admissible set theory without foun-
dation and explicit mathematics 95
III.1 Pseudo-hierarchies in admissible set theory . . . . . . . . . . . . . . . 96
III.1.1 Hierarchies and pseudo-hierarchies . . . . . . . . . . . . . . . 96
III.1.2 Admissible sets and the theories KPi0, KPir and KPm0 . . . . . 97
III.1.3 A pseudo-hierarchy principle for KPi0 . . . . . . . . . . . . . . 100
III.1.4 Extending theories by (psh0) . . . . . . . . . . . . . . . . . . . 101
III.1.5 Fixed point principles vs. iteration principles . . . . . . . . . . 105
III.1.6 On linearity, iteration and choice . . . . . . . . . . . . . . . . 110
III.2 Admissible sets and linearity . . . . . . . . . . . . . . . . . . . . . . . 111
III.2.1 The constructible hierarchy L . . . . . . . . . . . . . . . . . . 112
III.2.2 A (cid:1) formula expressing admissibility . . . . . . . . . . . . . . 115
0
III.2.3 On hyp . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120
III.2.4 Admissibles linearly ordered by 2 . . . . . . . . . . . . . . . . 127
III.2.5 Dependent choice in admissible set theory . . . . . . . . . . . 129
III.3 Pseudo-hierarchies in explicit mathematics . . . . . . . . . . . . . . . 140
III.3.1 Hierarchies and pseudo-hierarchies . . . . . . . . . . . . . . . 140
III.3.2 The theory EMA . . . . . . . . . . . . . . . . . . . . . . . . . 142
0
III.3.3 A pseudo-hierarchy principle for EMA . . . . . . . . . . . . . 143
0
III.3.4 Choice in explicit mathematics . . . . . . . . . . . . . . . . . 148
III.3.5 EMA , OMA and asymmetric interpretations . . . . . . . . . . 149
0
Contents v
Index 161

Introduction
Pseudo-hierarchies have become a powerful tool in several areas of mathematical
logic. They were (cid:12)rst applied in the context of hyperarithmetical theory by Spec-
tor [41], Gandy [16] and Feferman and Spector [13]. Especially in second order
arithmetic, the potent and (cid:13)exible technique of pseudo-hierarchy arguments seems
nowadays virtually indispensable. A typical application for speci(cid:12)c (cid:12)xed point def-
initions is given in Avigad [2], and a rich fund of important results obtained by
working with pseudo-hierarchies, e.g. the pairwise equivalence of (ATR), the Perfect
Set Theorem and (cid:6)0 determinacy, can be found in Simpson [40]. Without even
1
looking at the many more instances of pseudo-hierarchy arguments for second order
arithmetic gathered in [40] and the second chapter of this thesis, it is not presump-
tuous to wish for an equally potent device in subsystems of set theory or explicit
mathematics.
Making the uncon(cid:12)ned application of pseudo-hierarchy arguments possible in sub-
systems of admissible set theory without foundation and explicit mathematics has
been the leading goal of this thesis. After a careful analysis of pseudo-hierarchies is
second order arithmetic, (cid:12)rstly to understand this method thoroughly with regard
to its subsequent adaption to the aforementioned frameworks, and secondly to em-
phasis the versatility of this technique by applying it in order to achieve own results,
we pinpoint the obstacles on the road to success and provide e(cid:11)ective strategies to
sidestep these di(cid:14)culties. Besides, we research various (cid:12)xed point theories: Among
otherthings,weprovethatoperationsinducedbypositivearithmeticalformulaspos-
sess ingeneralno(cid:1)1 de(cid:12)nable (cid:12)xed points, althoughthere arewell-know methodsto
1
obtain (cid:6)1 as well as (cid:5)1 de(cid:12)nable (cid:12)xed points (see e.g. [1]). As a nice by-product of
1 1
thislineofresearch, weobtainanembedding ofID(cid:3) (thetheoryobtainedfromID by
1 1
restricting (cid:12)xed point induction to formulas that contain (cid:12)xed point constants only
positively) into (cid:6)1-DC, and thereby answer an old question ask in Feferman’s article
1
on Hancock’s conjecture [11] about the upper bound of ID(cid:3). Claiming (cid:12)xed points
1
of monotone, (cid:6) de(cid:12)nable operations on the entire universe over KPi0 leads to the
theory KPi0 +((cid:6)-fp0) with the same proof-theoretic ordinal as KPm0, the standard
theory of strength meta-predicative Mahlo in admissible set theory (cf. Ja(cid:127)ger and
Strahm [27]). This result depends crucially on the application of pseudo-hierarchy
1
2 Introduction
arguments. Further, we study a strengthening of KPi0 (cf. Ja(cid:127)ger [19]) where not
only the admissibles distinguished by the predicate Ad(x) are linearly ordered by
2, but all sets re(cid:13)ecting the Kripke-Platek axioms. Surprisingly, this extension has
already the strength of (cid:1)1-CA . Moreover, we elaborate on the connection between
2 0
iteration and dependent choice: We demonstrate how the axiom of KPm0 asserting
that the class fx : Ad(x)g is linearly ordered by 2 is used to embed trans(cid:12)nite de-
pendent choice, and that this axiom can be compensated by claiming (cid:5) re(cid:13)ection
2
on admissibles that additionally satisfy dependent choice, which nicely parallels the
situation in second order arithmetic where extending ACA by (cid:5)1-re(cid:13)ection on mod-
0 2
els of (cid:6)1-DC leads also to a theory of strength meta-predicative Mahlo (cf. Ru(cid:127)ede
1
[37],[38]). Finally, we propose a form of dependent choice suitable for subsystems of
explicit mathematics.
The concept of a hierarchy is based on the notion of an operation: In second order
arithmetic, a formula A(U;u) induces canonically an operation FA acting on the
powerset of the natural numbers and mapping X to fx : A(X;x)g. By iterating
such an operation along a well-ordering (cid:30), a proper hierarchy G is obtained whose
(cid:11)th level (G) := FA((G) ) is the FA image of the disjoint union of the levels
(cid:11) (cid:30)(cid:11)
(G) below (cid:11). Proper hierarchies on their own constitute a very powerful and
(cid:12)
useful concept which is implemented in Friedman’s well-known theory ATR (cf. e.g.
0
Friedman [14], Friedman/McAloon/Simpson[15], Steel [42]) and, at least in second
order arithmetic, are intrinsically tied to pseudo-hierarchies. A pseudo-hierarchy
looks locally like a proper hierarchy, so again (G) = FA((G) ), however, the
(cid:11) (cid:30)(cid:11)
underlying ordering (cid:30) is only a linear ordering but not well-founded.
In subsystems of second order arithmetic comprising arithmetical comprehension,
the existence of a pseudo-hierarchy for A follows already if FA can be iterated along
arbitrary well-orderings. Moreover, such a pseudo-hierarchy can inherit a chosen (cid:6)1
1
de(cid:12)nable property ofthe proper hierarchy: If A(U;u)is anarithmetical and B(U;V)
a (cid:6)1 formula of second order arithmetic such that
1
8X[Wo(X) ! 9F(HierA(F;X)^B(F;X))];
then the well-known fact that being a well-ordering cannot be expressed by a (cid:6)1
1
formula forces the existence of a linear ordering (cid:30) that is not a well-ordering and a
hierarchy G that meet HierA(G;(cid:30))^B(G;(cid:30)).
Each pseudo-hierarchy argument exploits that the (cid:12)eld of the underlying ordering (cid:30)
of the pseudo-hierarchy possesses a set K (cid:18) Field((cid:30)) that is open at the bottom, i.e.
K is non-empty, upward closed and has no (cid:30)-least element. To apply its simplest
form, we consider a pseudo-jumphierarchy G along the linear ordering (cid:30). Basically,
(G) is the collection of all sets that are (cid:5)0 in some level (G) for (cid:13) (cid:30) (cid:11). Then, for
(cid:11) 1 (cid:13)
a set K open at the bottom, the collection M of all sets that are contained in each
Introduction 3
level (G) for (cid:11) 2 K is closed under arithmetical comprehension: If the set X is
(cid:11)
arithmetical in some set Y 2_ M, then X is already (cid:5)0 in Y for some n 2 N. Given
n
a (cid:12)xed but arbitrary (cid:11) 2 K, there is a sequence (cid:11) (cid:31) (cid:11) (cid:31) ::: (cid:31) (cid:11) of elements of
0 0 1 n
K. Since Y is also in (G) , the de(cid:12)nition of the jumphierarchy yields that X is an
(cid:11)n
element of (G) . Hence, the set X is in each (G) for (cid:13) 2 K, thus also in M.
(cid:11)0 (cid:13)
Often, one can boost the above argument by imposing additional properties on the
pseudo-hierarchy. Avigad’s argument in [2] for instance, considers a monotone op-
eration FA induced by an arithmetical formula. Then, there is a monotone pseudo-
hierarchy G where (G) = FA( (G) ) and further, if x 2 (G) , then there is
(cid:11) (cid:12)(cid:30)(cid:11) (cid:12) (cid:11)
a (cid:30)-least level (cid:11) where x appears (cid:12)rst. For a set K open at the bottom, each
0
S
x 2 Z := (G) enters the hierarchy at a least level (cid:11) . Clearly, (cid:11) is not in K,
(cid:11)2K (cid:11) 0 0
otherwise there is a (cid:12) 2 K with (cid:12) (cid:30) (cid:11) , contradicting that x 2 Z. Therefore, each
0
T
x 2 Z enters the hierarchy already at some level below K, thus Z = (G) .
(cid:11)(cid:30)K (cid:11)
Regarding Z as a union, the monotonicity of the operation and the hierarchy imply
S
that (G) (cid:18) FA(Z) for all (cid:11) (cid:30) K, whereas regarding Z as an intersection yields
(cid:11)
FA(Z) (cid:18) (G) for all (cid:11) 2 K, thus FA(Z) = Z.
(cid:11)
A more general form of the condition imposed on the pseudo-hierarchy above is
to stipulate that the underlying ordering (cid:30) of a pseudo-hierarchy G, which is not
a well-ordering, looks like a well-ordering in a model D of ACA above G. If G is
a pseudo-jumphierarchy conform with this condition, then the collection M from
our (cid:12)rst example is even a model of (cid:6)1-AC: If 8x9XA(X;x) holds in M for some
1
arithmetical formula A(U;u), then
f(cid:11) 2 Field((cid:30)) : 8x(9X 2_ (G) )A(X;x)g
(cid:11)
is a superset of K in D. Therefore it has a least element (cid:11) , which is already below
0
K. Hence, Y := (G) 2_ M, and for each x there is a least index e such that
(cid:11)0 x
A((Y) ;x). Applying arithmetical comprehension in M yields a set Z 2_ M with
ex
(Z) = Y , thus 8xA((Z) ;x). The property that the underlying ordering (cid:30) of a
x ex x
pseudo-hierarchy G lookslike a well-ordering in a model D ofACA above G is indeed
so apt for pseudo-hierarchy arguments that we hardly ever need supplementary
conditions.
Our declared aim is to establish an analogue situation with respect to pseudo-
hierarchies in subsystems of admissible set theory without foundation and explicit
mathematics. In admissible set theory, for instance, there exist initial segments of
the constructible hierarchy L along arbitrary well-orderings, so we ask for a con-
structible hierarchy along a linear ordering that is not wellfounded. Obviously, this
wish is not to ful(cid:12)ll in a well-founded universe, which highlights that the existence
of pseudo-hierarchies is not provable without additional assumptions. In particular,
neither in admissible set theory nor in explicit mathematics one can disprove that
4 Introduction
being a well-ordering is expressible by a (cid:6) or a (cid:6)+ formula, respectively, which un-
dermines the argument applied in second order arithmetic to prove the existence of
pseudo-hierarchies and forces us to come up with new ideas.
In accordance with the setting in second order arithmetic, we base the concept of
a hierarchy in admissible set theory upon the notion of an operation: A formula
A(u;v) induces an operation fA, if for each set x there exists exactly one set y such
that A(x;y). The set y is then denoted by fA(x). When iterating an operation fA
along a well-ordering (cid:30), we obtain a proper hierarchy, namely a function g with
domain Field((cid:30)) where g((cid:11)) = fA(g(cid:22)(cid:11)) for each (cid:11) in the domain of g. Again, a
pseudo-hierarchy g looks locally like a proper hierarchy but its domain is a linear
ordering that is not well-founded. Motivated by the previous example and assured
by experience, we settle for the following pseudo-hierarchy principle:
For all (cid:6) operations fA, such that
8x[Wo(x) ! 9ghierA(g;x)];
there exists a linear ordering (cid:30) that is not a well-ordering and a hierarchy g that
meet hierA(g;(cid:30))^Wog+((cid:30)), where g+ denotes the least admissible above g.
This principle is not provable in a normal theory T of admissible sets, however,
the following strategy o(cid:11)ers an excellent workaround: We simply extend T by the
above pseudo-hierarchy principle. Provided that jTj < j(cid:5)1-CA j, then this extension
1 0
is consistent, and moreover, has still the same proof-theoretic ordinal. Namely,
if we can iterate a (cid:6) operation fA along arbitrary well-orderings but no pseudo-
hierarchy forfA exists, then 8x[Wo(x) $ 9g(HierfA(g;x)^Wog+(x)]. Exploiting the
universal character of the formula Wo(x) and applying (cid:1) separation then proves the
translation of each instance of ((cid:5)1-CA) and thus also each ordinal (cid:11) below j(cid:5)1-CA j,
1 1 0
in particular TI (U;(cid:11)). Consequently, the pseudo-hierarchy principle is derivable in
(cid:1)
Ty, the extension of T by the axiom :TI (U;jTj), stating that the relation U does
(cid:1)
not have a least element with respect to the underlying ordering (cid:1) of our notation
system. Finally, jTj = jTyj is a consequence of an extension of Schu(cid:127)tte’s famous
boundedness Theorem (cf. [39]) shown in Ja(cid:127)ger and Probst [25] and summarized in
subsection III.1.4.
In explicit mathematics, the setting is again slightly di(cid:11)erent, and additional prob-
lems have tobe dealt with. The canonic notion ofan operationspecifying the transi-
tion from one level of a hierarchy to the next is now an individual term (f : < ! <)
thatmapsnamestonames. Alsoahierarchyfortheoperationf alongtheordering(cid:30)
is represented by an individual term (h : Field((cid:30)) ! <) that assigns names of types
tothe elements ofthe(cid:12)eld of(cid:30) insuch away thath((cid:11)) = f(j(f(cid:12) : (cid:12) (cid:30) (cid:11)g;h)). Now
the statement that h is a hierarchy for the operation f along the ordering (cid:30) is a (cid:6)+
formula, a (cid:6) formula that contains the naming predicate < only positively. Unfor-
tunately, the lack of an appropriate form of (cid:1) separation prevents to infer ((cid:5)1-CA)
1