Hierarchies of predicates of finite types:
Gespeichert in:
| Beteilige Person: | |
|---|---|
| Format: | Hochschulschrift/Dissertation Buch |
| Sprache: | Englisch |
| Veröffentlicht: |
Providence, RI
American Mathematical Society
1964
|
| Schriftenreihe: | Memoirs of the American Mathematical Society
51 |
| Schlagwörter: | |
| Links: | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=001837091&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| Umfang: | 95 S. |
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| adam_text | HIERARCHIES OP PBEDICATES 3
SUMMARY AND TABLE OF CONTENTS
§1. COMPLETE PREDICATES AND THEIR DEGREES.
Let m,r,s,k be arbitrary non negative integers such that
H|S S r+1 and s 0. If r 0, and for example k = 2, let the predicate
Atfm (EocjH^HEfr ^TUtf^/KV^, |r l} of c^,xxxvilll be desig¬
nated by B+1NJ; (or by m+13^ I SA3 if T is recursive in the type s predicate
SA). If r = 0, proceed similarly using C 14,XXXII11 . For all k 0,
t m j£ CSA] are complete predicates in the sense (generalized) of
C 19 3 • The degree (see[l^, 11.9]) dg(m+1J^LSA3) depends only on
Sg = dg(BA). Let m+1^ = dg(n+1N£) and IB+1(SS)^ = dg^1^ C SA]). A
nuBber of preliminary results are obtained concerning the order relation
among the degrees so definable. A theorem asserting the non existence of
complete predicates for to r+1 and s r+1 with Un and 8A actually
present is proved, leaving an Interesting open problem.
1.1, 1.2 Notations and Conventions. P. 8
1.3. Type Restrictions. P. 9
I.**. The Enumeration Theorem. P.10
1.5 Generalized Jump Operators: b+1n£, ^J^A] . • • • p.11
1.6. Complete Predicates. •«••• P.12
1.7. The Hierarchy Theorem. P«l4
1.8. Strengthening the Hierarchy Theorem. p.15
1 9,1.10 The Degrees of Jump Operators and
Complete Predicates. p#15
1.11. The Existential Operator and its Dual. P.19
1 12. Elimination of Certain Iterated Jump
Operators. 20
^ D.A. CLARKE
I.13. Kleene s Functional mp™ Applied to
Degrees. • p. 23
1.1k. The Jump Operators Applied to Degrees. p. 23
1.15. Hierarchies of Degrees: Discussion
and Notations. p. 24
1.16. On the Existence of Other Jump
Operators. P 25
§2. TEE FINITE HIERARCHIES.
The class H inductively defined, (i) m+1g^ * H , (U) Sa C W —*¦
m+1{sg)^ W, (ill) only these, is the largest class of complete degrees
(degrees of complete predicates) definable at the present stage of know¬
ledge. The class ffl+^£ is defined as ls?^, but with fixed m. We considers
(1) When are two members of % equal? (il) What is the order relation in
%t A number of results are obtained and the answer Is given completely
for^= Um m+^£ The hierarchies (a) m+1(sa)^ ^{ ^l and I
(b) m+1(sa)r: m+1(In+1(Sa)^^ •••, m $ r, sg t£, appear to be funda |
mental for the following reason. Any increasing sequence of degrees of
dL (bounded with respect to type and order) is coflnal with one of the
hierarchies under (a) or (b).
2.1. The Class % of Complete Degrees of Finite
Type Definable by Finite Quantification. P« 27
2.2. The Linear Hierarchy B+^£ of Complete B+1degrees.
The Ramified Hierarchy ^= Jm m+1£ . P 27
2.3. Matrices Associated with Degrees ofW p28
Z.k. Reduced Matrloes P« 29 :
2.5, 2.6. The Order in ,£ Determined by an Ordering
of Reduced Matrices. P* 30
HIERARCHIES OF PREDICATES 5
2.7. The Cofinality of Chains in ad with
Certain Special Chains. p. 34
2.8. Some Miscellaneous Results. Examples
and Counter Examples. p. 36
§3. TRANSFINITE HIERARCHIES.
The hierarchies (a) and (b) can be extended Into the transflnlte
by forming infinite recursive Joins C19, 3.13 and repeating the pattern;
of. Cl3i §4; 14, 11.273. (For m = r+1, the degrees of (a) (of (b) after
the first) are equal (are undefined).) We may not form recursive Joins
of arbitrary increasing sequences (of. 113, XIII3), but only of those
which are suitably enumerated. Summing only recursively enumerated
sequences In Ll3, §43proved after the fact to be adequate. As Kleene
conjectured In C14, 11.27 3, this is not so for higher types. Expanding
Kleene s proposal, we extend (a) and (b) by simultaneously (and In¬
ductively) generating ordinal notations and predicates, using previously
defined predicates to define ordinal notations and Indexing new predicates
with previously defined ordinal notations. The resulting class of
notations (treating primarily the unrelatlvlzed case sa = ()) Is ^0r
Partially well ordered by r; the predicates (ordinals) indexed by
them are ^H1 (f|a|r), a £ ?0r, where k = 0 denotes the case based on (a).
The least ordinal not indexed by °0r Is called °0Jr. All (relevant) basic
Properties analogous to[ll, 6.1 6.3; 12,§§20, 21; 14, 8.51 are proved
(here and in 7.3).
3*1. Historical Background. •••• P« 39
3.2. The Problem Discussed. P« 41
3O 3 5. Definitions of Ordinal Notations and the
Transfinite Hierarchies. P« 43
6 D.A. CLARKE
3.6. Basic Properties of the Notations. p. 47
3.7,3.8. Generalizations of the Existential Operator. •• p.48
3.9. Enumeration and Explicit Definition of the
Predicates ^B*. p. 50
3.10,3.11 The Order Among £dg(B+^)J, a £ *0r. p. 52
3.12. Enumeration of the Set S(a r t ), b € or. « p. 55
hi. REDUCTION OF THE DEFINITION OF ™0r TO EXPLICIT DEFINITION.
We reduce to definition of ?0r to explicit definition. The
analogues of r.12, Theorem I; 13, XXIV 3 are false when r 0, thus leaving
unsettled the existence of hierarchies analogous to the hyperarlthmetloal
hierarchy in this sense.
k.l. *Hr, an Alternate Inductive Definition of ™0r. p. 59
li.2 k.lt. The Forms of the Predicates In the Inductive
Clauses of the Definition of ®Rr. p. 59
^.5. The Frege Dedekind Method Applied to the
Inductive Definition of ™Rr. p. 6l
§5. UNIQUENESS ORDINALS.
The degree of m+^ depends only on £|alr (cf. T. 27t Theorem 5 ] )»
and on the degrees of the predicates used in the definitions.
5 1. Recursive Index Constructions. P* 64
5.2. The Forms of the Predicates |alr = ™| t» ir and
£|a|r$£|b|r. The Main Theorem. P» 65
5 3,5.^. Lemmas to the Main Theorem. P» 70
5.5. Relatlvlzation. P 74
HIERARCHIES OF PREDICATES 7
£6. THE CASE OF °0r ETC. WHEN m = r.
The classes m0m and m0m (and 0 If m = 0 ) index the same ordinals
and degrees; ^0 , k 1, Indexes these same ordinals, but only a proper
subset of these degrees. Apart from this, the degrees m+ihr =
dg( kHa) °* = k a r» are a11 different and extend £. The order relation
is completely determined.
6.1. Recursive Index Constructions. p. 76
6.2. The Main Theorem. p. 77
6.3. The Case m = r = 0. p. 82
6A. A Suslin Kleene Type Theorem Fails to
Hold for r 0 • p. 83
$7. RELATIVELY RECURSIVE WELL ORDERINGS.
We Introduce well orderings recursive In a predicate of finite
type (e.g. in B+iHr, which, for all a c ? )r, give exactly the ordinals
K a
^O* ; an analogue of Markwald s Theorem), and use these (1) to show
that the analogue of C27, Theorem 6, Corollary 2 ] is false when r 0,
and (11) to establish the order relation among the ordinals £uJr.
7.1. The Definitions. P 85
7 2. The Analogue of a Theorem of Spector Fails
to Hold for r 0. P 86
7.3. Sums of Ordinal Notations. p. 86
l.b. The Forms of Certain Predicates which are
Well Orderings of Types £tor. P 87
7 5 An Analogue of Markwald s Theorem. p. 89
7.6. The Order among the Ordinals °tor. p. 90
BIBLIOGRAPHY P 93
|
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| discipline | Mathematik |
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| illustrated | Not Illustrated |
| indexdate | 2024-12-20T07:56:13Z |
| institution | BVB |
| language | English |
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| publishDate | 1964 |
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| publisher | American Mathematical Society |
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| series | Memoirs of the American Mathematical Society |
| series2 | Memoirs of the American Mathematical Society |
| spellingShingle | Clarke, Douglas A. Hierarchies of predicates of finite types Memoirs of the American Mathematical Society Number theory Recursive functions Hierarchie (DE-588)4024842-2 gnd Prädikat (DE-588)4225652-5 gnd Prädikatenlogik (DE-588)4046974-8 gnd Aussage (DE-588)4003809-9 gnd |
| subject_GND | (DE-588)4024842-2 (DE-588)4225652-5 (DE-588)4046974-8 (DE-588)4003809-9 (DE-588)4113937-9 |
| title | Hierarchies of predicates of finite types |
| title_alt | Hierarchies of predicates of arbitary finite types |
| title_auth | Hierarchies of predicates of finite types |
| title_exact_search | Hierarchies of predicates of finite types |
| title_full | Hierarchies of predicates of finite types by D. A. Clarke |
| title_fullStr | Hierarchies of predicates of finite types by D. A. Clarke |
| title_full_unstemmed | Hierarchies of predicates of finite types by D. A. Clarke |
| title_short | Hierarchies of predicates of finite types |
| title_sort | hierarchies of predicates of finite types |
| topic | Number theory Recursive functions Hierarchie (DE-588)4024842-2 gnd Prädikat (DE-588)4225652-5 gnd Prädikatenlogik (DE-588)4046974-8 gnd Aussage (DE-588)4003809-9 gnd |
| topic_facet | Number theory Recursive functions Hierarchie Prädikat Prädikatenlogik Aussage Hochschulschrift Finiter Typus |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=001837091&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| volume_link | (DE-604)BV008000141 |
| work_keys_str_mv | AT clarkedouglasa hierarchiesofpredicatesoffinitetypes AT clarkedouglasa hierarchiesofpredicatesofarbitaryfinitetypes |
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