An introduction to Gödel's theorems:
Gespeichert in:
| 1. Verfasser: | |
|---|---|
| Format: | Buch |
| Sprache: | English |
| Veröffentlicht: |
Cambridge [u.a.]
Cambridge Univ. Press
2009
|
| Ausgabe: | 4. printing with corr. |
| Schriftenreihe: | Cambridge introductions to philosophy
|
| Schlagworte: | |
| Online-Zugang: | Inhaltsverzeichnis |
| Beschreibung: | XIV, 361 S. |
| ISBN: | 9780521674539 9780521857840 |
Internformat
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| 100 | 1 | |a Smith, Peter |d 1944- |e Verfasser |0 (DE-588)173658121 |4 aut | |
| 245 | 1 | 0 | |a An introduction to Gödel's theorems |c Peter Smith |
| 250 | |a 4. printing with corr. | ||
| 264 | 1 | |a Cambridge [u.a.] |b Cambridge Univ. Press |c 2009 | |
| 300 | |a XIV, 361 S. | ||
| 336 | |b txt |2 rdacontent | ||
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| 338 | |b nc |2 rdacarrier | ||
| 490 | 0 | |a Cambridge introductions to philosophy | |
| 650 | 0 | 7 | |a Gödelscher Unvollständigkeitssatz |0 (DE-588)4021417-5 |2 gnd |9 rswk-swf |
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| 999 | |a oai:aleph.bib-bvb.de:BVB01-017630699 | ||
Datensatz im Suchindex
| _version_ | 1804139230512807936 |
|---|---|
| adam_text | Contents
Preface
xiii
1
What
Gödel s
Theorems say
1
Basic arithmetic
·
Incompleteness
·
More incompleteness
■
Some implica¬
tions?
·
The uiiprovability of consistency
·
More implications?
■
What s
next?
2
Decidability and enumerability
8
Functions
·
Effective decidability, effective eomputability
·
Enumerable
sets
-
Effective onumerability
·
Effectively enumerating pairs of numbers
3
Axiomatized formal theories
17
Formalizat
ion as an ideal
·
Formalized languages
·
Axiomatized formal
theories
■
More definitions
·
The effective eimmerability of theorems
·
ľsegaf
ion-complete theories are deeidable
4
Capturing numerical properties
28
Three remarks on notation
·
A remark about extensionality
·
The language
L
-A quick remark about truth
■
Expressing numerical properties and
relations
-
Capturing numerical properties and relations
·
Expressing vs.
capturing: keeping the distinction clear
5
The truths of arithmetic
37
Sufficiently expressive languages
·
More about effectively enumerable sets
•
The truths of arithmetic are not effectively enumerable
■
Incompleteness
6
Sufficiently strong arithmetics
43
The idea of a sufficiently strong theory
·
An undecidability theorem
·
Another incompleteness theorem
7
Interlude: Taking stock
47
Comparing incompleteness arguments
·
A road-map
8
Two formalized arithmetics
51
BA. Baby Arithmetic
·
BA is negation complete
-
Q. Robinson Arithmetic
■
Q
is not complete
·
Why
Q
is interesting
Vil
Contents
9
What Q can
prove
58
Systems
of logic
·
Capturing less-than-or-equal-to in
Q
·
Adding
<
to
Q
·
Q
is order-adequate
·
Defining the
Δο, Σι
and
Πι
wffs
·
Some easy results
■
Q
is Ei-complete
·
Intriguing corollaries
·
Proving
Q
is order-adequate
10
First-order Peano Arithmetic
71
Induction and the Induction Schema
·
Induction and relations
·
Arguing
using induction
·
Being more generous with induction
■
Summary overview
of PA
·
Hoping for completeness?
·
Where we ve got to
■
Is PA consistent?
11
Primitive recursive functions
83
Introducing the primitive recursive functions
■
Defining the p.r. functions
more carefully
·
An aside about extensionality
■
The p.r. functions are
computable
■
Not all computable numerical functions are p.r.
·
Defining
p.r. properties and relations
■
Building more p.r. functions and relations
·
Further examples
12
Capturing p.r. functions
99
Capturing a function
·
Two more ways of capturing a function
·
The idea
of p.r. adequacy
13
Q
is p.r. adequate
106
More definitions
·
Q
can capture all
Σ ι
functions
■
La can express all p.r.
functions: starting the proof
·
The idea of a /j-function
·
L. can express
all p.r. functions: finishing the proof
·
The p.r. functions are
Σι
·
The
adequacy theorem
■
Ganonically capturing
14
Interlude: A very little about
Principia
118
Principias logicism
·
Godeľs
impact
·
Another road-map
15
The arithmetization of syntax
124
Godei
numbering
■
Coding sequences
·
Term. Atom, Wff. Sent and Prf
are p.r.
·
Some cute notation
·
The idea of diagonalization
■
What s next?
■
The concatenation function
·
Proving that Term is p.r.
■
Proving that
Atom and Wff are p.r.
■
Proving Prf is p.r.
16
PA is incomplete
138
Reminders
·
G
is true if and only if it is improvable
·
PA is incomplete: the
semantic argument
·
G is of
Goldbach
type
·
Starting the syntactic argu¬
ment for incompleteness
·
^--incompleteness, ¡¿-inconsistency
■
Finishing
the syntactic argument
■
Godei
sentences and what they say
17 Gödel s
First Theorem
147
VIII
Contents
Generalizing the semantic argument
■
Incompletability
·
Generalizing the
syntactic argument
·
The First Theorem
18
Interlude: About the First Theorem
153
What we ve proved
·
The reach of
Gödelian
incompleteness
·
Some ways
to argue that Gr is true
■
What doesn t follow from incompleteness
·
What
does follow from incompleteness?
19
Strengthening the First Theorem
162
Broadening the scope of the incompleteness theorems
■
True Basic Arith¬
metic can t be axiomatized
·
Rossers improvement
·
l-consistency and
Σ
j
-soundness
20
The Diagonalization Lemma
169
Provability predicates
■
An easy theorem about provability predicates
·
G
and
Prov
·
Proving that
G
is equivalent to
-■Prov^G 1)
·
Deriving the
Diagonalization Lemma
21
Using the Diagonalization Lemma
175
The First Theorem again
·
An aside:
Godei
sentences
again
·
The
Gödel-
Rosser Theorem again
·
Capturing provability?
■
Tarskľs
Theorem
·
The
Master Argument
·
The length of proofs
22
Second-order arithmetics
187
Second-order arithmetical languages
■
The Induction Axiom
·
Neat arith¬
metics
■
Introducing PA2
■
Categoricity
■
Incompleteness and categoricity
■
Another arithmetic
·
Speed-up again
23
Interlude: Incompleteness and Isaacson s conjecture
200
Taking stock
■
Goodstein s Theorem
·
Isaacson s conjecture
·
Ever upwards
•
Ancestral arithmetic
24 Gödel s
Second Theorem for PA
213
Defining Con
·
The Formalized First Theorem in PA
■
The Second Theorem
for PA
·
On ¿¿¡-incompleteness and ¿¿-consistency again
■
How should we
interpret the Second Theorem?
·
How interesting is the Second Theorem
for PA?
■
Proving the consistency of PA
25
The derivability conditions
223
More notation
·
The
Hilbert-Bernays-Löb
derivability conditions
■
G, Con.
and
Godei
sentences
·
Incompletability and consistency extensions
·
The
equivalence of fixed points for -iProv
·
Theories that prove their own
inconsistency
·
Lob s Theorem
IX
Contents________________________________________________
26
Deriving the derivability conditions
233
Nice* theories
·
The second derivability condition
·
The third derivability
condition
·
Useful corollaries
■
The Second Theorem for weaker arithmetics
•
Jeroslow s Lemma and the Second Theorem
27
Reflections
241
The Second Theorem: the story so far
■
There are provable consistency
sentences
·
What does that show?
·
The reflection schema: some definitions
•
Reflection and PA
·
Reflection, more generally
■
The best and most
general version
·
Another route to accepting
a Godei
sentence?
28
Interlude: About the Second Theorem
253
Real vs ideal mathematics
·
A quick aside:
Gödel s
caution
·
Relating
the real and the ideal
·
Proving real-soundness?
·
The impact of
Godei
■
Minds and computers
·
The rest of this book: another road-map
29
μ
-Recursive
functions
266
Minimization and
μ
-recursive
functions
·
Another definition of
μ
-recursive-
ness
■
The
Ackermann-Péter
function
·
Ackermann-Péter
is ^-recursive but
not p.r.
·
Introducing Church s Thesis
·
Why can t we diagonalize out?
·
Using Church s Thesis
30
Undecidability and incompleteness
278
Q
is recursively adequate
·
Nice theories can only capture /¿-recursive func¬
tions
·
Some more definitions
·
Q
and PA are undecidable
·
The
Entschei¬
dungsproblem ■
Incompleteness theorems again
■
Negation-complete the¬
ories are recursively decidable
■
Recursively adequate theories are not
recursively decidable
·
True Basic Arithmetic is not r.e.
31
Turing machines
288
The basic conception
■
Turing computation defined more carefully
·
Some
simple examples
·
Turing machines and their states
32
Turing machines and recursiveness
299
/¡-Recursiveness entails Turing computability
·
μ
-Recursiveness
entails
Turing computability: the details
·
Turing computability entails
μ
-recurs-
iveiiess
■
Generalizing
33
Halting problems
306
Two simple results about Turing programs
·
The halting problem
·
The
Entscheidungsproblem
again
·
The halting problem and incompleteness
■
Another incompleteness argument
·
Kleene s Normal Form Theorem
·
Kleenex Theorem entails
Godeľs
First Theorem
Contents
34
The Church-Turing Thesis
316
From Euclid to Hubert
· 1936
and all that
·
What the Church-Turing
Thesis is not
·
The status of the Thesis
35
Proving the Thesis?
325
The project
·
Vagueness and the idea of computability
■
Formal proofs
and informal demonstrations
·
Squeezing arguments
·
The first premiss
for a squeezing argument
·
The other premisses, thanks to Kolmogorov
and Uspenskii
■
The squeezing argument defended
·
To summarize
36
Looking back
343
Further reading
345
Bibliography
347
Index
357
xi
|
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| language | English |
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| spelling | Smith, Peter 1944- Verfasser (DE-588)173658121 aut An introduction to Gödel's theorems Peter Smith 4. printing with corr. Cambridge [u.a.] Cambridge Univ. Press 2009 XIV, 361 S. txt rdacontent n rdamedia nc rdacarrier Cambridge introductions to philosophy Gödelscher Unvollständigkeitssatz (DE-588)4021417-5 gnd rswk-swf (DE-588)4151278-9 Einführung gnd-content Gödelscher Unvollständigkeitssatz (DE-588)4021417-5 s DE-604 Digitalisierung UB Regensburg application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=017630699&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Smith, Peter 1944- An introduction to Gödel's theorems Gödelscher Unvollständigkeitssatz (DE-588)4021417-5 gnd |
| subject_GND | (DE-588)4021417-5 (DE-588)4151278-9 |
| title | An introduction to Gödel's theorems |
| title_auth | An introduction to Gödel's theorems |
| title_exact_search | An introduction to Gödel's theorems |
| title_full | An introduction to Gödel's theorems Peter Smith |
| title_fullStr | An introduction to Gödel's theorems Peter Smith |
| title_full_unstemmed | An introduction to Gödel's theorems Peter Smith |
| title_short | An introduction to Gödel's theorems |
| title_sort | an introduction to godel s theorems |
| topic | Gödelscher Unvollständigkeitssatz (DE-588)4021417-5 gnd |
| topic_facet | Gödelscher Unvollständigkeitssatz Einführung |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=017630699&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
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