Goedels Theorem 1
Lecture Contents
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The notion of eective decidability
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Whats a formalised language?
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Whats a formal axiomatised theory?
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Whats negation incompleteness?
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Deductivism about basic arithmetic
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Two versions of Geodels First Incompletenes
Goedels Theorem 7
Lecture Contents
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A very little on Hilberts programme.
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Goedel coding
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The relation that holds between m and n when m codes for a
PA derivation of the w with code n is p.r (and so is
recursive).
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The (standard) notation p q to denote
Goedels Theorem 4
Lecture Contents
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The idea of a decidable theory
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The !-rule
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Induction: the induction axiom, the induction rule, the
induction schema
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First-order Peano Arithmetic
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The idea of
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Addendum: A consistent extension of Q is sound for 1
Philosophy of Mathematics
Neo-logicism
Owen Griffiths
oeg21@cam.ac.uk
St Johns College, Cambridge
10/11/15
Neo-logicism
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Neo-logicists hold that Frege was broadly correct but went
wrong when he rejected HP as an implicit definition of
number.
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Recall th
Marxism
Lecture 2 History
John Filling
jf582@cam.ac.uk
Overview
1. Precursors
2. The 1859 Preface
3. Modes, forces, and relations of production
4. Cohens functionalist interpretation
5. Questions for Cohen
6. Looking ahead
In every enquiry concerning the
Philosophy of Mathematics
Structuralism
Owen Griffiths
oeg21@cam.ac.uk
St Johns College, Cambridge
17/11/15
Neo-Fregeanism
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Last week, we considered recent attempts to revive Fregean
logicism.
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Analytic logicists try to show that HP is an analytic truth
Realism and Idealism
Michael Dummett
Owen Griffiths
oeg21@cam.ac.uk
St Johns College, Cambridge
22/10/15
Realism and antirealism
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Last week, we saw that external realism a highly intuitive
view in all sorts of areas faces a serious challenge from
Putnams
Goedels Theorem 9
Lecture Contents
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Provability predicates
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The Diagonalisation Lemma
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Incompleteness from the Diagonalisation Lemma
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Tarskis Theorem
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The Master Argument
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Rossers Theorem
Provability Predicates
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PrfT stands in for a T -w that captu
Marxism
Lecture 1 Introduction
John Filling
jf582@cam.ac.uk
Overview
1. What?
2. Who and when?
3. How?
4. What?, again
5. Looking ahead
What?
1. An historical movement?
2. A set of questions?
3. A set of answers?
4. A distinctive method?
Overview
1. What?
Philosophy of Mathematics
Kant and Frege
Owen Griffiths
oeg21@cam.ac.uk
St Johns College, Cambridge
27/10/15
Last week
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Kant argued that mathematical truths are synthetic and a
priori.
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They are synthetic since they are non-trivial, ampliative,
depend o
Philosophy of Mathematics
Freges logicism
Owen Griffiths
oeg21@cam.ac.uk
St Johns College, Cambridge
3/11/15
Frege and Kant
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Kant took arithmetic and geometry to be bodies of synthetic
a priori truths.
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Frege agreed about geometry, but disagreed about a
Goedels Theorem 8
Lecture Contents
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The idea of diagonalisation
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How to construct a canonical Goedel sentence
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If an axiomatised theory is sound, it is negation incomplete
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Applying that to PA
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!-incompleteness, !-inconsistency
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If PA is !-consisten
Realism and Idealism
Internal realism
Owen Griffiths
oeg21@cam.ac.uk
St Johns College, Cambridge
12/11/15
Easy answers
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Last week, we considered the metaontological debate between
Quine and Carnap.
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Quine put forward a methodology for approaching ontolo
Goedels Theorem 3
Lecture Contents
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The idea of a decidable theory
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Baby Arithmetic is negation complete
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Robinson Arithmetic, Q
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A simple proof that Robinson Arithmetic is not complete
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Adding to Robinson Arithmetic
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Why Robinson Arithmetic is inte
Philosophy of Mathematics
Nominalism
Owen Griffiths
oeg21@cam.ac.uk
St Johns College, Cambridge
24/11/15
Structuralism
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Last week, we looked at structuralism as a philosophy of
mathematics.
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We focussed on a Shapiros platonist structuralism.
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This week
Philosophy of Mathematics
Intuitionism
Owen Griffiths
oeg21@cam.ac.uk
St Johns College, Cambridge
01/12/15
Classical mathematics
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Consider the Pythagorean argument that
2 is irrational:
1. Assume that 2 is rational, so 2 = mn , where m and n are
coprime.
Marxism
Lecture 3 Ideology
John Filling
jf582@cam.ac.uk
Overview
1. What is ideology?
2. What do we want from a theory of ideology?
3. Models of ideology
4. Fetishism
5. Looking ahead
What ideology is not
Ideology in the
1. conventional sense
e.g. social
Goedels Theorem 2
Lecture Contents
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The idea of a decidable theory
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Any consistent, negation-complete, axiomatised formal theory
is decidable
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Expressing and capturing properties, relations and functions
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The idea of a sufficiently strong theory
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No
Goedels Theorem 5
Lecture Contents
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What is a primitive recursive function?
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How to prove results about all p.r. functions
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The p.r. functions are computable . . .
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. . . but not all computable functions are p.r.
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Recursive functions
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The idea of a
Goedels Theorem 10
Lecture Contents
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ConT , a canonical consistency sentence for T
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The Formalised First Theorem
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The Second Theorem
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Why the Second Theorem matters
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What it takes to prove it
Definitions
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? is T s absurdity constant if it has one, o
Philosophy of Mathematics
Kant
Owen Griffiths
oeg21@cam.ac.uk
St Johns College, Cambridge
20/10/15
Immanuel Kant
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Born in 1724 in K
onigsberg, Prussia.
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Enrolled at the University of K
onigsberg in 1740 and remained
there his whole career.
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Published T
Goedels Theorem 6 (abridged)
Lecture Contents
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LA can express all recursive (and p.r.) functions
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The role of the -function trick in proving that result
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In fact, 1 ws suffice to express all recursive functions
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Q can capture all recursive functions
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Realism and Idealism
Ontology
Owen Griffiths
oeg21@cam.ac.uk
St Johns College, Cambridge
5/11/15
The story so far
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So far, weve considered realism (understood as external
realism or semantic realism) and antirealism (understood in
Dummetts way).
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In the
Realism and Idealism
External Realism
Owen Griffiths
oeg21@cam.ac.uk
St Johns College, Cambridge
8/10/15
What is metaphysics?
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Metaphysics is the attempt to:
give a general description of the whole of the Universe
(Moore)
describe the most general struct
Philosophy of Mathematics
Introduction
Owen Griffiths
oeg21@cam.ac.uk
St Johns College, Cambridge
13/10/15
Questions in the philosophy of mathematics
Ontological Do mathematical objects like numbers and sets exist?
Metaphysical What is the nature of mathe
Realism and Idealism
Antirealism
Owen Griffiths
oeg21@cam.ac.uk
St Johns College, Cambridge
29/10/15
Antirealism
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Last week, we considered two of Michael Dummetts most
celebrated arguments against realism: the acquisition
argument and the manifestation a
Realism and Idealism
Just More Theory
Owen Griffiths
oeg21@cam.ac.uk
St Johns College, Cambridge
15/10/15
External realism
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Last week, we saw that Putnams model-theoretic argument
presents external realism with a serious problem.
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We characterised exter
Goedels Theorem 11
Lecture Contents
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Currys Paradox
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Loebs Theorem
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Loebs Theorem implies the Second Incompleteness Theorem
again
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ConT as an undecidable sentence
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Consistent theories that prove their own inconsistency
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Whats still to come. . .
Curr