# Syllabus Philosophy 420A Topics in Symbolic Logic

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Syllabus

Philosophy 420A

Topics in Symbolic Logic

Gödel’s Logic

Second Term 2012
-
2013

INSTRUCTOR:

John Woods

MW 10:30
-
12:00

john.woods@ubc.ca

BUCH
D325

www.johnwoo
ds.ca

OFFICE HOURS

BUCH E 276
Tuesday
9:30
-
11:00

COURSE DESCRIPTION

We
will be examining

the Gödel incompleteness theorems for formalized
arithmetic. Arguably logic’s most important limitation results, the incompleteness
theorems are widely cons
idered the capstone of the Golden Age of mathematical logic.

The first of these theorems asserts that when the Peano axioms for arithmetic are
embedded in the logic of
Principia Mathematica
, it is demonstrable that there are true
sentences of formal arith
metic which the system, if consistent, can’t prove. If the system
is omega
-
consistent (to be explained), it is also provable that there exist sentences such
that neither they nor their negations are provable. The second incompleteness theorem
asserts that
no formal system of this kind has resources enough to prove its own
consistency.

Students will be expected to achieve a technically accurate and conceptually
mature command of these theorems and of their importance, or otherwise, for the
philosophy of log
ic and mathematics, and for certain foundational questions in
metaphysics, the philosophy of mind, and artificial intelligence.

BACKGROUND REQUIREMENTS

Students are required to have completed Philosophy 320 or an equivalent course,
preferably with a gra
de of B or higher. A facility with logic’s technical methods is also
necessary, as is an openness to see and explore the philosophical relevance of the
theorems. It is an easy matter to devise a formal system some of whose true sentences are
unprovable. Wh
at is it, then, about the Gödel results that made them so important both
technically and philosophically? Accordingly it is desirable that students bring to the
course a solid philosophical background.

2

Francesco Berto,
There is Something

About Gödel: A Complete Guide to the
Incompleteness Theorem
, Oxford: Wiley
-
Blackwell, 2009.

This is a beautifully clear and accurate presentation of the material, with no technical
demands beyond what is required for accuracy, and filled with interesting

philosophical
suggestions. The book is available as a moderately priced paperback.

Instructor’s

Notes on Gödel
, to be posted online at regular intervals.

Kurt Gödel,
On Formally Undecidable Propositions of Principia

Mathematica
and Related Systems,
translated by B. Beltzer and with an Introduction by R.B.
Braithwaite, New York: Dover, 1992. This is an unaltered reprint of the Basic
Books edition of 1962. Originally published in German as an article in
Monatschefte fü
r Mathematik und Physik,
38 (1931), 173
-
198.

It is perhaps a bit strange to list this English translation of Gödel’s original paper as

supplementary reading. But more recent expositions are more accessible. Even so,
anyone

interested in the history of log
ic should certainly read it. Inexpensive

paperback
.

Raymond M. Smullyan,
Gödel’s Incompleteness Theorems
, volume 22, Oxford
Logic Guides, New York: Oxford University Press, 1992.

This elegant book is a masterpiece for which a “standard one
-
semester cours
e in
mathematical logic is more than enough for the understanding of [it].” This advice is
rather more hopeful than accurate. Berto’s book is more user
-
friendly and still
mathematically spot on. But if one is up to it, Smullyan’s is a must. Expensive.

Her
bert B. Enderton,
A Mathematical Introduction to Logic,
2
nd

edition, San

As the author writes, “The book is intended to serve as a textbook for an introductory
mathematics course in logic at the junior
-
senior level

The book
is intended for the
reader who had not studied logic previously, but who has some experience in
mathematical reasoning.” Enderton’s is a highly regarded text, whose treatment of
Gödel’s theorems and mathematically related matters is excellent. Anyone wanti
ng a first
class review of the basic concepts and results in proof, truth and computability will read
this book with profit. But as an introductory text, is more suitable for mathematicians
than philosophers. Fairly expensive.

Peter Smith,
An Introduction

to Gödel’s Theorems
, New York: Cambridge
University Press, 2007.

3

This too will appeal to philosophy students with some familiarity with the formal
apparatus of first order logic. We have here an agreeable mix of technical accuracy and
expository attracti
veness. The book is available as a moderately priced paperback.

Kurt Gödel,
The Collected Works,
Solomon Feferman, editor
-
in
-
chief, John W.
Dawson, Warren Goldfarb, Charles Parsons
, Wilfred Sieg and Robert Solova
y,
editors, New York: Oxford University Pre
ss, in five volumes, 1986
-
2003.

A magnificent scholarly achievement, and the
ideally
perfect Christmas present
.

Ernest Nagel and James R. Newman,
Gödel’s Proof,
revised edition, edited and
with a new foreword by Douglas R. Hofstader, New York: New York U
niversity
Press, 2001.

This is the golden oldie of user
-
friendly expositions for the smart general reader,
appearing originally as an article in
Scientific American
in 1956. The new edition is
graced with a charming foreword by Douglas Hofstader author of

Gödel, Escher, Bach:
An Eternal Golden Braid
. Inexpensive

paperback
.

John Pollock ,
Technical Methods in Philosophy,
Boulder: Westview Press, 1990.

This slim volume is a mathematically correct and philosophically agreeable introduction
to mathematical l
ogic. It contains a compact and accessible treatment of what is
essentially Gödel’s first incompleteness theorem. Inexpensive

paperback
.

Mark van Atten and Juliette Kennedy, “Gödel’s logic”, in Dov M. Gabbay and
John Woods, editors,
Logic From Gödel to Ch
urch
, pages 449
-
509, volume 5 of
Gabbay and Woods,
Handbook of the History of Logic,
Amsterdam: North
-
Holland, 2009.

This is an elegant survey of Gödel’s contributions to mathematical logic other than his
work in set theory. (Gödel’s set theoretic

signifi
cance is discussed in various places in
volume 6 of the same
Handbook,

entitled
Sets and Extensions in the Twentieth Century
).
The coverage here is broader than is required for the incompleteness results, but it
supplies a helpful context in which to refle
ct on them. Very expensive. But available in
the Library.

Martin Davis, editor,
The U
ndecidable: Basic Papers on Undecidable
Propositions, Unsolvable Problems and Computable Functions,
Mineola, NY:
Doveer, 1993. First published in 1965 by Raven Press.

Th
is is a classic anthology of the fundamental papers on undecidability and unsolvability
published in the period from 1931 to 1946 by the giants in the field: Gödel, Church,
Turing, Rosser and Kleene.

ASSIGNMENTS

4

1.

In
-
class test, Monday 21

January 2013. Wor
th 20% of the course’s final mark.

2.

In
-
class test,

Wednesday, 20

February 2013. Worth 20% of the course’s final
mark.

3.

In
-
c
lass test, Monday 25

March 2013. Worth 20% of the course’s final mark.

4.

Final examination TBA. Worth 40% of the final mark.

LE

A+ 90
-
100

A 85
-
89

A
-

80
-
84

B+ 76
-
79

B 72
-
75

B
-

68
-
71

C+ 64
-
67

C 60
-
63

C
-

55
-
59

D 50
-
54

F 0
-
49 (fail)

Please note that students are expected to come to class well
-