Topics in Symbolic Logic
Second Term 2012
BUCH E 276
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
, it is demonstrable that there are true
sentences of formal arith
metic which the system, if consistent, can’t prove. If the system
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
no formal system of this kind has resources enough to prove its own
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.
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
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.
There is Something
About Gödel: A Complete Guide to the
, Oxford: Wiley
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
suggestions. The book is available as a moderately priced paperback.
Notes on Gödel
, to be posted online at regular intervals.
SUPPLEMENTARY DISCRETIONARY READING
On Formally Undecidable Propositions of Principia
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
r Mathematik und Physik,
38 (1931), 173
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,
interested in the history of log
ic should certainly read it. Inexpensive
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
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.
bert B. Enderton,
A Mathematical Introduction to Logic,
Diego: Academic Press, 2001.
As the author writes, “The book is intended to serve as a textbook for an introductory
mathematics course in logic at the junior
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.
to Gödel’s Theorems
, New York: Cambridge
University Press, 2007.
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
veness. The book is available as a moderately priced paperback.
The Collected Works,
Solomon Feferman, editor
chief, John W.
Dawson, Warren Goldfarb, Charles Parsons
, Wilfred Sieg and Robert Solova
editors, New York: Oxford University Pre
ss, in five volumes, 1986
A magnificent scholarly achievement, and the
perfect Christmas present
Ernest Nagel and James R. Newman,
revised edition, edited and
with a new foreword by Douglas R. Hofstader, New York: New York U
This is the golden oldie of user
friendly expositions for the smart general reader,
appearing originally as an article in
in 1956. The new edition is
graced with a charming foreword by Douglas Hofstader author of
Gödel, Escher, Bach:
An Eternal Golden Braid
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
Mark van Atten and Juliette Kennedy, “Gödel’s logic”, in Dov M. Gabbay and
John Woods, editors,
Logic From Gödel to Ch
, pages 449
509, volume 5 of
Gabbay and Woods,
Handbook of the History of Logic,
This is an elegant survey of Gödel’s contributions to mathematical logic other than his
work in set theory. (Gödel’s set theoretic
cance is discussed in various places in
volume 6 of the same
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
Martin Davis, editor,
ndecidable: Basic Papers on Undecidable
Propositions, Unsolvable Problems and Computable Functions,
Doveer, 1993. First published in 1965 by Raven Press.
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.
class test, Monday 21
January 2013. Wor
th 20% of the course’s final mark.
February 2013. Worth 20% of the course’s final
lass test, Monday 25
March 2013. Worth 20% of the course’s final mark.
Final examination TBA. Worth 40% of the final mark.
Please note that students are expected to come to class well
prepared and on time, ready
to participate in class discussio