**Stephen G. Simpson**

**March 21, 2012**

Here is a list of typographical errors in *Subsystems of Second
Order Arithmetic*, Stephen G. Simpson, Springer-Verlag, 1999, XIV +
445 pages. It contains all errors which were discovered through 2006.
All of these errors were corrected in the second edition which was
published in 2009.

- Proof of Theorem I.9.1, displayed formula, replace
by

. - Theorem I.10.3, in item 8, replace ``set of sentences'' by
``consistent set of sentences''.
- Proof of Lemma III.2.5, the subscripts need to be repaired.
The proof should be as follows:
We first consider the case of a finite product . In this case, for each , let the smallest such that . Put and let be an enumeration of . Then attests to the compactness of .

In the case of a countably infinite product , for each let smallest such that . Put and let be an enumeration of . Again attests to the compactness of . This completes the proof of the lemma.

- Proof of Theorem III.3.2, main paragraph, replace
by

, and replace by . Also, replace all three occurrences of by . - Proof of Theorem III.4.3, last line of third paragraph, replace
and .
- Proof of Theorem III.6.5, middle of page 120, replace
by
.
- Proof of Theorem III.6.5, first line of last paragraph on page
120, replace
by
.
- Proof of Lemma IV.1.4, end of proof, replace
by

. - Exercise IV.2.10, replace ``closed set'' by ``closed set ''.
- Definition V.4.5, replace A by .
- Remark V.10.1, replace - by
-.
- Theorem VI.2.6, replace ``Over'' by ``over''.
- Proof of Sublemma VI.3.3, in the definition of , replace
by .
- Definition VII.3.2.4, replace by .
- Proof of Theorem VII.3.31, first line, replace T by .
- Proof of Theorem VII.6.9, first paragraph, second last sentence,
replace ``The'' by ``Then''.
- Remark VIII.1.16, replace ``as do
and
-'' by ``while
and
- prove the same sentences.''
- Proof of Lemma VIII.2.16, first line of page 323, replace by
.
- Proof of Lemma IX.2.4, second line, replace
by
.
- Theorem X.2.9, items 5, 6, and 7, replace by
.
- Section X.3, just before Definition X.3.1, replace ``Ramsey's
theorem'' by ``Ramsey's theorem for exponent 3''.
- Definition X.4.1, replace ``atomic formula'' by ``numerical term''.
- Bibliography, item 268, replace ``borel'' by ``Borel''.
- Index, atomic formula, replace ``2, 410'' by ``2''.
- Index, formula, atomic, replace ``2, 410'' by ``2''.
- Index, GKT basis theorem, replace ``325-326'' by ``325-326,
354-356''.
- Index, numerical term, replace ``23'' by ``2, 23, 410''.
- Index, term, numerical, replace ``2, 23'' by ``2, 23, 410''.
- Index, add index entry, weak -, 342-347.
- Index, universal , replace ``333, 356'' by ``252,
333, 356''.

Here is a list of errors in *Subsystems of Second Order
Arithmetic*, Second Edition, Stephen G. Simpson, Association for
Symbolic Logic, Cambridge University Press, 2009, XVI + 444 pages.
This list is current as of today, March 21, 2012.

- The proof of Theorem IV.8.2 contains two typographical errors.
In displayed equation (15), should be
. In line 7 from the bottom of the page,
should be .
- David Madore has pointed out that part 18 of Lemma VII.3.7 is
incorrect. The error is not typographical but mathematical.
Namely,
does not include the Axiom of
Regularity, so for instance it is consistent with
that there are proper-class-many sets
such that , and of course all such sets are
hereditarily finite.
There are two ways to repair this.

- Preferred way: Move the Axiom of Regularity from Definition VII.3.8 (the axioms of ) to Definition VII.3.3 (the axioms of ).
- Another way: Change part 6 of Definition VII.3.6
to read as follows:
Trans is

Thus regularity is incorporated into the definitions of transitivity and hereditary finiteness.*transitive*, i.e., .

Under either 1 or 2, part 7 of Definition VII.3.6 can be simplified by omitting the clause which begins with .

- In line 2 of the proof of Theorem VII.3.9, ``Then'' should be
``The''.
- David Madore has pointed out another mathematical error.
Namely, in the Notes for §VII.5 on page 293, the stated
characterization of
is correct for but
incorrect for .
A correct characterization of reads as follows. the least admissible ordinal which is the limit of smaller admissible ordinals.

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