FOM: RE: conservative extension

Thu May 30 09:42:17 EDT 2002

On 29 May 2002, Neil Tennant wrote:

``Can anyone point me to a definitive survey article about everything that
is known on the topic of conservative extensions of logics or theories?''

I did not find such a survey.  However, I can provide you a list (after my
signature) of the MR numbers I found for articles on this topic and some
closely related topics.  (These are obtained by a search in the MathSciNet
database.  Probably, your University Library subscribes to this service, or
else perhaps your mathematics department does.  If not, I recommend it.)

Some of these look quite interesting, but without more specific information
about what you are looking for, it is difficult to narroa the search
criteria in the database.  By searching only for the phrase ``conservative
extension'' only in the titles, I came up with 36 matches  (When I tried
searching for ``conservative extension'' anywhere in the review, I got way
too many matches.)

I have not read these papers, but some of them have very interesting
reviews.  For example, according to the reviews,

MR: 98d:03056 discusses a linear logic formulation of ZF without regularity,
and shows it is a conservative extension of Zermelo Fraenkel Set Theory,

in MR: 95k:03007 ``The authors extend a sequent calculus presentation of
first-order intuitionistic logics by adjoining "excluded middle sequents"
and second-order quantifiers, and show that if a sequent is provable in the
resulting system then it is provable without second-order cuts. Applications
are given to formal program development and modelling techniques in formal
metatheory.'', (Review by J.A. Kalman),

etc.

and the one that looks a little more promiising, in terms of something
closer to a ``survey article'' is MR: 88h:03047.  Personally, I would
probably get the articles by Andreas Blass.  I know him, and have seen some
of his articles.  They seem to be easy enough to read (although, of course
not trivial), but also especially mathematically efficient, in terms of
getting more information across in less verbosity.

I hope this helps.

Matt Insall

[21] 81j:03088 Hajek, Petr On partially conservative extensions of
arithmetic. Logic Colloquium '78 (Mons, 1978), pp. 225--234, Stud. Logic
Foundations Math., 97, North-Holland, Amsterdam-New York, 1979. (Reviewer:
M. Yasuhara) 03F30
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[22] 80j:03083 Guaspari, D. Partially conservative extensions of arithmetic.
Trans. Amer. Math. Soc. 254 (1979), 47--68. (Reviewer: C. Smory\'nski) 03F30
(03F25 03F40 03H15)
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[23] 80d:03033 Tamthai, Mark A note on conservative extensions. Southeast
Asian Bull. Math. 2 (1978), no. 1, 53--54. (Reviewer: Perry Smith) 03C65
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[24] 58 #10448 Boffa, Maurice A finitary proof that Godel-Bernays set theory
(including the global axiom of choice) is a conservative extension of ZFC.
Bull. Acad. Polon. Sci. Ser. Sci. Math. Astronom. Phys. 26 (1978), no. 3,
207--209. (Reviewer: John W. Dawson, Jr.) 02K20
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[25] 58 #10346 Levin, A. M. A conservative extension of formal mathematical
analysis with the scheme of dependent choice. (Russian) Mat. Zametki 22
(1977), no. 1, 61--68. (Reviewer: G. E. Minc) 02D99
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[26] 57 #16059 Phillips, R. G. Omitting types in arithmetic and conservative
extensions. Victoria Symposium on Nonstandard Analysis (Univ. Victoria,
Victoria, B.C., 1972), pp. 195--202. Lecture Notes in Math., Vol. 369,
Springer, Berlin, 1974. (Reviewer: J. M. Plotkin) 02H20
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[27] 57 #16041 Williams, P. M. On the conservative extensions of semantical
systems: a contribution to the problem of analyticity. Synthese 25 (1973),
398--416. 02G99
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[28] 54 #12515 Blass, Andreas End extensions, conservative extensions, and
the Rudin-Frolik ordering. Trans. Amer. Math. Soc. 225 (1977), 325--340.
(Reviewer: S. R. Kogalovskii) 02H13 (04A05)
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[29] 52 #2826 Meyer, Robert K. Conservative extension in relevant
implication. Studia Logica 31 (1973), 39--48. (Reviewer: M. Rogava) 02C10
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[30] 52 #72 Baldwin, John T. Conservative extensions and the two cardinal
theorem for stable theories. Fund. Math. 88 (1975), no. 1, 7--9. (Reviewer:
J. M. Plotkin) 02H05 (02H13)
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[31] 45 #6615 Harris, John H. Ordinal theory in a conservative extension of
predicate calculus. Notre Dame J. Formal Logic 12 (1971), 423--428.
(Reviewer: P. Aczel) 02K35
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[32] 39 #50 Kreisel, G. Axiomatizations of nonstandard analysis that are
conservative extensions of formal systems for classical standard analysis.
1969 Applications of Model Theory to Algebra, Analysis, and Probability
(Internat. Sympos., Pasadena, Calif., 1967) pp. 93--106 Holt, Rinehart and
Winston, New York (Reviewer: J. Mycielski) 02.57
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[33] 1 886 063 Aceto, Luca; Fokkink, Wan; Verhoef, Chris Conservative
extension in structural operational semantics. Current trends in theoretical
computer science, 504--524, World Sci. Publishing, River Edge, NJ, 2001.
68Q55
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[34] 1 727 889 Aceto, Luca; Fokkink, Wan; Verhoef, Chris Conservative
extension in structural operational semantics. Bull. Eur. Assoc. Theor.
Comput. Sci. EATCS No. 69 (1999), 110--132. 68Q85
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[35] 1 150 322 Blikle, Andrzej; Tarlecki, Andrzej; Thorup, Mikkel On
conservative extensions of syntax in system development. Images of
programming. Theoret. Comput. Sci. 90 (1991), no. 1, 209--233. 68Q50 (68N05
68Q55)
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[36] 949 736 Blair, Howard A. Canonical conservative extensions of logic
program completions. 1987 Symposium on Logic Programming (San Francisco, CA,
1987), 154--161, IEEE Comput. Soc. Press, Washington, DC, 1987. 68T25

[1] 1 860 733 (Review) Aghaei, Mojtaba; Ardeshir, Mohammad Gentzen-style
axiomatizations for some conservative extensions of basic propositional
logic. Studia Logica 68 (2001), no. 2, 263--285. 03F05 (03B60)
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[2] 2001m:03048 Hajek, Petr; Paris, Jeff; Shepherdson, John Rational Pavelka
predicate logic is a conservative extension of \L ukasiewicz predicate
logic. J. Symbolic Logic 65 (2000), no. 2, 669--682. (Reviewer: Radim B\v
elohlavek) 03B50
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[3] 2001h:03042 Mares, Edwin D. CE is not a conservative extension of E. J.
Philos. Logic 29 (2000), no. 3, 263--275. (Reviewer: P. C. Gilmore) 03B47
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[4] 98d:03056 Shirahata, Masaru A linear conservative extension of
Zermelo-Fraenkel set theory. Studia Logica 56 (1996), no. 3, 361--392.
(Reviewer: Wim Veldman) 03E70 (03B60 03F50)
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[5] 98b:68044 D'Argenio, Pedro R.; Verhoef, Chris A general conservative
extension theorem in process algebras with inequalities. Theoret. Comput.
Sci. 177 (1997), no. 2, 351--380. 68Q10 (68Q60)
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[6] 97m:68050 Wong, Limsoon Normal forms and conservative extension
properties for query languages over collection types. 12th Annual ACM
SIGACT-SIGMOD-SIGART Symposium on Principles of Database Systems (PODS)
(Washington, DC, 1993). J. Comput. System Sci. 52 (1996), no. 3, part 2,
495--505. 68P15
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[7] 95k:03007 Basin, David; Matthews, Sean A conservative extension of
first-order logic and its applications to theorem proving. Foundations of
software technology and theoretical computer science (Bombay, 1993),
151--160, Lecture Notes in Comput. Sci., 761, Springer, Berlin, 1993.
(Reviewer: J. A. Kalman) 03B15 (03B35 03F05 68Q60 68T15)
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[8] 95c:03133 Judah, Haim; Marshall, M. Victoria Kelley-Morse + types of
well order is not a conservative extension of Kelley-Morse. Arch. Math.
Logic 33 (1994), no. 1, 13--21. (Reviewer: Renling Jin) 03E70 (03E35)
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[9] 92c:03059 Kim, S. M. The consistency and conservative extension problem
for the impredicative extension of ${\rm VBG}$: modifications of ${\rm ZF}$
equiconsistent with ${\rm ZF}\sp {\rm KM}$. An. \c Stiin\c t. Univ. Al. I.
Cuza Ia\c si Sec\c t. I a Mat. 35 (1989), no. 3, 199--201. (Reviewer: Eva
Coplakova) 03E30
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[10] 91d:03007 Giambrone, Steve; Meyer, Robert K. Completeness and
conservative extension results for some Boolean relevant logics. Studia
Logica 48 (1989), no. 1, 1--14. (Reviewer: Miros\l aw Szatkowski) 03B46
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[11] 90m:03073 Cegielski, Patrick La theorie des corps reels-clos inductifs
est une extension conservative de l'arithmetique de Peano. (French) [The
theory of inductive real-closed fields is a conservative extension of Peano
arithmetic] C. R. Acad. Sci. Paris Ser. I Math. 310 (1990), no. 5, 239--242.
03C60 (03F30 12D15 12L12)
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[12] 90j:03027 Kuzichev, A. A.; Kuzichev, A. S. A conservative extension of
a formal arithmetic. (Russian) Vestnik Moskov. Univ. Ser. I Mat. Mekh. 1988,
, no. 6, 77--78; translation in Moscow Univ. Math. Bull. 43 (1988), no. 6,
56--58 03B40 (03F30)
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[13] 90g:03019 Kondo, Michiro ${\rm A}1$ is not a conservative extension of
${\rm S}4$ but of ${\rm S}5$. J. Philos. Logic 18 (1989), no. 3, 321--323.
(Reviewer: Chris Brink) 03B45 (03B60)
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[14] 88h:03047 Enayat, Ali Conservative extensions of models of set theory
and generalizations. J. Symbolic Logic 51 (1986), no. 4, 1005--1021.
(Reviewer: James H. Schmerl) 03C62
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[15] 87i:03106 Flagg, R. C. Epistemic set theory is a conservative extension
of intuitionistic set theory. J. Symbolic Logic 50 (1985), no. 4, 895--902
(1986). (Reviewer: P. C. Gilmore) 03E70 (03B20 03F25 03F55)
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[16] 87f:03044 Meyer, Robert K.; Urbas, Igor Conservative extension in
relevant arithmetic. Z. Math. Logik Grundlag. Math. 32 (1986), no. 1,
45--50. (Reviewer: Wies\l aw Dziobiak) 03B45 (03F30)
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[17] 87d:03048 Plyushkyavichyus, Regimantas A conservative extension of the
quantified modal logic ${\rm S}5$. (Russian) Mat. Logika Primenen. No. 4
(1985), 25--38, 137. (Reviewer: Wies\l aw Dziobiak) 03B45
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[18] 85g:03084 Goodman, Nicolas D. Epistemic arithmetic is a conservative
extension of intuitionistic arithmetic. J. Symbolic Logic 49 (1984), no. 1,
192--203. (Reviewer: Henry Africk) 03F50 (03F05)
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[19] 82d:03096 Friedman, Harvey A strong conservative extension of Peano
arithmetic. The Kleene Symposium (Proc. Sympos., Univ. Wisconsin, Madison,
Wis., 1978), pp. 113--122, Stud. Logic Foundations Math., 101,
North-Holland, Amsterdam-New York, 1980. (Reviewer: Wilfried Sieg) 03F50
(03F35)
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[20] 82c:03051 Blass, Andreas Conservative extensions of models of
arithmetic. Arch. Math. Logik Grundlag. 20 (1980), no. 3-4, 85--94.
(Reviewer: A. J. Wilkie) 03C62
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