Contents 1 Subfields and scope 2 History 2.1 Early history 2.2 19th century 2.2.1 Foundational theories 2.3 20th century 2.3.1
3 Formal logical systems 3.1 First-order logic 3.2 Other classical logics 3.3 Nonclassical and modal logic 3.4 Algebraic logic 4 Set theory
5 Model theory
6
6.1 Algorithmically unsolvable problems 7
13.1 Undergraduate texts 13.2 Graduate texts 13.3 Research papers, monographs, texts, and surveys 13.4 Classical papers, texts, and collections 14 External links Subfields and scope[edit]
The Handbook of Mathematical
set theory model theory recursion theory, and proof theory and constructive mathematics (considered as parts of a single area). Each area has a distinct focus, although many techniques and results
are shared among multiple areas. The borderlines amongst these fields,
and the lines separating mathematical logic and other fields of
mathematics, are not always sharp. Gödel's incompleteness theorem
marks not only a milestone in recursion theory and proof theory, but
has also led to
L ω 1 , ω displaystyle L_ omega _ 1 ,omega . In this logic, quantifiers may only be nested to finite depths, as in first-order logic, but formulas may have finite or countably infinite conjunctions and disjunctions within them. Thus, for example, it is possible to say that an object is a whole number using a formula of L ω 1 , ω displaystyle L_ omega _ 1 ,omega such as ( x = 0 ) ∨ ( x = 1 ) ∨ ( x = 2 ) ∨ ⋯ . displaystyle (x=0)lor (x=1)lor (x=2)lor cdots . Higher-order logics allow for quantification not only of elements of
the domain of discourse, but subsets of the domain of discourse, sets
of such subsets, and other objects of higher type. The semantics are
defined so that, rather than having a separate domain for each
higher-type quantifier to range over, the quantifiers instead range
over all objects of the appropriate type. The logics studied before
the development of first-order logic, for example Frege's logic, had
similar set-theoretic aspects. Although higher-order logics are more
expressive, allowing complete axiomatizations of structures such as
the natural numbers, they do not satisfy analogues of the completeness
and compactness theorems from first-order logic, and are thus less
amenable to proof-theoretic analysis.
Another type of logics are fixed-point logics that allow inductive
definitions, like one writes for primitive recursive functions.
One can formally define an extension of first-order logic — a notion
which encompasses all logics in this section because they behave like
first-order logic in certain fundamental ways, but does not encompass
all logics in general, e.g. it does not encompass intuitionistic,
modal or fuzzy logic.
Knowledge representation and reasoning List of computability and complexity topics List of first-order theories List of logic symbols List of mathematical logic topics List of set theory topics Notes[edit] ^ Undergraduate texts include Boolos, Burgess, and Jeffrey (2002),
Enderton (2001), and Mendelson (1997). A classic graduate text by
Shoenfield (2001) first appeared in 1967.
^ Jozef Maria Bochenski, A Precis of Mathematical
"Die Ausführung dieses Vorhabens hat eine wesentliche Verzögerung dadurch erfahren, daß in einem Stadium, in dem die Darstellung schon ihrem Abschuß nahe war, durch das Erscheinen der Arbeiten von Herbrand und von Gödel eine veränderte Situation im Gebiet der Beweistheorie entstand, welche die Berücksichtigung neuer Einsichten zur Aufgabe machte. Dabei ist der Umfang des Buches angewachsen, so daß eine Teilung in zwei Bände angezeigt erschien." Translation: "Carrying out this plan [by Hilbert for an exposition on proof theory for mathematical logic] has experienced an essential delay because, at the stage at which the exposition was already near to its conclusion, there occurred an altered situation in the area of proof theory due to the appearance of works by Herbrand and Gödel, which necessitated the consideration of new insights. Thus the scope of this book has grown, so that a division into two volumes seemed advisable." So certainly Hilbert was aware of the importance of Gödel's work by 1934. The second volume in 1939 included a form of Gentzen's consistency proof for arithmetic. ^ A detailed study of this terminology is given by Soare (1996). ^ Ferreirós (2001) surveys the rise of first-order logic over other formal logics in the early 20th century. ^ Soare, Robert Irving (22 December 2011). "Computability Theory and Applications: The Art of Classical Computability" (PDF). Department of Mathematics. University of Chicago. Retrieved 23 August 2017. ^ "Jan Salamucha", http://pl.wikipedia.org/wiki/Jan_Salamucha . ^ "Stanislaw Schayer", http://pl.wikipedia.org/wiki/Stanislaw_Schayer . ^ Jozef Maria Bochenski, A Precis of Mathematical Logic, rev. and trans., Albert Menne, ed. and trans., Otto Bird, Dordrecht, South Holland: Reidel, Sec. 0.3, p. 2. ^ Jozef Maria Bochenski, A Precis of Mathematical Logic, rev. and trans., Albert Menne, ed. and trans., Otto Bird, Dordrecht, South Holland: Reidel, Sec. 0.3, p. 2. References[edit] Undergraduate texts[edit] Walicki, Michał (2011), Introduction to Mathematical Logic, Singapore: World Scientific Publishing, ISBN 978-981-4343-87-9 . Boolos, George; Burgess, John; Jeffrey, Richard (2002), Computability
and
Graduate texts[edit] Andrews, Peter B. (2002), An Introduction to Mathematical
Research papers, monographs, texts, and surveys[edit] Augusto, Luis M. (2017). Logical consequences. Theory and
applications: An introduction. London: College Publications.
ISBN 978-1-84890-236-7.
Cohen, P. J. (1966), Set Theory and the Continuum Hypothesis, Menlo
Park, CA: W. A. Benjamin .
Cohen, Paul Joseph (2008) [1966].
Classical papers, texts, and collections[edit] Burali-Forti, Cesare (1897), A question on transfinite numbers , reprinted in van Heijenoort 1976, pp. 104–111. Dedekind, Richard (1872), Stetigkeit und irrationale Zahlen . English translation of title: "Consistency and irrational numbers". Dedekind, Richard (1888), Was sind und was sollen die Zahlen? Two English translations: 1963 (1901). Essays on the Theory of Numbers. Beman, W. W., ed. and
trans. Dover.
1996. In From Kant to Hilbert: A Source
Fraenkel, Abraham A. (1922), "Der Begriff 'definit' und die Unabhängigkeit des Auswahlsaxioms", Sitzungsberichte der Preussischen Akademie der Wissenschaften, Physikalisch-mathematische Klasse, pp. 253–257 (German), reprinted in English translation as "The notion of 'definite' and the independence of the axiom of choice", van Heijenoort 1976, pp. 284–289. Frege, Gottlob (1879), Begriffsschrift, eine der arithmetischen
nachgebildete Formelsprache des reinen Denkens. Halle a. S.: Louis
Nebert. Translation: Concept Script, a formal language of pure thought
modelled upon that of arithmetic, by S. Bauer-Mengelberg in Jean Van
Heijenoort, ed., 1967. From Frege to Gödel: A Source
External links[edit] Hazewinkel, Michiel, ed. (2001) [1994], "Mathematical logic",
Encyclopedia of Mathematics,
Classical
In the London
Mathematical Logic
Set Theory & Further Logic
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