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Consistency (English Wikipedia)

Analysis of information sources in references of the Wikipedia article "Consistency" in English language version.

refsWebsite
Global rank English rank
2doi.org
2nd place
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1ams.org
451st place
277th place
1zbmath.org
1,923rd place
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ams.org (Global: 451st place; English: 277th place)

mathscinet.ams.org

  • Carnielli, Walter; Coniglio, Marcelo Esteban (2016). Paraconsistent logic: consistency, contradiction and negation. Logic, Epistemology, and the Unity of Science. Vol. 40. Cham: Springer. doi:10.1007/978-3-319-33205-5. ISBN 978-3-319-33203-1. MR 3822731. Zbl 1355.03001.

doi.org (Global: 2nd place; English: 2nd place)

  • Tarski 1946 states it this way: "A deductive theory is called consistent or non-contradictory if no two asserted statements of this theory contradict each other, or in other words, if of any two contradictory sentences … at least one cannot be proved," (p. 135) where Tarski defines contradictory as follows: "With the help of the word not one forms the negation of any sentence; two sentences, of which the first is a negation of the second, are called contradictory sentences" (p. 20). This definition requires a notion of "proof". Gödel 1931 defines the notion this way: "The class of provable formulas is defined to be the smallest class of formulas that contains the axioms and is closed under the relation "immediate consequence", i.e., formula c of a and b is defined as an immediate consequence in terms of modus ponens or substitution; cf Gödel 1931, van Heijenoort 1967, p. 601. Tarski defines "proof" informally as "statements follow one another in a definite order according to certain principles … and accompanied by considerations intended to establish their validity [true conclusion] for all true premises – Reichenbach 1947, p. 68]" cf Tarski 1946, p. 3. Kleene 1952 defines the notion with respect to either an induction or as to paraphrase) a finite sequence of formulas such that each formula in the sequence is either an axiom or an "immediate consequence" of the preceding formulas; "A proof is said to be a proof of its last formula, and this formula is said to be (formally) provable or be a (formal) theorem" cf Kleene 1952, p. 83. Tarski, Alfred (1946). Introduction to Logic and to the Methodology of Deductive Sciences (Second ed.). New York: Dover. ISBN 0-486-28462-X. {{cite book}}: ISBN / Date incompatibility (help) Gödel, Kurt (1 December 1931). "Über formal unentscheidbare Sätze der Principia Mathematica und verwandter Systeme I". Monatshefte für Mathematik und Physik. 38 (1): 173–198. doi:10.1007/BF01700692. Gödel, Kurt (1 December 1931). "Über formal unentscheidbare Sätze der Principia Mathematica und verwandter Systeme I". Monatshefte für Mathematik und Physik. 38 (1): 173–198. doi:10.1007/BF01700692. van Heijenoort, Jean (1967). From Frege to Gödel: A Source Book in Mathematical Logic. Cambridge, MA: Harvard University Press. ISBN 0-674-32449-8. (pbk.) Reichenbach, Hans (1947). Elements of Symbolic Logic. New York: Dover. ISBN 0-486-24004-5. {{cite book}}: ISBN / Date incompatibility (help) Tarski, Alfred (1946). Introduction to Logic and to the Methodology of Deductive Sciences (Second ed.). New York: Dover. ISBN 0-486-28462-X. {{cite book}}: ISBN / Date incompatibility (help) Kleene, Stephen (1952). Introduction to Metamathematics. New York: North-Holland. ISBN 0-7204-2103-9. {{cite book}}: ISBN / Date incompatibility (help) 10th impression 1991. Kleene, Stephen (1952). Introduction to Metamathematics. New York: North-Holland. ISBN 0-7204-2103-9. {{cite book}}: ISBN / Date incompatibility (help) 10th impression 1991.
  • Carnielli, Walter; Coniglio, Marcelo Esteban (2016). Paraconsistent logic: consistency, contradiction and negation. Logic, Epistemology, and the Unity of Science. Vol. 40. Cham: Springer. doi:10.1007/978-3-319-33205-5. ISBN 978-3-319-33203-1. MR 3822731. Zbl 1355.03001.

zbmath.org (Global: 1,923rd place; English: 1,068th place)

  • Carnielli, Walter; Coniglio, Marcelo Esteban (2016). Paraconsistent logic: consistency, contradiction and negation. Logic, Epistemology, and the Unity of Science. Vol. 40. Cham: Springer. doi:10.1007/978-3-319-33205-5. ISBN 978-3-319-33203-1. MR 3822731. Zbl 1355.03001.