Classical papers, texts, and collections edit Burali-Forti, Cesare (1897 A question on transfinite numbers, reprinted in van Heijenoort 1976,. . Matematisk-naturvidenskabelig Klasse, 6 : 136. Boston: Kluwer Academic Publishers, isbn. The immediate criticism of the method led Zermelo to publish a second exposition of his result, directly addressing criticisms of his proof ( Zermelo 1908a ).
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Gödel, Kurt (1929 Über die Vollständigkeit des Logikkalküls, doctoral dissertation, University Of Vienna. From Frege to Gödel: A Source Book in Mathematical Logic. 4 Symbolic logic edit Leopold Löwenheim ( 1915 ) and Thoralf Skolem ( 1920 ) obtained the LöwenheimSkolem theorem, which says that first-order logic cannot control the cardinalities of infinite structures. Two English translations: 1963 (1901). In 1891, he published a new proof of the uncountability of the real numbers that introduced the diagonal argument, and used this method to prove Cantor's theorem that no set can have the same cardinality as its powerset. Several deduction systems are commonly considered, including Hilbert-style deduction systems, systems of natural deduction, and the sequent calculus developed by Gentzen. 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,.g. Katz, Robert (1964 Axiomatic Analysis, Boston, MA:. In his work on the incompleteness theorems in 1931, Gödel lacked a rigorous concept of an effective formal system; he immediately realized that the new definitions of computability could be used for this purpose, allowing him to state the incompleteness theorems in generality that could. Richard Swineshead (1498 Calculationes Suiseth Anglici, Papie: Per Franciscum Gyrardengum. Another type of logics are fixed-point logics that allow inductive definitions, like one writes for primitive recursive functions. In From Kant to Hilbert: A Source Book in the Foundations of Mathematics, 2 vols, Ewald, William.,., Oxford University Press : 787832.