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Single Idea 9188

[filed under theme 5. Theory of Logic / A. Overview of Logic / 7. Second-Order Logic ]

Full Idea

Gödel proved the completeness of standard formalizations of first-order logic, including Frege's original one. However, an implication of his famous theorem on the incompleteness of arithmetic is that second-order logic is incomplete.

Gist of Idea

Gödel proved that first-order logic is complete, and second-order logic incomplete

Source

report of Kurt Gödel (works [1930]) by Michael Dummett - The Philosophy of Mathematics 3.1

Book Ref

'Philosophy 2: further through the subject', ed/tr. Grayling,A.C. [OUP 1998], p.136

A Reaction

This must mean that it is impossible to characterise arithmetic fully in terms of first-order logic. In which case we can only characterize the features of abstract reality in general if we employ an incomplete system. We're doomed.

The 33 ideas with the same theme [logic extending variables to predicates and relations]:

11033 Predications of predicates are predications of their subjects [Aristotle]
9188 Gödel proved that first-order logic is complete, and second-order logic incomplete [Gödel, by Dummett]
8789 Various strategies try to deal with the ontological commitments of second-order logic [Hale/Wright on Quine]
10014 Quine rejects second-order logic, saying that predicates refer to multiple objects [Quine, by Hodes]
10828 Quantifying over predicates is treating them as names of entities [Quine]
13639 Quine says higher-order items are intensional, and lack a clearly defined identity relation [Quine, by Shapiro]
10794 The nominalist is tied by standard semantics to first-order, denying higher-order abstracta [Marcus (Barcan)]
10769 Second-order logic isn't provable, but will express set-theory and classic problems [Tharp]
13842 Second-order completeness seems to need intensional entities and possible worlds [Hacking]
14249 Boolos reinterprets second-order logic as plural logic [Boolos, by Oliver/Smiley]
10830 Second-order logic metatheory is set-theoretic, and second-order validity has set-theoretic problems [Boolos]
10225 Monadic second-order logic might be understood in terms of plural quantifiers [Boolos, by Shapiro]
10736 Boolos showed how plural quantifiers can interpret monadic second-order logic [Boolos, by Linnebo]
10780 Any sentence of monadic second-order logic can be translated into plural first-order logic [Boolos, by Linnebo]
10616 Second-order arithmetic can prove new sentences of first-order [Smith,P]
10015 Higher-order logic may be unintelligible, but it isn't set theory [Hodes]
17791 Only second-order logic can capture mathematical structure up to isomorphism [Mayberry]
15944 Second-order logic is better than set theory, since it only adds relations and operations, and nothing else [Shapiro, by Lavine]
13629 Broad standard semantics, or Henkin semantics with a subclass, or many-sorted first-order semantics? [Shapiro]
13650 Henkin semantics has separate variables ranging over the relations and over the functions [Shapiro]
13645 In standard semantics for second-order logic, a single domain fixes the ranges for the variables [Shapiro]
13649 Completeness, Compactness and Löwenheim-Skolem fail in second-order standard semantics [Shapiro]
10299 If the aim of logic is to codify inferences, second-order logic is useless [Shapiro]
10298 Some say that second-order logic is mathematics, not logic [Shapiro]
10594 Henkin semantics is more plausible for plural logic than for second-order logic [Maddy]
19296 If second-order variables range over sets, those are just objects; properties and relations aren't sets [Hale]
11024 Semantics must precede proof in higher-order logics, since they are incomplete [Read]
21657 Since properties can have properties, some theorists rank them in 'types' [Hofweber]
10704 We can formalize second-order formation rules, but not inference rules [Potter]
15926 Second-order logic presupposes a set of relations already fixed by the first-order domain [Lavine]
10751 Second-order logic needs the sets, and its consequence has epistemological problems [Rossberg]
10759 There are at least seven possible systems of semantics for second-order logic [Rossberg]
10757 Henkin semantics has a second domain of predicates and relations (in upper case) [Rossberg]
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