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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.
| 17892 | For clear questions posed by reason, reason can also find clear answers [Gödel] |
| 9188 | Gödel proved that first-order logic is complete, and second-order logic incomplete [Gödel, by Dummett] |
| 10620 | Originally truth was viewed with total suspicion, and only demonstrability was accepted [Gödel] |
| 17883 | Gödel's Theorems did not refute the claim that all good mathematical questions have answers [Gödel, by Koellner] |
| 17885 | Gödel eventually hoped for a generalised completeness theorem leaving nothing undecidable [Gödel, by Koellner] |
| 10614 | The real reason for Incompleteness in arithmetic is inability to define truth in a language [Gödel] |