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Full Idea
The concept of truth of sentences in a language cannot be defined in the language. This is the true reason for the existence of undecidable propositions in the formal systems containing arithmetic.
Gist of Idea
The real reason for Incompleteness in arithmetic is inability to define truth in a language
Source
Kurt Gödel (works [1930]), quoted by Peter Smith - Intro to Gödel's Theorems 21.6
Book Ref
Smith,Peter: 'An Introduction to Gödel's Theorems' [CUP 2007], p.182
A Reaction
[from a letter by Gödel] So they key to Incompleteness is Tarski's observations about truth. Highly significant, as I take it.
| 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] |