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Full Idea
There are two routes to Incompleteness results. One goes via the semantic assumption that we are dealing with sound theories, using a result about what they can express. The other uses the syntactic notion of consistency, with stronger notions of proof.
Gist of Idea
Two routes to Incompleteness: semantics of sound/expressible, or syntax of consistency/proof
Source
Peter Smith (Intro to Gödel's Theorems [2007], 18.1)
Book Ref
Smith,Peter: 'An Introduction to Gödel's Theorems' [CUP 2007], p.156
| 19391 | We can assign a characteristic number to every single object [Leibniz] |
| 10621 | Gödel's First Theorem sabotages logicism, and the Second sabotages Hilbert's Programme [Smith,P on Gödel] |
| 17888 | The undecidable sentence can be decided at a 'higher' level in the system [Gödel] |
| 17883 | Gödel's Theorems did not refute the claim that all good mathematical questions have answers [Gödel, by Koellner] |
| 10770 | If completeness fails there is no algorithm to list the valid formulas [Tharp] |
| 10609 | Two routes to Incompleteness: semantics of sound/expressible, or syntax of consistency/proof [Smith,P] |
| 17894 | We have no argument to show a statement is absolutely undecidable [Koellner] |
| 15353 | The first incompleteness theorem means that consistency does not entail soundness [Horsten] |
| 10755 | A deductive system is only incomplete with respect to a formal semantics [Rossberg] |
| 13852 | Axioms are ω-incomplete if the instances are all derivable, but the universal quantification isn't [Engelbretsen/Sayward] |