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Full Idea
Where Gödel's First Theorem sabotages logicist ambitions, the Second Theorem sabotages Hilbert's Programme.
Gist of Idea
Gödel's First Theorem sabotages logicism, and the Second sabotages Hilbert's Programme
Source
comment on Kurt Gödel (On Formally Undecidable Propositions [1931]) by Peter Smith - Intro to Gödel's Theorems 36
Book Ref
Smith,Peter: 'An Introduction to Gödel's Theorems' [CUP 2007], p.343
A Reaction
Neo-logicism (Crispin Wright etc.) has a strategy for evading the First Theorem.
| 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] |