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Full Idea
Gödel was quick to point out that his original incompleteness theorems did not produce instances of absolute undecidability and hence did not undermine Hilbert's conviction that for every precise mathematical question there is a discoverable answer.
Gist of Idea
Gödel's Theorems did not refute the claim that all good mathematical questions have answers
Source
report of Kurt Gödel (works [1930]) by Peter Koellner - On the Question of Absolute Undecidability Intro
Book Ref
-: 'Philosophia Mathematica' [-], p.1
A Reaction
The normal simplistic view among philosophes is that Gödel did indeed decisively refute the optimistic claims of Hilbert. Roughly, whether Hilbert is right depends on which axioms of set theory you adopt.
Related Idea
Idea 17885 Gödel eventually hoped for a generalised completeness theorem leaving nothing undecidable [Gödel, by Koellner]
| 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] |