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Full Idea
It is a lesson of the first incompleteness theorem that consistency does not entail soundness. If we add the negation of the gödel sentence for PA as an extra axiom to PA, the result is consistent. This negation is false, so the theory is unsound.
Clarification
PA is Peano Arithmetic
Gist of Idea
The first incompleteness theorem means that consistency does not entail soundness
Source
Leon Horsten (The Tarskian Turn [2011], 04.3)
Book Ref
Horsten,Leon: 'The Tarskian Turn' [MIT 2011], p.52
| 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] |