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Full Idea
To the extent that we are justified in accepting Peano Arithmetic we are justified in accepting its consistency, and so we know how to expand the axiom system so as to overcome the limitation [of Gödel's Second Theorem].
Gist of Idea
PA is consistent as far as we can accept, and we expand axioms to overcome limitations
Source
Peter Koellner (On the Question of Absolute Undecidability [2006], 1.1)
Book Ref
-: 'Philosophia Mathematica' [-], p.4
A Reaction
Each expansion brings a limitation, but then you can expand again.
| 17884 | Mathematical set theory has many plausible stopping points, such as finitism, and predicativism [Koellner] |
| 17887 | PA is consistent as far as we can accept, and we expand axioms to overcome limitations [Koellner] |
| 17891 | Arithmetical undecidability is always settled at the next stage up [Koellner] |
| 17890 | There are at least eleven types of large cardinal, of increasing logical strength [Koellner] |
| 17893 | 'Reflection principles' say the whole truth about sets can't be captured [Koellner] |
| 17894 | We have no argument to show a statement is absolutely undecidable [Koellner] |