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Single Idea 17891

[filed under theme 6. Mathematics / B. Foundations for Mathematics / 4. Axioms for Number / g. Incompleteness of Arithmetic ]

Full Idea

The arithmetical instances of undecidability that arise at one stage of the hierarchy are settled at the next.

Gist of Idea

Arithmetical undecidability is always settled at the next stage up

Source

Peter Koellner (On the Question of Absolute Undecidability [2006], 1.4)

Book Ref

-: 'Philosophia Mathematica' [-], p.10

The 6 ideas from 'On the Question of Absolute Undecidability'

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]