more on this theme | more from this text
Full Idea
Some of the standard large cardinals (in order of increasing (logical) strength) are: inaccessible, Mahlo, weakly compact, indescribable, Erdös, measurable, strong, Wodin, supercompact, huge etc. (...and ineffable).
Gist of Idea
There are at least eleven types of large cardinal, of increasing logical strength
Source
Peter Koellner (On the Question of Absolute Undecidability [2006], 1.4)
Book Ref
-: 'Philosophia Mathematica' [-], p.10
A Reaction
[I don't understand how cardinals can have 'logical strength', but I pass it on anyway]
| 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] |