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Full Idea
No logic which can axiomatise arithmetic can be compact or complete.
Gist of Idea
No logic which can axiomatise arithmetic can be compact or complete
Source
John Mayberry (What Required for Foundation for Maths? [1994], p.412-1)
Book Ref
'Philosophy of Mathematics: anthology', ed/tr. Jacquette,Dale [Blackwell 2002], p.412
A Reaction
I take this to be because there are new truths in the transfinite level (as well as the problem of incompleteness).