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Full Idea
Full second-order logic has all the expressive power needed to do mathematics, but has an unworkable model theory.
Gist of Idea
Second-order logic has the expressive power for mathematics, but an unworkable model theory
Source
Stewart Shapiro (Higher-Order Logic [2001], 2.1)
Book Ref
'Blackwell Guide to Philosophical Logic', ed/tr. Goble,Lou [Blackwell 2001], p.34
A Reaction
[he credits Cowles for this remark] Having an unworkable model theory sounds pretty serious to me, as I'm not inclined to be interested in languages which don't produce models of some sort. Surely models are the whole point?
| 18841 | Categoricity implies that Dedekind has characterised the numbers, because it has one domain [Rumfitt on Dedekind] |
| 10833 | Many concepts can only be expressed by second-order logic [Boolos] |
| 10294 | Second-order logic has the expressive power for mathematics, but an unworkable model theory [Shapiro] |
| 10164 | Peano Arithmetic can have three second-order axioms, plus '1' and 'successor' [Reck/Price] |
| 13701 | A single second-order sentence validates all of arithmetic - but this can't be proved axiomatically [Sider] |
| 10979 | Although second-order arithmetic is incomplete, it can fully model normal arithmetic [Read] |
| 10980 | Second-order arithmetic covers all properties, ensuring categoricity [Read] |
| 17899 | Second-order induction is stronger as it covers all concepts, not just first-order definable ones [George/Velleman] |
| 10674 | A plural language gives a single comprehensive induction axiom for arithmetic [Hossack] |