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Full Idea
Late nineteenth century mathematicians said that, although plus, minus and 0 could not be precisely defined, they could be partially 'implicitly defined' as a group. This nonsense was rejected by Frege and others, as expressed in Russell 1903.
Gist of Idea
The idea that groups of concepts could be 'implicitly defined' was abandoned
Source
Wilfrid Hodges (Model Theory [2005], 2)
Book Ref
'Stanford Online Encyclopaedia of Philosophy', ed/tr. Stanford University [plato.stanford.edu], p.7
A Reaction
[compressed] This is helpful in understanding what is going on in Frege's 'Grundlagen'. I won't challenge Hodges's claim that such definitions are nonsense, but there is a case for understanding groups of concepts together.
| 10473 | Model theory studies formal or natural language-interpretation using set-theory [Hodges,W] |
| 10475 | A 'structure' is an interpretation specifying objects and classes of quantification [Hodges,W] |
| 10474 | |= should be read as 'is a model for' or 'satisfies' [Hodges,W] |
| 10476 | The idea that groups of concepts could be 'implicitly defined' was abandoned [Hodges,W] |
| 10478 | Since first-order languages are complete, |= and |- have the same meaning [Hodges,W] |
| 10477 | |= in model-theory means 'logical consequence' - it holds in all models [Hodges,W] |
| 10480 | First-order logic can't discriminate between one infinite cardinal and another [Hodges,W] |
| 10481 | Models in model theory are structures, not sets of descriptions [Hodges,W] |