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Full Idea
First-order logic is hopeless for discriminating between one infinite cardinal and another.
Gist of Idea
First-order logic can't discriminate between one infinite cardinal and another
Source
Wilfrid Hodges (Model Theory [2005], 4)
Book Ref
'Stanford Online Encyclopaedia of Philosophy', ed/tr. Stanford University [plato.stanford.edu], p.11
A Reaction
This seems rather significant, since mathematics largely relies on first-order logic for its metatheory. Personally I'm tempted to Ockham's Razor out all these super-infinities, but mathematicians seem to make use of them.
| 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] |