green numbers give full details | back to texts | unexpand these ideas
10476 | The idea that groups of concepts could be 'implicitly defined' was abandoned |
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. | |||
From: Wilfrid Hodges (Model Theory [2005], 2) | |||
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. |
10478 | Since first-order languages are complete, |= and |- have the same meaning |
Full Idea: In first-order languages the completeness theorem tells us that T |= φ holds if and only if there is a proof of φ from T (T |- φ). Since the two symbols express the same relationship, theorist often just use |- (but only for first-order!). | |||
From: Wilfrid Hodges (Model Theory [2005], 3) | |||
A reaction: [actually no spaces in the symbols] If you are going to study this kind of theory of logic, the first thing you need to do is sort out these symbols, which isn't easy! |
10477 | |= in model-theory means 'logical consequence' - it holds in all models |
Full Idea: If every structure which is a model of a set of sentences T is also a model of one of its sentences φ, then this is known as the model-theoretic consequence relation, and is written T |= φ. Not to be confused with |= meaning 'satisfies'. | |||
From: Wilfrid Hodges (Model Theory [2005], 3) | |||
A reaction: See also Idea 10474, which gives the other meaning of |=, as 'satisfies'. The symbol is ALSO used in propositional logical, to mean 'tautologically implies'! Sort your act out, logicians. |
10474 | |= should be read as 'is a model for' or 'satisfies' |
Full Idea: The symbol in 'I |= S' reads that if the interpretation I (about word meaning) happens to make the sentence S state something true, then I 'is a model for' S, or I 'satisfies' S. | |||
From: Wilfrid Hodges (Model Theory [2005], 1) | |||
A reaction: Unfortunately this is not the only reading of the symbol |= [no space between | and =!], so care and familiarity are needed, but this is how to read it when dealing with models. See also Idea 10477. |
10481 | Models in model theory are structures, not sets of descriptions |
Full Idea: The models in model-theory are structures, but there is also a common use of 'model' to mean a formal theory which describes and explains a phenomenon, or plans to build it. | |||
From: Wilfrid Hodges (Model Theory [2005], 5) | |||
A reaction: Hodges is not at all clear here, but the idea seems to be that model-theory offers a set of objects and rules, where the common usage offers a set of descriptions. Model-theory needs homomorphisms to connect models to things, |
10473 | Model theory studies formal or natural language-interpretation using set-theory |
Full Idea: Model theory is the study of the interpretation of any language, formal or natural, by means of set-theoretic structures, with Tarski's truth definition as a paradigm. | |||
From: Wilfrid Hodges (Model Theory [2005], Intro) | |||
A reaction: My attention is caught by the fact that natural languages are included. Might we say that science is model theory for English? That sounds like Quine's persistent message. |
10475 | A 'structure' is an interpretation specifying objects and classes of quantification |
Full Idea: A 'structure' in model theory is an interpretation which explains what objects some expressions refer to, and what classes some quantifiers range over. | |||
From: Wilfrid Hodges (Model Theory [2005], 1) | |||
A reaction: He cites as examples 'first-order structures' used in mathematical model theory, and 'Kripke structures' used in model theory for modal logic. A structure is also called a 'universe'. |
10480 | First-order logic can't discriminate between one infinite cardinal and another |
Full Idea: First-order logic is hopeless for discriminating between one infinite cardinal and another. | |||
From: Wilfrid Hodges (Model Theory [2005], 4) | |||
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. |