By C. J. Ash;J. N. Crossley;C. J. Brickhill;J. C. Stillwell;N. H. Williams
This advent to the most rules and result of mathematical common sense is a major therapy aimed at non-logicians. beginning with a old survey of good judgment in precedent days, it strains the 17th-century improvement of calculus and discusses sleek theories, together with set conception, the continuum speculation, and different principles. 1972 version.
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Perhaps I can dwell a little on this, since symbolism became so important from this point onwards. A little description of what symbolism can look like will help. Purely logical connectives, such as and, or, not are given symbols such as &, V, ; need symbols (x, y, z and so on) for variables and also symbols P, Q, R for predicates (or properties or relations). Out of these we make formulae such as this: P(x) V Q(x), which is read as saying that x has property P or x has property Q, and this can be quantified by expressing ‘for all x’ by ∀x and ‘there exists an x’ by x.
For suppose that Σ′ is a finite subset of Σ*. Then Σ′ contains, apart from some of the sentences of Σ, finitely many sentences E(ci, cj). These will involve only finitely many of the constant letters ci, which will, for some n, be among c1, …, cn Now, by assumption, Σ has a normal model 〈A, …〉 with at least n elements, so choosing elements a1, a2, … of A with a1, …, an distinct, it is easy to see that 〈A, …,a1, a2, …〉 is a model for Σ′, where a1, a2, … are the interpretations of c1, c2, … Thus Σ* has a model, and so has a normal model 〈B, … b1 b2, …〉 where b1, b2, … are the interpretations of c1c2, … .
We have to say: what does ‘countable’ mean in relation to a model? And this will be treated in Chapter 3. But this was very thought-provoking and it led to exceptionally good theorems later on. All the time up until this (1920), and up to 1930 in fact, people had used predicate calculus for making logical deductions without finding out for sure whether they obtained all of the valid statements this way. The argument that was developed for the Löwenheim–Skolem theorem is similar, in some presentations at least, to a proof of the completeness of the predicate calculus.
What is Mathematical Logic? by C. J. Ash;J. N. Crossley;C. J. Brickhill;J. C. Stillwell;N. H. Williams