Ideas from 'First-Order Logic' by Wilfrid Hodges [2001], by Theme Structure

[found in 'Blackwell Guide to Philosophical Logic' (ed/tr Goble,Lou) [Blackwell 2001,0-631-20693-0]].

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5. Theory of Logic / A. Overview of Logic / 1. Overview of Logic
Logic is the study of sound argument, or of certain artificial languages (or applying the latter to the former)
5. Theory of Logic / I. Semantics of Logic / 1. Semantics of Logic
A formula needs an 'interpretation' of its constants, and a 'valuation' of its variables
I |= φ means that the formula φ is true in the interpretation I
There are three different standard presentations of semantics
5. Theory of Logic / J. Model Theory in Logic / 3. L÷wenheim-Skolem Theorems
Up L÷wenheim-Skolem: if infinite models, then arbitrarily large models
Down L÷wenheim-Skolem: if a countable language has a consistent theory, that has a countable model
5. Theory of Logic / K. Features of Logics / 6. Compactness
If a first-order theory entails a sentence, there is a finite subset of the theory which entails it
6. Mathematics / B. Foundations for Mathematics / 6. Mathematics as Set Theory / a. Mathematics is set theory
A 'set' is a mathematically well-behaved class