Ideas from 'FirstOrder Logic' by Wilfrid Hodges [2001], by Theme Structure
[found in 'Blackwell Guide to Philosophical Logic' (ed/tr Goble,Lou) [Blackwell 2001,0631206930]].
Click on the Idea Number for the full details 
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5. Theory of Logic / A. Overview of Logic / 1. Overview of Logic
10282

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
10285

I = φ means that the formula φ is true in the interpretation I

10283

A formula needs an 'interpretation' of its constants, and a 'valuation' of its variables

10284

There are three different standard presentations of semantics

5. Theory of Logic / J. Model Theory in Logic / 3. LöwenheimSkolem Theorems
10288

Down LöwenheimSkolem: if a countable language has a consistent theory, that has a countable model

10289

Up LöwenheimSkolem: if infinite models, then arbitrarily large models

5. Theory of Logic / K. Features of Logics / 6. Compactness
10287

If a firstorder theory entails a sentence, there is a finite subset of the theory which entails it

6. Mathematics / B. Foundations for Mathematics / 5. Mathematics as Set Theory / a. Mathematics is set theory
10286

A 'set' is a mathematically wellbehaved class
