Ideas of Wilfrid Hodges, by Theme

[British, b.1941, Of Bedford College, then Queen Mary and Westfield, London.]

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2. Reason / D. Definition / 7. Contextual Definition
The idea that groups of concepts could be 'implicitly defined' was abandoned
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 / A. Overview of Logic / 5. First-Order Logic
Since first-order languages are complete, |= and |- have the same meaning
5. Theory of Logic / B. Logical Consequence / 4. Semantic Consequence |=
|= in model-theory means 'logical consequence' - it holds in all models
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
There are three different standard presentations of semantics
I |= φ means that the formula φ is true in the interpretation I
5. Theory of Logic / I. Semantics of Logic / 4. Satisfaction
|= should be read as 'is a model for' or 'satisfies'
5. Theory of Logic / J. Model Theory in Logic / 1. Logical Models
Model theory studies formal or natural language-interpretation using set-theory
A 'structure' is an interpretation specifying objects and classes of quantification
Models in model theory are structures, not sets of descriptions
5. Theory of Logic / J. Model Theory in Logic / 3. L÷wenheim-Skolem Theorems
Down L÷wenheim-Skolem: if a countable language has a consistent theory, that has a countable model
Up L÷wenheim-Skolem: if infinite models, then arbitrarily large models
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 / A. Nature of Mathematics / 5. The Infinite / i. Cardinal infinity
First-order logic can't discriminate between one infinite cardinal and another
6. Mathematics / B. Foundations for Mathematics / 6. Mathematics as Set Theory / a. Mathematics is set theory
A 'set' is a mathematically well-behaved class