Ideas from 'Higher-Order Logic' by Stewart Shapiro [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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4. Formal Logic / F. Set Theory ST / 4. Axioms for Sets / j. Axiom of Choice IX
The axiom of choice is controversial, but it could be replaced
5. Theory of Logic / A. Overview of Logic / 5. First-Order Logic
First-order logic is Complete, and Compact, with the L÷wenheim-Skolem Theorems
5. Theory of Logic / A. Overview of Logic / 7. Second-Order Logic
If the aim of logic is to codify inferences, second-order logic is useless
Some say that second-order logic is mathematics, not logic
5. Theory of Logic / B. Logical Consequence / 1. Logical Consequence
Logical consequence can be defined in terms of the logical terminology
5. Theory of Logic / G. Quantification / 5. Second-Order Quantification
Second-order variables also range over properties, sets, relations or functions
5. Theory of Logic / J. Model Theory in Logic / 3. L÷wenheim-Skolem Theorems
The L÷wenheim-Skolem Theorems fail for second-order languages with standard semantics
The L÷wenheim-Skolem theorem seems to be a defect of first-order logic
Downward L÷wenheim-Skolem: if there's an infinite model, there is a countable model
Up L÷wenheim-Skolem: if natural numbers satisfy wffs, then an infinite domain satisfies them
6. Mathematics / B. Foundations for Mathematics / 3. Axioms for Number / e. Peano arithmetic 2nd-order
Second-order logic has the expressive power for mathematics, but an unworkable model theory
8. Modes of Existence / B. Properties / 11. Properties as Sets
Logicians use 'property' and 'set' interchangeably, with little hanging on it