Ideas from 'HigherOrder Logic' by Stewart Shapiro [2001], by Theme Structure
[found in 'Blackwell Guide to Philosophical Logic' (ed/tr Goble,Lou) [Blackwell 2001,0631206930]].
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4. Formal Logic / F. Set Theory ST / 4. Axioms for Sets / j. Axiom of Choice IX
10301

The axiom of choice is controversial, but it could be replaced

5. Theory of Logic / A. Overview of Logic / 5. FirstOrder Logic
10588

Firstorder logic is Complete, and Compact, with the LöwenheimSkolem Theorems

5. Theory of Logic / A. Overview of Logic / 7. SecondOrder Logic
10299

If the aim of logic is to codify inferences, secondorder logic is useless

10298

Some say that secondorder logic is mathematics, not logic

5. Theory of Logic / B. Logical Consequence / 1. Logical Consequence
10300

Logical consequence can be defined in terms of the logical terminology

5. Theory of Logic / G. Quantification / 5. SecondOrder Quantification
10290

Secondorder variables also range over properties, sets, relations or functions

5. Theory of Logic / J. Model Theory in Logic / 3. LöwenheimSkolem Theorems
10296

The LöwenheimSkolem Theorems fail for secondorder languages with standard semantics

10297

The LöwenheimSkolem theorem seems to be a defect of firstorder logic

10292

Downward LöwenheimSkolem: if there's an infinite model, there is a countable model

10590

Up LöwenheimSkolem: if natural numbers satisfy wffs, then an infinite domain satisfies them

6. Mathematics / B. Foundations for Mathematics / 3. Axioms for Number / e. Peano arithmetic 2ndorder
10294

Secondorder logic has the expressive power for mathematics, but an unworkable model theory

8. Modes of Existence / B. Properties / 11. Properties as Sets
10591

Logicians use 'property' and 'set' interchangeably, with little hanging on it
