Ideas from 'Believing the Axioms I' by Penelope Maddy [1988], by Theme Structure

[found in 'Journal of Symbolic Logic' (ed/tr -) [- ,]].

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4. Formal Logic / F. Set Theory ST / 4. Axioms for Sets / a. Axioms for sets
New axioms are being sought, to determine the size of the continuum
4. Formal Logic / F. Set Theory ST / 4. Axioms for Sets / b. Axiom of Extensionality I
Extensional sets are clearer, simpler, unique and expressive
The Axiom of Extensionality seems to be analytic
4. Formal Logic / F. Set Theory ST / 4. Axioms for Sets / f. Axiom of Infinity V
The Axiom of Infinity states Cantor's breakthrough that launched modern mathematics
Infinite sets are essential for giving an account of the real numbers
4. Formal Logic / F. Set Theory ST / 4. Axioms for Sets / g. Axiom of Powers VI
The Power Set Axiom is needed for, and supported by, accounts of the continuum
4. Formal Logic / F. Set Theory ST / 4. Axioms for Sets / h. Axiom of Replacement VII
Replacement was added when some advanced theorems seemed to need it
4. Formal Logic / F. Set Theory ST / 4. Axioms for Sets / j. Axiom of Choice IX
Efforts to prove the Axiom of Choice have failed
Modern views say the Choice set exists, even if it can't be constructed
A large array of theorems depend on the Axiom of Choice
4. Formal Logic / F. Set Theory ST / 5. Conceptions of Set / e. Iterative sets
The Iterative Conception says everything appears at a stage, derived from the preceding appearances
4. Formal Logic / F. Set Theory ST / 5. Conceptions of Set / f. Limitation of Size
Limitation of Size is a vague intuition that over-large sets may generate paradoxes