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
The Axiom of Extensionality seems to be analytic
Extensional sets are clearer, simpler, unique and expressive
4. Formal Logic / F. Set Theory ST / 4. Axioms for Sets / f. Axiom of Infinity V
Infinite sets are essential for giving an account of the real numbers
The Axiom of Infinity states Cantor's breakthrough that launched modern mathematics
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
A large array of theorems depend on the Axiom of Choice
Modern views say the Choice set exists, even if it can't be constructed
Efforts to prove the Axiom of Choice have failed
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