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
13011

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
13013

The Axiom of Extensionality seems to be analytic

13014

Extensional sets are clearer, simpler, unique and expressive

4. Formal Logic / F. Set Theory ST / 4. Axioms for Sets / f. Axiom of Infinity V
13022

Infinite sets are essential for giving an account of the real numbers

13021

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
13023

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
13028

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
13024

Efforts to prove the Axiom of Choice have failed

13026

A large array of theorems depend on the Axiom of Choice

13025

Modern views say the Choice set exists, even if it can't be constructed

4. Formal Logic / F. Set Theory ST / 5. Conceptions of Set / e. Iterative sets
13019

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
13018

Limitation of Size is a vague intuition that overlarge sets may generate paradoxes
