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
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

13024

Efforts to prove the Axiom of Choice have failed

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
