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

[axiom for a vast set based on successors]

12 ideas
We may assume that there are infinite collections, as there is no logical reason against them [Russell]
Infinity says 'for any inductive cardinal, there is a class having that many terms' [Russell]
Infinity: there is an infinity of distinguishable individuals [Ramsey]
The axiom of infinity is not a truth of logic, and its adoption is an abandonment of logicism [Kneale,W and M]
Frege, unlike Russell, has infinite individuals because numbers are individuals [Bostock]
Infinity: ∃x (0 ∈ x ∧ ∀y ∈ x (S(y) ∈ x) [Kunen]
The misnamed Axiom of Infinity says the natural numbers are finite in size [Mayberry]
Infinite sets are essential for giving an account of the real numbers [Maddy]
The Axiom of Infinity states Cantor's breakthrough that launched modern mathematics [Maddy]
Axiom of Infinity: completed infinite collections can be treated mathematically [Maddy]
Infinity: There exists a set of the empty set and the successor of each element [Clegg]
Infinity: There is at least one limit level [Potter]