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Single Idea 18104

[filed under theme 4. Formal Logic / F. Set Theory ST / 4. Axioms for Sets / f. Axiom of Infinity V ]

Full Idea

Frege was able to prove that there are infinitely many individuals by taking the numbers themselves to be individuals, but this course was not open to Russell.

Gist of Idea

Frege, unlike Russell, has infinite individuals because numbers are individuals

Source

report of Gottlob Frege (Grundlagen der Arithmetik (Foundations) [1884]) by David Bostock - Philosophy of Mathematics 5.2

Book Ref

Bostock,David: 'Philosophy of Mathematics: An Introduction' [Wiley-Blackwell 2009], p.135

The 13 ideas with the same theme [axiom for a vast set based on successors]:

22288 We have the idea of self, and an idea of that idea, and so on, so infinite ideas are available [Dedekind, by Potter]
18104 Frege, unlike Russell, has infinite individuals because numbers are individuals [Frege, by Bostock]
14440 We may assume that there are infinite collections, as there is no logical reason against them [Russell]
14447 Infinity says 'for any inductive cardinal, there is a class having that many terms' [Russell]
13430 Infinity: there is an infinity of distinguishable individuals [Ramsey]
10051 The axiom of infinity is not a truth of logic, and its adoption is an abandonment of logicism [Kneale,W and M]
13037 Infinity: ∃x (0 ∈ x ∧ ∀y ∈ x (S(y) ∈ x) [Kunen]
17800 The misnamed Axiom of Infinity says the natural numbers are finite in size [Mayberry]
13021 The Axiom of Infinity states Cantor's breakthrough that launched modern mathematics [Maddy]
13022 Infinite sets are essential for giving an account of the real numbers [Maddy]
18191 Axiom of Infinity: completed infinite collections can be treated mathematically [Maddy]
10878 Infinity: There exists a set of the empty set and the successor of each element [Clegg]
13044 Infinity: There is at least one limit level [Potter]