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Single Idea 22288

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

Full Idea

Dedekind had an interesting proof of the Axiom of Infinity. He held that I have an a priori grasp of the idea of my self, and that every idea I can form the idea of that idea. Hence there are infinitely many objects available to me a priori.

Gist of Idea

We have the idea of self, and an idea of that idea, and so on, so infinite ideas are available

Source

report of Richard Dedekind (Nature and Meaning of Numbers [1888], no. 66) by Michael Potter - The Rise of Analytic Philosophy 1879-1930 12 'Numb'

Book Ref

Potter,Michael: 'The Rise of Anaytic Philosophy 1879-1930' [Routledge 2020], p.83

A Reaction

Who said that Descartes' Cogito was of no use? Frege endorsed this, as long as the ideas are objective and not subjective.

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]