[Math] Has Dedekind’s proof of existence of infinite sets been analyzed by historians

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This pdf by David Joyce notes that in paragraph 66 of his famous essay, Dedekind claims to prove the existence of an infinite set.

The proof exploits the assumption that there exists a set $S$ of all things, and that a mathematical thing is an object of our thought. Then if $s$ is such a thing, then the thought, denoted $s'$, that $s$ is a mathematical object is a thing distinct from $s$. Denoting the passage from $s$ to $s'$ by $\phi$, Dedekind gets a self-map $\phi$ of $S$ which is some kind of blend of the successor function and the brace-forming operation. From this Dedekind concludes that $S$ is infinite, QED. This is quite remarkable. It would be interesting to analyze the hypotheses/axioms that would be used in a formalisation of such a proof. Ferreiros alludes to this proof on page 111 of his book Labyrinth of thought but does not analyze it.

Question. Has this proof been analyzed by historians, mathematicians, or philosophers?

Best Answer

See pages 107 and following and pages 244 and following of Labyrinth of Thought: A History of Set Theory and Its Role in Modern Mathematics by José Ferreirós (2008).

  • Dedekind and the set-theoretical approach
    1. The algebraic origins of Dedekind's set theory, 1856-58
    2. A new fundamental notion for algebra: fields
    3. The emergence of algebraic number theory
    4. Ideals and methodology
    5. Dedekind's infinitism
    6. The diffusion of Dedekind's views
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