[This pdf by David Joyce](http://aleph0.clarku.edu/~djoyce/numbers/dedekind.pdf) 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?



Has Dedekind's proof of existence of infinite sets been analyzed by historians?
See [pages 107](https://books.google.com/books?id=vrQLbbxGNMsC&pg=PA107) and following and [pages 244](https://books.google.com/books?id=vrQLbbxGNMsC&pg=PA244) and following of [Labyrinth of Thought: A History of Set Theory and Its Role in Modern Mathematics](https://books.google.com/books?id=vrQLbbxGNMsC) 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


