TITLE: Has Dedekind's proof of existence of infinite sets been analyzed by historians?
QUESTION [12 upvotes]: 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?

REPLY [3 votes]: You might want to take a look at Greg Oman's preprint 

"Unifying Some Notions Of Infinity In $ZC$ and $ZF$", 

It was to appear in Reports on Mathematical Logic, but you can find the preprint in the Publications section of his Homepage.
In that preprint, you might want to take a look at his "Problem 1":

Let $ZC$$-$$I$ denote Zermelo set theory with Choice minus Infinity (extensionality, pairing, union, power set, and choice remain).  Study the sentences ($\exists$$x$)$\varphi$($x$) for which $ZC$$-$$I$$+$($\exists$$x$)$\varphi$($x$)$\vDash$"There exists a Dedekind (Tarski) infinite set." 

As regards your question (and Dedekind's 'proof' of the existence of an infinite set), you might want to make '$\varphi$' resemble what you describe in your synopsis of Dedekind's 'proof' and see if it (plus $ZC$$-$$I$) satisfies "There exists a Dedekind (Tarski) infinite set."  If your '$\varphi$' does satisfy "There exists a Dedekind (Tarski) infinte set.", then one might rightfully claim Dedekind's 'proof' was successful (the 'realm of his thoughts' notwithstanding).
Oman (in his preprint) also states Zermelo's Axiom of Infinity:

There exists a set $Y$ with the following properties:
  (1) $\emptyset$$\in$$Y$, and (2) For all $y$, {$y$}$\in$$Y$.

In what sense does $Y$ resemble Dedekind's 'realm of thoughts'?