# Can Dedekind's 'proof' of the existence of infinite sets be properly formulated and carried out in positive set theory?

This question is related to Mikhail Katz's recent mathoverflow question, "Has Dedekind's proof of the existence of existence of infinite sets been analyzed by historians?". Dedekind's 'proof' seems (at least to me) to boil down to (the realm of his thoughts notwithstanding) the following question:

Can $\omega$ be proven to be Dedekind-infinite (i.e., that there is a proper subset of $\omega$ that can be put in 1-1 correspondence with $\omega$), and that no finite set of $\omega$ can be put in 1-1 correspondence with any of its proper subsets?

Positive set theory is a consistent fragment of Naive Set Theory consisting of:

Extensionality: $x$=$y$ $\leftrightarrow$($\forall$$a)(a$$\in$$x$$\leftrightarrow$$a$$\in$$y) Comprehension for positive formulas: (\forall$$w_1$,...,$w_n$)($\exists$$S)(\forall$$x$)($x$$\in$$S$$\leftrightarrow$$\varphi$($x$,$w_1$,...,$w_n$), where $\varphi$ is contained in the smallest class of formulas closed under $\land$, $\lor$, $\exists$, $\forall$, =, $\in$.

My questions are these:

(i). Can the notion of finite set be adequately defined in positive set theory?

(ii). Can it be proven in positive set theory that no finite set can be put in 1-1 correspondence with any of its proper subsets?

Since it is reasonable to assume that $\omega$ can be defined in positive set theory and that 1-1 correspondences of $\omega$ to proper subsets of $\omega$ can also be defined in positive set theory, this would seem to show forth the basic validity of Dedekind's proof of the existence of infinite sets (the realm of his thoughts notwithstanding).

• Ferreiros goes into the details of Dedekind's proof. Surely the proof makes sense; one needs merely to isolate which axioms he implicitly relies upon. Did you read Ferreiros? – Mikhail Katz Jun 9 '16 at 12:48
• @MikhailKatz: Yes. He didn't seem to have much use for Dedekind's proof, though. My Google search didn't give any results of papers by philosophers of mathematics regarding his proof, either. I wonder why? – Thomas Benjamin Jun 9 '16 at 12:54
• There is discussion of Dedekind's proof in contemporary authors like Cantor, Hilbert, Cassirer, and others. – Mikhail Katz Jun 9 '16 at 13:05
• @MikhailKatz: Did Cantor, Hilbert, and Cassirer (for example) find the proof essentially correct? – Thomas Benjamin Jun 9 '16 at 14:10
• I would get in touch with Ferreiros. He seems to know a lot about it. – Mikhail Katz Jun 9 '16 at 14:10