Expressing a numerical and contingently true proposition with Wittgenstein’s definition of number

Max Black, on numbers in the Tractatus, writes:

It may be possible to construct a formal arithmetic along the lines that Wittgenstein here indicated. But it might reasonably be objected that Wittgenstein’s sketch does not begin to do justice to the ‘logic’ of numbers in ordinary language. Wittgenstein here seems to be adhering to Russell’s programme of substituting, a constructed concept, for the less tidy numerical concepts already in use. It is hard to see how, on his conception, we are to make sense of, say, a proposition like ‘Tom was the third man to speak’, involving a number but contingently true. Russell, whose analytic Wittgenstein rejects (6.031 a), bore the applications of arithmetic to counting in mind; Wittgenstein seems to have made no provision for such application (313–4).

As a reminder, 6.02 in the Tractatus states:

And this is how we arrive at numbers. I give the following definitions:

$$x = \Omega^{0}{'}x \text{ Def.}$$

$$\Omega'\Omega^{v}{'}x = \Omega^{v+1}{'}x \text{ Def.}$$

I thought it would be fun to try to represent the phrase ‘Tom was the third man to speak’ using Wittgenstein’s definition of number. Here it is:

\begin{aligned} P^{3'}x =_{\text{def.}} & \,\exists\varphi \\ & \quad\left( \left( Px \rightarrow P^{1'}x \right) \leftrightarrow \left( \neg\left( \neg Px \land P^{1'}x \right) \right) \right) \\ & \quad\quad\quad \rightarrow \left( \left( P^{1'}x \rightarrow Py \right) \leftrightarrow \left( \neg\left( \neg P^{1'}x \land Py \right) \right) \right) \\ & \quad\quad\quad\quad\quad \rightarrow \left( Py \rightarrow \left( P'P^{1'}x =_{\text{def.}} P^{2'}x \right) \right) \\ & \quad\quad\quad\quad\quad\quad\quad \rightarrow \left( \left( P^{2'}x \rightarrow Pz \right) \leftrightarrow \left( \neg\left( \neg P^{2'}x \land Pz \right) \right) \right) \\ & \quad\quad\quad\quad\quad\quad\quad\quad\quad \rightarrow \left( Pz \rightarrow \left( P'P'P^{1'}x =_{\text{def.}} P^{3'}x\right) \right) \end{aligned}

Py and Pz, of course, are second and third men to speak. We use x to keep track of their term in the sequence of speakers. We use $\varphi$ to represent some instance of sequence.

I think I had Carnap’s very elegant definition of the number two in the back of my mind while writing this. Requisite to grasping fully his definition of the number is his definition of $\mathrm{2}_{m}(f)$, which means “at least two objects fall under the concept f.” Thus the definition:

2_m =_{\text{def.}} ((\exists x) \wedge (\exists y)) (\neg (x = y) \wedge f(x) \wedge f(y))

So the number two is defined:

2(f) =_{\text{def.}} \mathrm{2}_{m}(f) \wedge \neg \mathrm{3}_{m}(f)