In his paper, "Completed versus Incomplete Infinity in Arithmetic" (look under "web.math.princeton.edu/$\sim$nelson/papers.html" under the subheading "Infinity"), the late Edward Nelson defines the notion of 'counting number' as follows:

0 is a counting number

if $y$ is a counting number, so is $y{'}$ [ $^{'}$ is the successor operation--my comment]

The next sentence reads as follows:

This is all that we assume about the notion, and in particular we do not postulate that all numbers are counting numbers.

On page 7 of this paper, Nelson refers to the postulate that all numbers are counting numbers as a "Platonic postulate". It seems clear from his paper that Nelson believes that "All numbers are counting numbers" is a postulate that is definitely false.

But is it? I believe that the following formal statement (rightly or wrongly--you decide) in the language of $PA$ captures the intuitive notion of the aforementioned statement:

$\alpha$: ($\forall$$x$)($x$$\neq$0.$\supset$.($\exists$$y$)($y^{'}$= x)) (Note: $\alpha$ was found by me in someone's class notes online.)

Question: Is '$\alpha$' independent of $PA$?

Suppose, to the contrary, that $PA$$\vdash$$\alpha$. Then the following question seemingly arises:

If $PA$$\vdash$$\alpha$, does this make $PA$ susceptible to the criticisms Nelson holds to concerning $PA$? If not, why not?