In his paper, "Completed versus Incomplete Infinity in Arithmetic" (look under "www.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?