In his paper, "Completed versus Incomplete Infinity in Arithmetic" (which can be found [here](http://web.math.princeton.edu/%7Enelson/papers/e.pdf)), the late Edward Nelson defines the notion of 'counting number' as follows:



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> 0 is a counting number
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> if $y$ is a counting number, so is $y{'}$ [ $^{'}$ is the successor operation--my comment]
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The next sentence reads as follows:



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



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> If $PA$$\vdash$$\alpha$, does this make $PA$ susceptible to the criticisms Nelson holds to concerning $PA$? If not, why not?
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Is the statement "All numbers are counting numbers" independent of $PA$?
The statement asserting that every number is a counting number is $\forall n\ C(n)$, and this is definitely independent of PA, if PA is understood to include induction only in the usual language of arithmetic, without the predicate $C$. To see this, we can simply observe that the statement is true in the standard model of arithmetic, but in any non-standard model of PA, we may take $C(x)$ to hold of exactly the standard numbers, and these satisfy the property about counting numbers that you mentioned. Indeed, one can take the counting numbers to be those in any closed-under-successor cut of a nonstandard model of PA. 


If one wants to subject the predicate $C$ to the induction scheme, however, then it is clear that every number will be a counting number, by induction. In particular, in second-order PA we will be able to prove that every number is a counting number.


Meanwhile, the perspective of the argument above suggests that we may imagine the counting numbers of Nelson to be referring to a possibly proper cut in the natural numbers, but the cut is not definable in the language of arithmetic or in any language in which we have induction.


