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If $f(x)=px $ then $ff(x)=p^{2}x$ ,$fff(x)=p^{3}x $ and so on. The n-1 composition raises p to the power n. If 0 exists, then we can obtain 1. From 1, we can obtain 2 and so on.Can we define natural numbers as the exponent of p in $f(x)=px$.

Can we construct natural numbers ( without zero ) like this ?

Aristotle Stagiritis
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    You can just define that $1$ exists and that gives you all the naturals. – Ross Millikan Jul 28 '20 at 05:01
  • How will we know how many compositions of $f$ we've taken if we haven't defined the natural numbers yet? – Greg Martin Jul 28 '20 at 05:07
  • @Greg Zero composition gives us p raised to 1, then we apply 1 composition and get p raised to power 2 and so on. But l see where you are going. – Aristotle Stagiritis Jul 28 '20 at 05:12
  • @Ross, l think n-1 doesn't look good. It's better to start with 1. – Aristotle Stagiritis Jul 28 '20 at 05:13
  • Ok I think there's something here: in the usual construction (start with the empty set and repeatedly apply the function $A\mapsto A\cup{A}$), we don't have the natural numbers yet but we can still get the entire set of them. I suppose the same is true of this construction. (The function $f$, after all, is just a set of ordered pairs.) – Greg Martin Jul 28 '20 at 05:13
  • @AristotleStagiritis: The usual practice in PA is to use $s$ standing for successor instead of $p$, but that is just symbolism. You can start wherever you want. If you start from $1$ the exponent $n$ gives the natural $n+1$ – Ross Millikan Jul 28 '20 at 05:16
  • @Greg I think this construction is just a variation of a method used. When they construct natural numbers using concatenation . So x=1, xx=2 , xxx=3 and so on. – Aristotle Stagiritis Jul 28 '20 at 05:19
  • Yes. But (1) I don't see any particular advantage to this variation, and (2) a function (defined on real numbers?) is, set-theoretically, a much more complicated object than the empty set and its relatives. Indeed, constructions of the real numbers usually require the natural numbers to have been constructed already! – Greg Martin Jul 28 '20 at 07:38

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