Set $s$ is a natural number if $s$ is transitive and for every $x$, $y$ and $z$

  • $y\in{s}\rightarrow(y$ is transitive$)$, and if
  • $x\in{P}s\wedge(x$ is transitive$)\wedge{z}\in{P}x\wedge(z$ is transitive$)$ $\rightarrow{z}\in{x}\cup\{x\}$, and if
  • $x\in{P}s\wedge(x$ is transitive$)\wedge\bigcup{x}\neq{x}$ $\rightarrow x=\bigcup{x}\cup\{\bigcup{x}\}$.

Then $\emptyset$ is the first natural number, the successor of natural number $s$ is defined by $s\cup\{s\}$ and the predecessor of number $s$ is defined by $\bigcup{s}$. All Peano axioms are straightforward proved, including the principle of mathematical induction. Also the natural number arithmetic is proved, for example if $a$ and $b$ are natural numbers then with induction is proved $a+b$ and $a\cdot{b}$ are natural numbers. The set $\omega$ of natural numbers is then postulated as $s\in\omega\leftrightarrow(s$ is a natural number).

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    $\begingroup$ What does that $P$ denote? $\endgroup$ – Asaf Karagila Mar 15 '14 at 21:39
  • $\begingroup$ $Ps$ denotes the power-set of set $s$. $\endgroup$ – d.h.homan Mar 16 '14 at 8:55
  • $\begingroup$ I see. It would be clearer if you'd write $x\subseteq s$ and so on. $\endgroup$ – Asaf Karagila Mar 16 '14 at 9:32
  • $\begingroup$ You are right, there is no compelling reason not to use it. $\endgroup$ – d.h.homan Mar 16 '14 at 12:34

This definition is not for a natural number, but rather for an ordinal. Let's see that $s=\omega$ satisfies this definition.

  1. $\omega$ is transitive.
  2. If $y\in\omega$ then $y$ is transitive.
  3. If $x\subseteq\omega$ and transitive, and $z\subseteq x$ and transitive (in particular, both $x$ and $z$ are natural numbers), then $z\subseteq x\cup\{x\}$.
  4. If $x$ is a transitive subset of $\omega$ and $\bigcup x\neq x$ then $x=\bigcup x\cup\{\bigcup x\}$. This is true for every ordinal, because if $x$ is not a successor then $\bigcup x=x$. So the implication holds vacuously.

In order to limit the definition to finite ordinals you need to require that every $x\subseteq s$ which is transitive and non-empty satisfies, $x\neq\bigcup x$. This means that every ordinal $\leq s$ is zero or a successor.

  • $\begingroup$ Thank you for the reply. However, I do not want to use the set $\endgroup$ – d.h.homan Mar 16 '14 at 16:23
  • $\begingroup$ However, I do not want to use the set $\omega$. I claim the Peano axioms of the natural numbers can be proved with this definition: if $s$ satisfies the definition also $s\cup\{s\}$ satisfies the definition thus is a natural number, etc. Also the principle of mathematical induction can be proved. $\endgroup$ – d.h.homan Mar 16 '14 at 16:36
  • $\begingroup$ @d.h.homan: If you don't want $s=\omega$ to satisfy your definition of "natural number", then you have to put something in the axioms to make that happen; it's not good enough to just say "I don't want $\omega$ to be a natural number". (maybe I've misunderstood your objection; it's not completely clear) $\endgroup$ – Hurkyl May 20 '14 at 8:10

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