It is said that all natural numbers follow the rule of induction: if a said property holds for one number and for its successor, it holds for all natural numbers.
But, let us define the following property: n < 10
It holds for 1, as 1<10, and it holds for 2, as 2<10.
But it clearly does not hold for all natural numbers.
I believe it is quite unlikely I've proven Peano Arithmetic wrong, so could somebody point the error in my reasoning?
On
Induction is founded on a base case (true for $1$ or $0$ - definitions differ a little, but it doesn't matter in the end), and a case concerned with the successor function - that if the statement is true for $n$ it is also true for $n+1$ however large $n$ is.
Now $n+1\lt x$ is only true when $n\lt x-1$, so $n\lt x$ is not strong enough to establish the successor part of the induction for all $n$.
The correct statement is "if said property holds for some number then it holds for its successor".
In this case, $\color{red}{n<10\implies n+1<10}$, which is clearly false for $n=9$.
A correct example could be with the property $n>10$.
Indeed, $\color{green}{n>10\implies n+1>11\implies n+1>10}$.
Also note that the property must be established independently for some value called the base case. In the second example, the base case could be $n=11$, and the the property is established for all $n\ge11$.