How do we know two is bigger than one?
How do we know that two is bigger than one? Apart from being told that it is, how do we know..? There's no reason for the numberline to be in order just like the alphabet has no reason to be in order.
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$\begingroup$On the real numbers the order relation $x<y$ is defined in the following way:
$a<b$ if and only if there exists a positive number c such that $a+c = b$.
As $1+1 = 2$ (and $1$ is a positive number) we conclude that $1 < 2$.
$\endgroup$ $\begingroup$If we consider $\Bbb{N}$ by Peano axioms we know that there is always an element called $1$. Now when we perform addition operation on this $\Bbb{N}$ which is $1+1$ we define this new number to be $2$.
If we want intuition think we are walking on a road. The starting step which we have taken let it be 1. Now when I take two $1$ steps I define this to be two. So we have gone farther away from the starting step. So we can say that the new step which we have got is bigger than the earlier step.
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