A recent question on propositional logic posted on Philosophy Stack Exchange yielded an answer which states, in part, that,
The fact that $4$ is greater than $2$ is not a "logical fact" but and [sic] arithmetical one: it depends on the axioms of arithmetic.
The context of the question was a cited proposition from a book on logic for which the author claims that knowing that a creature has four legs is not enough to prove that it has more than two legs.
Is there a set of mathematical axioms or form of mathematics in which $4>2$ is meaningful but false?
Some commenters mentioned the possibility of modular (modulo) arithmetic, but admitted that $4>2$ in those systems would either be true or meaningless.
Yes, there is a "model" for which this assertion is true, in a weaker sense ; in the framework of 2-adic numbers (https://mathworld.wolfram.com/p-adicNorm.html), we have :
$$\underbrace{|4|_2}_{1/2^2=1/4} < \ \ \ \underbrace{|2|_2}_{1/2}$$
where $| \cdots|_2$ is the 2-adic "valuation", which is a kind of absolute value, or norm.
Explanation : this norm on integers is based on the (unique) prime factor decomposition of any integer .
Let us take an example ; consider $360$ ; as it is decomposed into
$$360=2^3 \times 3^2 \times 5$$
the $2$-adic norm is obtained by inverting the first factor (containing the power of $2$) :
$$|360|_2=\dfrac{1}{2^3}=\dfrac{1}{8}$$
"The more a number is divisible by $2$, the smaller it is for this norm".
This norm and the associated distance verify the distance axioms, but in an almost paradoxical way where triangle inequality
$$d(x,z) \le d(x,y)+d(y,z) \ \ \text{is a consequence of} \ \ d(x,z) \le \max(d(x,y),d(y,z))$$
which is much stricter condition ("ultrametric distance").
Remarks :
1) p-adic numbers have played a rôle in the proof or Fermat's last theorem, as well as for a recent Field medalist : see the nice article here
2) A brief and accurate account on the discovery of $p$-adic numbers in general (p any prime number, not especially $2$) and the difficulty to give them a rigorous status in the 1900/1940s is given here.