Laws of Thermodynamics

There are 4 fundamental laws in thermodynamics that governs everything. But the first law is called Zeroth Law due to its been discovered the last but its super fundamental than all of the rest so have been moved to the front.

Zeroth Law

This is the foundation of Temperature.

$$A \sim B \quad \text{and} \quad B \sim C \implies A \sim C$$

If systems $A$ and $B$ are in thermal equilibrium, and $B$ and $C$ are in thermal equilibrium, Then $A$ and $C$ are also in the same equilibrium.

This transitivity property is required for the Temperature ($T$) to be a mathematically valid property. If $A=300^\circ{C}$, and $B=300^\circ{C}$ we say no heat transfer. If $C=300^\circ{C}$, then when $A$ touched with $C$ and some heat transferred, then the Temperature Property is mathematically invalid.

Also thermometer can be used because of this. If transitivity property was not valid, even when Thermometer show $50^\circ \text{C}$ in two cases, we cannot say they are equal. The only way to truly say they are equal would be to make them touch and see if there is heat transfer.