The Zeroth Law of thermodynamics, what is the 3rd object of measuring temperature of our body ( one is human body and other is thermometer)? We know that The Zeroth law of thermodynamics is-

If an object A and another object B are both in equilibrium with each other; then other object C which is in thermal equilibrium with object B will also be in equilibrium with object A and the temperature of object C is equal to the temperature of object A.

Now, when we measure our body temperature using a medical thermometer, We know 1st object is our "Human body" and 2nd object is " The thermometer" but what is the 3rd object that is equilibrium with the body and thermometer?
I heard people saying, if the first object is human body and 2nd object is the thermometer (which we use to measure the temperature for fever), so the third object must be the environment(surroundings). But I have issues with that. Why it has to be the environment. Thermometer bulb, Thermal substance(here:Mercury), and our body can be that 3rd system. only if we need to. But 2 systems can also be in thermal equilibrium with each other.
Thank you.
 A: The "Zeroth law" of thermodynamics is just needed to make thermal equilibrium "well defined", i.e., an equivalence relation. In other words, it is needed to make the idea of "thermal equilibrium" to be like an "equivalence" between objects at the same temperature.
There are actually three conditions we need


*

*An objects is in thermal equilibrium with itself

*If object A is in thermal equilibrium with object B, then object B is in thermal equilibrium with object A

*The Zeroth Law


Notice that these are just like conditions for equality between values. We are just so used to these being true for equivalence that we think of them more as properties of equivalence rather than requirements.


*

*$a=a$

*If $a=b$ then $b=a$

*If $a=b$ and $b=c$, then $a=c$
Note that the third requirement doesn't mean you need "three different values" in order for two numbers to be equal. It is just needed to "finish" this idea of equality. If you have three numbers and you know two pairs of them contain equal numbers, then it is guaranteed that the final pair contains equal numbers as well.$^*$
In the same way, in order to use the concept of thermal equilibrium, you don't need three objects. It is totally fine to talk about two objects being in thermal equilibrium. i.e. you can just say your body and the thermometer are in thermal equilibrium. If you had some third object that you knew was in thermal equilibrium with your body, then you could safely conclude the third object is also in thermal equilibrium with the thermometer.

$^*$
An example of a relation where the first two conditions hold but the third does not could be the idea of meeting someone. Certainly you have met yourself, and if you have met me I have met you. But just because you have met me and I have met my advisor, we cannot for sure say that you have met my advisor. Therefore, "meeting" cannot be thought of as equivalence.
