Foundations — Zeroth law — transitivity of thermal equilibrium

Before you can feel why that matters, you need to own every symbol the parent note throws at you. This page defines each one from absolute zero — plain words, then a picture, then the reason the topic can't live without it. Nothing here assumes you've met the notation before.

The cast of symbols (built in order)

1. A "system" — the letters $A$, $B$, $C$, $D$

Why the topic needs it: the Zeroth Law is a statement about three or more separate objects. Without name tags we couldn't say "this one matches that one." The letters keep the players straight.

2. Thermal contact and "heat"

Why the topic needs it: the whole law is about the moment heat stops flowing. To talk about "no net flow" you must first have a picture of flow. See Thermal equilibrium and heat.

3. The relation symbol $\sim$ ("is in thermal equilibrium with")

Why the topic needs it: the entire law is written in terms of $\sim$. It is the single verb of the sentence "$A$ shakes hands with $C$, and $B$ shakes hands with $C$, therefore $A$ shakes hands with $B$."

4. The three properties of a "good match": R, S, T

A relation is just "a rule that either holds or doesn't between two things" (like $\sim$, or "is the same height as"). Three questions decide whether that rule is well-behaved.

Why the topic needs it: a relation that is Reflexive and Symmetric and Transitive is called an equivalence relation. The Zeroth Law is precisely the claim that $\sim$ is transitive — the only one of the three that isn't "obvious." See Equivalence relations (Mathematics).

5. From "matching" to a number: temperature $T$

Here is the payoff that makes every symbol worth defining.

Why the topic needs it: $T$ is the output of the whole derivation. The parent note doesn't assume temperature exists — it earns it from transitivity. See Temperature and its measurement and Thermometers and temperature scales.

6. Intensive vs extensive — why $T$ is not energy

Why the topic needs it: it kills the classic trap "equal temperature = equal heat content." Equilibrium equalises the intensive $T$, never the extensive energy. See Intensive vs extensive properties.

7. The logic arrows $⟹$ and $⟺$

Why the topic needs it: the law is stated as $(A\sim C)\text{ and }(B\sim C)⟹(A\sim B)$ — a one-way street. That one-wayness is exactly why the mistake "both not-equal-to $C$ ⇒ not-equal to each other" is wrong: you can't drive backwards down a $⟹$.

How these foundations feed the topic

Equipment checklist

Test yourself — cover the right side.

Connections

• Thermal equilibrium and heat
• Equivalence relations (Mathematics)
• Temperature and its measurement
• Thermometers and temperature scales
• Intensive vs extensive properties
• First law of thermodynamics
• Second law of thermodynamics