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From Wikipedia:

The kelvin is defined as the fraction 1⁄273.16 of the thermodynamic temperature of the triple point of water.

That presupposes that we can take a fraction of temperature.

Now, taking a fraction of mass, distance, or duration is well defined. Is taking a fraction of temperature well defined? Doesn't this depend on operational definitions of temperature - such as change in length?

This is similar to the objection many physicists had with measuring human perceived loudness: What does it mean to say that humans perceive loudness as proportional to log of pressure? We can order perceived loudness, but how do you add two perceptions of loudness, so that any type of proportionality makes sense?

Related question: If $K$ cannot be defined in terms of basic units, why do we say $mgs$ and not $mgsK$?

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Temperature is a continuous property... there is no reason it needs to be defined only for the set of integers. Similarly, no measurement device is constrained to only measure whole number multiples of the triple point of water. –  zhermes Feb 12 '13 at 18:36
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1 Answer

The quote you gave uses "thermodynamic temperature". This temperature is defined based on the second law. The idea is that we define the efficiency of a Carnot engine to be $1 - \frac{T_1}{T_2}$

The zeroth law essentially shows that the idea of temperature makes sense, without telling us exactly what it is.

To investigate temperature, suppose we have a three reservoirs at temperatures $T_1$, $T_2$, and $T_3$. We have reversible heat engines between 1-2 and 2-3. This is equivalent to a single reversible engine between 1-3. The second law can be used to demonstrate that all such reversible engines have the same ratios of heat and work inputs and outputs when operating between the same reservoirs.

For any given engine between temperatures $T_{hot}$ and $T_{cold}$, define the ratio of heat output to the cold reservoir to heat absorbed from the hot reservoir as $\frac{Q_{out}}{Q_{in}} \equiv f(T_{hot},T_{cold})$. Then our two engines being equivalent to a third engines says that $f(T_1,T_2) f(T_2,T_3) = f(T_1,T_3)$.

That condition is satisfied for $f(T_{hot},T_{cold}) = F(\frac{T_{cold}}{T_{hot}})$ for any $F$.

To define temperature, we simply take $F(x) = x$, which fixes temperature up to a constant multiplier. To measure the ratio of temperatures between two objects, construct a reversible heat engine between them and measure the ratio of heat dumped into the colder to heat absorbed from the warmer.

It turns out that at low pressures, gases obey the ideal gas equation $PV \propto T$, where $T$ is the thermodynamic temperature defined above. This is now an observed result, not a definition of temperature. It suggests that the natural unit for temperature is energy, but for historical reason we have invented a new unit called the Kelvin, and introduced a constant $k_B = 1.38*10^{-23} J/K$ to convert.

ref: this is from ch. 1 of Kardar's Statistical Physics of Particles, and there are more details there

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