I've just bought CBP Finn's book "Thermal Physics" and on
page 5 just after his statement of the 0th Law ("If each of two systems is in
thermal equilibrium with a third, they are in equilibrium with another") there
is the following statement :
"The temperature of a system is a property that determines whether or not that
system is in thermal equilibrium with another.
More formal mathematical arguments may be developed to show the existence of
temperature but we shall not go into them here..." He then gives references to
"Equilibrium Thermodynamics" by CJ Adkins and "Heat and Thermodynamics" by MW
Zemansky and RH Dittman.
I would just go to the references, but unfortunately I don't currently have
ready access to a library containing them and I wondered if anyone here would
care to take the time to sketch an elementary mathematical argument that proves
the existence of temperature as property of systems ?