I'm confused about the definition of temperature, and I would like to make some points clearer since I did not find an explanation.
Firstly I have heard of three adjectives regarding temperature (scales): empirical, absolute and thermodynamical.
I think I understood the difference between empirical and thermodynamical, but what is exactly meant by absolute?
Is this the same as "indipendent from the particular system used to measure temperature", or does this adjective indicates that the scale can only have positive numbers, with the $0 K$ placed, indeed, at absolute zero?
For example, ideal gas constant volume thermometer gives absolute temperature, because the measurement is not affected by the use of different gases, as long as they are ideal? Or is it because in the definition $$T=\mathrm{lim}_{p_0->0} 273.16 \frac{p}{p_0} \, K$$
it is implicit that $T=0K$ is a temperature that can be reached only if $p=0 Pa$ (a limiting condition) and hence the scale has the zero in a very particular place, that cannot be reached?
My second question is about the ideal gas thermometer itself: in the previous definition it is used the fact that, "for an ideal gas $T \propto p$ at contant volume". That's how textbook introduces the ideal gas thermometer.
What I do not get is: how can we use the relation $T \propto p$ in ideal gases before even knowing what is temperature, i.e. to define (which means to measure) the temperature itself?
I do not get if the relation $T \propto p$ is imposed by the definition of temperature, for make things easier or if there is something behind it that guarantees that, before knowing how to even measure temperature, the temperature is proportional to pressure at constant volume. This last option does not makes a lot of sense, so if this is the case I surely misunderstood the reasoning behind the use of ideal gas thermometer.
And this holds also for any thermometric property $X$, where the relation to get the temperature $\theta$ from the property $X$ itself, say $\theta(X)$ is assumed to be linear, which means $$\theta(X)=C \, X$$
Is this really an assumption, in the sense that it is imposed to be true, or is in some way known before the definition of $\theta$?