Photoelectron spectra of the argon 2p core level for free argon clusters of up to 4000 atoms are compared to detailed calculations. The comparison shows that the size-dependent shifts of the core level binding energy can be explained in a pure polarization-screening model. Important differences arise between the shifts for the bulk (interior) and the surface atoms. The agreement between experiment and theory allows the extrapolation of the cluster data to the "infinite" solid. In this way we obtain the shifts of the core level binding energy between the free atom, the surface atom and the bulk of argon. The relation between these shifts and those of the first ionization potential is discussed.