Bert,
You are correct in that you should be reading a low resistance on the terminals of a horn. If you do not, and instead are reading infinite resistance (an open), it does not necessarily mean that the coil is bad. Normally when the coil is energized, it pulls on the diaphragm of the horn, which in turn breaks the circuit and the coil ceases pulling on the diaphragm. This happens over and over rapidly at the frequency of the sound you hear. The set of contacts that are rapidly opened and closed in the process may need cleaning. If they have become corroded and pitted through use, it can be the cause of an infinite resistance reading, instead of the coil.
After dismantling the horn, you can check the resistance of the coil itself.
A relay burnishing tool is the proper way to clean the contacts. Never use contact cleaner on electromechanical parts, as they contain anti-oxidizing compounds that at these level of currents can burn and cause this problem, or even a fire.