Radioactive dating paper
The first is that atoms have always decayed at the same rate.
And this isn’t really an assumption as the decay rates have been tested in the laboratory for a hundred years or so, we have an example of a natural nuclear reactor where we can measure the various products and determine the decay rates (and the fine structure constant), and we can observe the past by looking deep into the past of the universe. The sigh isn’t for the effort of writing, it’s for the effort of finding all the references.
They used pottery and other materials in sites to date 'relatively'.
They thought that sites which had the same kinds of pots and tools would be the same age.
These include the main stable isotope (12C) and an unstable isotope (14C).
When an organism dies, it contains the standard ratio of 14C to 12C, but as the 14C isotope decays, the proportion of carbon 14 decreases at a known constant rate.
An important aspect of the study was a series of controlled tests comparing the radiocarbon content of un-cleaned, possibly contaminated samples, with chemically cleaned equivalents.