So, from what I understand, living things maintain (or at least prior to the industrial revolution did maintain) a predictable ratio of C-14 to C-12. I’m not super familiar with the mechanics of this, I imagine it’s a case of the amount of C-14 lost matching the rate it was replaced via respiration.

Once the organism dies, it stops controlling that ratio and we can measure the decay using a sample of the material.

It’s not that it breaks down differently (in fact, we rely on it being consistent), it’s how much it has broken down. Carbon-14 has a half-life of 5700 +/- 30 years (per Wikipedia), so if you see 1/2 the expected amount of carbon-14 then something would be around 5700 years old, with 1/4 the expected amount you’d predict 11400 years old, etc.

This relies on the amount of carbon-14 originally being predictable. This worked well in the past for living things (which from what I understand tended to maintain a consistent ratio of C-14 to C-12) or objects made from their organic material, but stopped being true around the industrial revolution when we started pumping the atmosphere full of carbon.

We use other isotopes, or other techniques in general, for very recent objects, or for things more than 50-60 thousand years old.

In fact, the entire foundation of math – its system of axioms – has had to be fixed due to contradictions existing in previous iterations. The most well known perhaps being Russell’s paradox in naive set theory: “Let X be the set of all sets that do not contain themselves. Does X contain itself?”

In fact, there have been many paradoxes that had to be resolved by the set theory we use today.