Since a sample of C14 decays by half every 5700 years, anything that contains measurable amounts of C14 cannot be more than about 60,000 years old. Therefore, diamonds which contain measurable C14 cannot be over that age (when they first formed from carbon), whereas conventional science claims they are a billion years old.
Where do you get that idea? At 60,000 years, there would be around 0.5% of the original C-14 left. So, the math does not support your claim. But half-lives are not strictly mathematical. It's a statistical measurement, so it's no longer valid once the sample size is below the threshold of statistical validity. If I count a million humans, odds are that half a million will be male, and half a million will be female. If I take a sample of five humans, there's no way to predict how many will be male and how many female - if I happened to select from NFL players, the sample could contain five males and no females.
Atoms that are radioactively decaying in proximity to each other influence the rate at which their neighbors decay. Once you get below a certain density, that no longer occurs, and the statistical half-life no longer applies. Atoms may take longer and longer to decay.
Of course, the claims fall even flatter since no one has actually detected measurable C-14 in any diamond. Oh, a geiger counter will click if near a diamond, but it will also click if near a glass window, a cup of tea, a plaster wall, your hand, or absolutely anything else, since there's this thing called "background radiation" that results from cosmic rays. It's called background radiation because it's always present, everywhere on earth. The levels of C-14 that folks are claiming
to have detected are within the range of background radiation, which means that it's literally impossible to actually measure anything there. Once you get down to that level, background radiation eliminates all possibility of getting an accurate result, because you cannot separate your result from the background. So, anyone claiming to detect C-14 at those levels is being dishonest.
Additionally, Creationists will make claims regarding other materials (oil, etc.) containing C-14 levels indicating young age. That ignores both the fact that C-14 is regularly produced by natural irradiation due to radioactive mineral deposits (uranium, etc.) and, perhaps even more importantly, that they are generally measuring K-40, not C-14. K-40 has a half-life of well over a billion
years, as opposed to C-14's 5730 years. Doesn't take much K-40 to screw up the measurement when the levels under discussion are low.
Incidentally, much as the C-12 to C-14 ratio is useful for dating biological samples, the K-40 to Ar-40 ratio is useful for dating geological samples. Argon does not naturally occur within minerals, so any argon that is present must necessarily be due to radioactive decay of K-40. A "fresh" geological sample of a potassium-containing mineral would have some amount of K-40, and no argon of any sort. Over time, some K-40 becomes Ar-40. The amount of argon in the sample will yield its age.
Because of K-40's extremely long half-life, Potassium-argon dating does not even begin to be useful until the sample is more than a hundred thousand years old. That's the minimum age that it can measure, and it's already older than the 60,000 you claim as a maximum.
The various isotope dating methods are useful in varying ranges and types of material, but the end result is that it can be proven that the planet is billions of years old, because rocks of that age have been dated.