I have selected twenty isotopes of interest. These are isotopes whose half-lives have been determined to lie within a specific range, and are not known to be produced in the Earth's crust by any major synthesis processes. Except for the various Technetium isotopes, which can arise if Molybdenum isotopes are coincident with Uranium isotopes in certain rocks, however this exception is rare and well documented.
The reason I have chosen these isotopes is simple. Because, they would all be present in measurable quantities in the Earth's crust, and detectable using modern mass spectrometry, if the planet was only 6,000 years old. This is because the half-lives of all these radionuclides are a good deal longer than 6,000 years. So, what happens when we search for these isotopes in Earth rocks?
NONE of them are present in measurable quantities!
The isotopes in question, in increasing atomic mass order, are;
Al26 : 730,000 years
Cl36 : 301,000 years
Ca41 : 102,000 years
Mn53 : 3,740,000 years
Fe60 : 2,500,000 years
Kr81 : 213,000 years
Zr93 : 1,530,000 years
Nb92 : 34,700,000 years
Tc98 : 4,200,000 years
Pd107 : 6,500,000 years
Sn126 : 100,000 years
Cs135 : 2,300,000 years
Sm146 : 103,000,000 years
Gd150 : 1,790,000 years
Dy154 : 3,000,000 years
Hf182: 9,000,000 years
Re186m : 200,000 years
Pb205 : 15,200,000 years
Np237 : 2,140,000 years
Cm247 : 15,600,000 years
The most recognised dating technique is radiocarbon dating. This is generally accepted to be useful for ten half-lives. This is because any measurable amount of carbon -14 detected after this is extremely small. I use radiocarbon dating as a reference to demonstrate two things, half-live compatibility and it's use on only organic samples.
For this demonstration I’m using twenty half-lives and non organic samples. So even for isotopes of common elements, this represents a vanishingly small amount of material that would test even the world's best mass spectrometer labs to detect in a sample.
So, what does the observation of no measurable quantity of the above isotopes mean? It means that at least 20 half-lives of the requisite isotopes must have elapsed for those isotopes to decay. Taking each isotope in turn, this means that:
Sn126, being absent, must have decayed over a period of 20 half lives = 20 × 100,000 years = 2,000,000 years. Therefore the Earth must be at least 2,000,000 years old for all the Sn126 to have decayed.
Ca41, being absent, must have decayed over a period of 20 half lives = 20 × 102,000 years = 2,040,000 years. Therefore the Earth must be at least 2,040,000 years old for all the Ca41 to have decayed.
Re186m, being absent, must have decayed over a period of 20 half lives = 20 × 200,000 years = 4,000,000 years. Therefore the Earth must be at least 4,000,000 years old for all the Re186m to have decayed.
Kr81, being absent, must have decayed over a period of 20 half lives = 20 × 213,000 years = 4,260,000 years. Therefore the Earth must be at least 4,260,000 years old for all the
Kr81 to have decayed.
Cl36, being absent, must have decayed over a period of 20 half lives = 20 × 301,000 years = 6,020,000 years. Therefore the Earth must be at least 6,020,000 years old for all the Cl36 to have decayed.
Al26, being absent, must have decayed over a period of 20 half lives = 20 × 730,000 years = 14,600,000 years. Therefore the Earth must be at least 14,800,000 years old for all the Al26 to have decayed.
Fe60, being absent, must have decayed over a period of 20 half lives = 20 × 2,500,000 years = 52,000,000 years. Therefore the Earth must be at least 30,000,000 years old for all the Fe60 to have decayed.
Zr93, being absent, must have decayed over a period of 20 half lives = 20 × 1,530,000 years = 30,600,000 years. Therefore the Earth must be at least 30,600,000 years old for all the Zr93 to have decayed.
Gd150, being absent, must have decayed over a period of 20 half lives = 20 × 1,790,000 years = 35,800,000 years. Therefore the Earth must be at least 35,800,000 years old for all the Gd150 to have decayed.
Np237, being absent, must have decayed over a period of 20 half lives = 20 × 2,140,000 years = 42,400,000 years. Therefore the Earth must be at least 42,400,000 years old for all the Np237 to have decayed.
Cs135, being absent, must have decayed over a period of 20 half lives = 20 × 2,300,000 years = 46,000,000 years. Therefore the Earth must be at least 46,000,000 years old for all the Cs135 to have decayed.
Dy154, being absent, must have decayed over a period of 20 half lives = 20 × 3,000,000 years = 60,000,000 years. Therefore the Earth must be at least 60,000,000 years old for all the Dy154 to have decayed.
Mn53, being absent, must have decayed over a period of 20 half lives = 20 × 3,740,000 years = 74,800,000 years. Therefore the Earth must be at least 74,800,000 years old for all the Mn53 to have decayed.
Tc98, being absent, must have decayed over a period of 20 half lives = 20 × 4,200,000 years = 84,000,000 years. Therefore the Earth must be at least 84,000,000 years old for all the Tc98 to have decayed.
Pd107, being absent, must have decayed over a period of 20 half lives = 20 × 6,500,000 years = 130,000,000 years. Therefore the Earth must be at least 130,000,000 years old for all the Pd107 to have decayed.
Hf182, being absent, must have decayed over a period of 20 half lives = 20 × 9,000,000 years = 180,000,000 years. Therefore the Earth must be at least 180,000,000 years old for all the Hf182 to have decayed.
Pb205, being absent, must have decayed over a period of 20 half lives = 20 × 15,200,000 years = 304,000,000 years. Therefore the Earth must be at least 304,000,000 years old for all the Pb205 to have decayed.
Cm247, being absent, must have decayed over a period of 20 half lives = 20 × 15,600,000 years = 312,000,000 years. Therefore the Earth must be at least 312,000,000 years old for all the Cm247 to have decayed.
Nb92, being absent, must have decayed over a period of 20 half lives = 20 × 34,700,000 years = 694,000,000 years. Therefore the Earth must be at least 694,000,000 years old for all the Nb92 to have decayed.
Sm146, being absent, must have decayed over a period of 20 half lives = 20 × 103,000,000 years = 2,060,000,000 years. Therefore the Earth must be at least 2,060,000,000 years old for all the Sm146 to have decayed.
This is an inescapable conclusion from observational reality, given that these isotopes are not found in measurable quantities in the Earth and would be found in measurable quantities if the Earth was only 6,000 years old, indeed, hardly any of the Sm146 would have disappeared in just 6,000 years, and it would form a significant measurable percentage of the naturally occurring Samarium that is present in crustal rocks. The fact that NO Sm146 is found places a minimum limit on the age of the earth of 2,060,000,000 years. Over two billion years and of course, dating using other isotopes with longer half-lives that can be measured precisely has established that the age of the Earth is approximately 4.5 billion years.
Since the decay of these isotopes obeys a precise mathematical law, and this law has been established through decades of observation of material of known starting composition originating from nuclear reactors specifically for the purpose of determining precise half-lives, which is one of the tasks that the UK National Physical Laboratory performs on a continuous basis in order to maintain scientific databases, the credibility of all of this is beyond question.
The majority of those isotopes are nowadays ONLY obtained by synthesis within nuclear reactors, and research of known samples of these materials confirms again and again, that not only does the precise mathematical law governing radionuclide decay apply universally to all of these isotopes, but that the half-lives obtained are valid. The laws of nuclear physics would have to be completely rewritten for any other scenario to be even remotely valid. So rewriting of the laws of nuclear physics would impact upon the very existence of stable isotopes, including stable isotopes of the elements that make up each and every one of us.