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The assumptions behind the evidences presented here cannot be proved, but the fact that such a wide range of different phenomena all much younger ages than are currently generally accepted, provides a strong case for questioning those accepted ages (currently 13.77 billion years for the universe and 4.543 billion years for the solar system).
For example, the amount of cratering on the moon, based on currently observed cratering rates, would suggest that the moon is quite old.
Ages of millions of years are all calculated by assuming the rates of change of processes in the past were the same as we observe today—called the principle of uniformitarianism.
If the age calculated from such assumptions disagrees with what they think the age should be, they conclude that their assumptions did not apply in this case, and adjust them accordingly.
If the calculated result gives an acceptable age, the investigators publish it.
Examples of ages listed here are also obtained by applying the same principle of uniformitarianism.
Additionally, I followed the link for the first claim, that the presence of DNA in 'ancient samples' refuted the samples' supposed age.