prove that there is any basis to believe it's only 6,000 years
Such a small age for earth is typically accomplished by biblical reckoning... and it is achieved by adding up all of the time that elapses over the acocunted generations, plus the time it says that certain things took, it appears, by biblical reckoning only, that the world could not have been created any longer ago than in the vicinity of 6,000 to 7,000 years. This reckoning, however, assumes very important things, which one may or may not accept as definitively true. One, it assumes that the bible is valid as a historical document with regards to events of antiquity that are greater than written history itself, and two... and this one is probably even more important, even if one were to accept that the Bible were true, it assumes that the biblical accounting of events of that time is actually exhaustive, and that the chronicle could not possibly have been compressed from what actually happened for the purpose of writing it all down, and since such minutia was not considred important for purposes of conveying the message that the chronicle was intended to tell.
Note, you see... you asked for proof that there is any basis to believe that it only 6,000 years old, not proof that it actually *IS* 6,000 years old, nor proof that the biblical account is necessarily even accurate. If one already believes the Bible to be true, then the accounting in the bible is, in fact, a not wholly unreasonable basis to come to the conclusion that the earth is that old (even if that conclusion is not indisputable).