sin and carbon 14 dating?

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It should be possible to find wood from an independently dated whore house and church. We could, for example, do a tree ring comparison on the timbers and corroborate this with the historical dates of construction of the buildings.

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You didn't say, but is your assumption that more sinning is taking place in the cat house than the church? Most people would, anyway. I am going to take the contrarian view. I do not have personal experience with whore houses but I do have lots of experiences with churches and my observation is that a LOT of sinning goes on there.Try gossiping, back biting, abuse of power, and lying among others.
My suspicion is that in fact, the your average whore house is less sinful than your average church, all things equal.

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But it would probably be useless for scientists because, by and large, the scientific community is predisposed to reject such things out of hand, and would probably not allow any such studies to be published in their journals.

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A convenient self serving little rationalization for creationists but a common misunderstanding of the scientific process. Science is based on the notion of FALSIFICATION - in other words, before ANY scientific claim is accepted, it must be shown to be falsifiable and then the data must in fact be consistent with the claim. And if a claim is not falsifiable - well then it is not a scientific claim and any studies regarding that claim are therefore not scientific studies and do not belong in a scientific journal. What part of "it is not science" do you not understand?