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We should remember that the Church persecuted Galileo for having the audacity to suggest that the earth was not the center of the universe - something we now know to be 100% true (and the church admits this as well.) If the church was wrong about that, then it stands to reason that it could be wrong about any of the theories of the universe which it puts forth. Take that logic into consideration with the mounds of scientific evidence that show that the earth, not to mention the universe, is far older than that, and it is difficult to find merit in the church's claim.
Additionally, if the universe were only 10,000 years old, we should only be able to see stars that are 10,000 light years away. any further than that, would mean the universe is older, since a light year is the distance light travels in one year. In fact we can see on the order of 12 billion light years away. this would take quite an elaborate trick by God to make something that's only 10,000 years old look like it emitted light 12 billion years ago.
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