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We can see light from stars more (way more) than 6,000 light years away, therefore the universe cannot be a mere 6,000 years old.
The furthest objects visible, quasars, have been detected 13 billion light years away.
The idea was first systematically advanced by creationist Barry Setterfield in his 1981 book The Velocity of Light and the Age of the Universe.
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However, they are all united by a desperate need to shoehorn an absurdly young age for the universe into a reality that says otherwise.
To solve the starlight problem, some creationists have proposed a change in the speed of light; this proposition became known as c-decay.
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The starlight problem states that if the universe was only 6,000 years old — as Biblical literalism and Young Earth Creationism (YEC) state — then there would not be sufficient time for distant starlight to reach Earth.
The methods of measuring distances to the billions of light years are rather complicated, but there are direct measurements well beyond the limits of YEC, using only parallax.
There are the measurements of the supernova SN1987A at about 168,000 light years, and the Gaia space mission should obtain many distances of objects up to about 30,000 light years.