Any model that accounts for cosmological redshift will limit our seeing distance, so that by itself doesn't prove that the universe is 13.7 billion years old. There are other factors, such as the time dilation of light curves from distant type 1a SNe. Also the uniformity of redshift from distant objects across the whole spectrum (e.g. why are 21 cm radio waves redshifted in the same proportion as 120 nm Lyman alpha photons?)

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