We don’t. We have no idea if the universe is finite or infinite. We can’t see beyond the horizon. As for how we know we’re right about things, we don’t. You can never know with 100% certainty that your explanation is perfect. We do know the distance across the visible universe because we can observe the light from distant objects and see how redshifted it is.
What we do know is that the scientific method will always get us closer to the truth. We form a hypothesis about how something works, do an experiment to test it, and record our observations. If the results of the experiment match our predictions, we think the hypothesis is correct. That doesn’t mean we won’t run into a situation where the hypothesis doesn’t work. It does mean we then have to adjust the hypothesis and do more experiments, until all the results match our predictions.
For example: quantum mechanics has never had an experiment done that disagrees with it. As far as we can tell, it accurately explains the universe at very small scales. It may be that some day we’ll run into a situation where it doesn’t work, but until then, all the evidence points to it being correct.
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