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Imagine that the Universe had a temporal beginning but no temporal end. At the beginning the Universe has a finite size, and as time passes its size increases exponentially. And the number of observers is proportional to the size of the Universe, so as time passes the number of observers also increases exponentially. If I'm a random observer, then I should expect to be temporally located infinitely far away from the beginning. And then if, instead of travelling forward in time like everyone does, I could travel backward in time, I would never reach the beginning of the Universe in any finite amount of time.

Would it be logically possible that the Universe has a beginning in time but we're temporally located infinitely far away from the beginning?

By infinite I'm not merely saying that there is an infinite amount of instants that separates us from the beginning, by viewing time as continuous or dense. I'm saying that there is an infinite amount of seconds that separates us from the beginning. Time has to be viewed as the Natural numbers, not the Real numbers.

And I use the word infinite in the mathematical sense, so it doesn't just mean "a huge number" like a googol.

I don't really care much about relativity, the Big Bang, etc. I'm not asking whether it is actually the case that our Universe is like that, I'm just asking if this is philosophically possible or if it breaks the rules of logic.

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Yes, you can take any ordinal starting with ω and model time on that. Then infinite time has elapsed at any time after ω. You can even have infinite time elapse infinitely many times by taking anything after ω², see ordinal numbers.

– Conifold – 2020-03-22T07:04:47.587if you have a 'beginning of time' doesn't that imply there was a 'time' before the beginning? Time is endless, forward and backward.. – Swami Vishwananda – 2020-03-22T10:47:56.557

Does this answer your question? Infinite past with a beginning?

– curiousdannii – 2020-03-22T22:41:40.813@curiousdannii: No, because by "infinite" it means "an infinite amount of instants", so it really means continuous or dense, i.e. in the Real Numbers there are an infinite amount of points between 0 and 1. My question deals with a completely different meaning of "infinite", an infinite amount of seconds, i.e. there are an infinite amount of negative integers. – user50746 – 2020-03-22T23:29:05.080

1@SwamiVishwananda: Not necessarily. Time could have a beginning, and there would be no time before the beginning. E.g. imagine a sphere like the surface of the Earth: there is nothing North of the North pole. – user50746 – 2020-03-22T23:30:51.193

Ah you're right, sorry. Maybe this one instead then: https://philosophy.stackexchange.com/q/2351/13641 There have been quite a lot of questions about actual infinities before.

– curiousdannii – 2020-03-23T00:01:57.377I defy anyone to make sense of the idea that time has a beginning. – None – 2020-03-23T12:42:47.347