Age of the universe and the singularity at the Big Bang Using the standard model of cosmology we calculate the Hubble time to obtain an estimate of the age of the universe.
This model assumes a beginning of time in the past. But that point is a true singularity in the sense that even if we switch to another coordinate system it will still be a singularity.
Thus that event in space-time cannot be reached through any path. Geodesics simply end there. One can even note that the beginning of time is not a point of the manifold and does not belong to it.
Now the question is that how is it possible to estimate a finite age of the universe even though the beginning of time is a singularity at infinity and not even included in the topology of space-time?
That point is at infinity and traveling to the past we approach it asymptomatically yet we had assumed a beginning and a finite age. Isn't this a contradiction?
 A: Strictly speaking the FLRW metric doesn't specify that time starts at the Big Bang. It specifies only that the Big Bang is a singular point so it is impossible to analytically continue a geodesic back in time past the Big Bang.
If it helps to make things clearer, exactly the same happens with an object falling into a black hole. A geodesic that crosses the horizon must reach the singularity and it cannot be analytically continued forward in time past the singularity. Hence the claim common in the popular science press that time stops at the centre of a black hole, just as it's claimed that time started at the Big Bang. Neither statement is especially helpful.
In both cases the proper length of the geodesic is finite. Since the elapsed time for an observer on the geodesic is just the length divided by $c^2$ this means the time from the Big Bang until now is finite, just as the proper time to hit the singularity in a black hole is finite.
A: Using the standard model of cosmology we calculate the Hubble time to obtain an estimate of the age of the universe.
Yes, 13.8 billion years. But IMHO there's an issue worth discussing, to do with something John said in another answer: "A distant observer sees falling objects slow as they approach the event horizon and asymptotically approach zero speed at the horizon". It's related to the "coordinate" speed of light being zero at the event horizon.   
This model assumes a beginning of time in the past. But that point is a true singularity in the sense that even if we switch to another coordinate system it will still be a singularity.
I'm confident the universe is expanding, I can't understand why Einstein didn't predict it. But I'm not confident that the universe began as some point-singularity.  
Thus that event in space-time cannot be reached through any path. Geodesics simply end there. One can even note that the beginning of time is not a point of the manifold and does not belong to it.
The thing is, you could even say the same about crossing the event horizon. In Kevin Brown's Formation and Growth of Black Holes, you can read this: "In both of these interpretations we find that an object goes to future infinity (of Schwarzschild coordinate time) as it approaches an event horizon, and its rate of proper time as a function of coordinate time goes to zero". The given explanation says the event horizon is a mere coordinate artefact rather than a true singularity, and there's no problem getting past the future infinity. But IMHO the question to ask is this: is it there yet? And the answer is no. It's always no. Hence I favour the frozen-star interpretation. Kevin Brown refers to it but doesn't like it. Some cosmologists don't even know about it. But I think it's right, and that it tells us something about the expanding universe. And inflation, but that's one for another day.    
Now the question is that how is it possible to estimate a finite age of the universe even though the beginning of time is a singularity at infinity and not even included in the topology of space-time?
Liken the expanding universe to moving away from the event horizon. Forget about gravity, swap space for time, and make yourself the gedanken observer. At the event horizon you're subject to infinite time dilation, your clock doesn't tick. As you move away your clock starts ticking, and runs faster and faster. But you don't notice that it runs faster, because you run faster too. And now here you are, and your clock reading is 13.8 billion years. But I'm like the distant gedanken observer. I've been watching you from my God's eye view, and my clock reading is 13.8 trillion years. Or more. Maybe a lot more, if you catch my drift. 
That point is at infinity and traveling to the past we approach it asymptomatically yet we had assumed a beginning and a finite age. Isn't this a contradiction?
I don't think so. Your proper time is finite. It's 13.8 billion years. But there could be some other proper time that isn't.   
