By the existence of time and universe, we have a number of alternatives:
ALTERNATIVE 1) Time existed form an infinite past, and universe existed from an infinite past (TU).
ALTERNATIVE 2) Time existed form an infinite past, and universe existed from a finite past (Tu).
Note I: I THINK THERE IS AN ARGUMENT OR PROOF FOR THAT 'IT IS IMPOSSIBLE THAT TIME EXISTED FROM A FINITE PAST' AND THEREFORE THE ALTERNATIVES 3 AND 4 ARE NOT TRUE (BUT ANYWAY WE WILL EXAMINE THEM HERE): Consider the state 'A' which is that time does not exist. Then consider the state 'B' which is that time exists. Every change should happen through time. For a change from the state 'A' to the state 'B', there must be time existing at 'A' and 'B'. Therefore the time existed at the state of 'A'. Saying that 'there was a time that time did not exist' is contradictory. Therefore it is impossible that time existed from a finite past.
Note II: Also since I am not aware of any strong non-mathematical example that my 'intuition' about that is wrong, I think, by my intuition, the existence of time does not depend on the existence of universe and 'time existed from an infinite past'.
ALTERNATIVE 3) Time existed from a finite past, and universe existed from a finite past (tu).
ALTERNATIVE 4) Time existed from a finite past, and universe existed from an infinite past (tU).
EXAMINING ALTERNATIVE 1) That 'the time existed from an infinite past and universe existed from an infinite past (TU)', by our assumptions or logic is impossible. Reasons are:
A) If the time or universe always existed, we can't imagine a zero start moment, because for any point of time that we assume as the zero moment, there is an earlier; and we can't imagine that the time or universe does not have a zero or n moment.
B) That the time or universe existed from an infinite past, it means infinite passed time, and infinite passed time may be by logic impossible.
EXAMINING ALTERNATIVE 2) That 'the time existed from an infinite past, and universe existed from a finite past (Tu)', it means that 'the universe came into existence from nothing', and because by logic, a 'substance' (or matter or universe) cannot come from nothing, I think the ALTERNATIVE 2 is clearly a 'violation of logic' or magic.
Also in ALTERNATIVE 2 the impossibility reasons of ALTERNATIVE 1 (A and B) also remain.
AN UNNECESSARY PART
EXAMINING ALTERNATIVE 3) If 'the time existed from a finite past and universe existed form a finite past (tu)', we have three alternatives:
C) Time and universe began to exist at the same time: Imagine with the start of time, a line segment (as a part of universe) started from length 0, increased through time, and now is reached to certain length 5. We can argue that if the current length is 5, since the length is not greater than 5, it means that before the 5 length before, the line segment did not exist (if it existed, the length would be greater than 5). By referring to before the existence of universe and time, we can prove that one of the two CASES is true: 1. CONTRADICTION: That the time and universe began at the same time is false, because it is contradictory, since it leads to the result that 'we can refer to a time before the existence of time'. 2. NO CONTRADICTION: If we can truly refer to a time before the existence of universe, we can say the universe 'came into existence from nothing' which is by logic impossible.
D) Time began before the universe began: This means universe came into existence from nothing, which is illogical.
E) Time began later than the universe began: This statement is contradictory, because 'later' means that time existed and has passed before that time begins, and therefore it is not true.
EXAMINING ALTERNATIVE 4) In the statement 'the time existed from a finite past and universe existed from an infinite past (tU)', that time started at some point, is contradictory, because the phrase 'at some point' means 'at some time' and shows that time already existed.
Note III: The Item E or ALTERNATIVE 4 may be related to the 'Big Bang theory'.
BACK TO THE MAIN PART
CONCLUSION: All alternatives, by our assumptions or logic are impossible.
Note IV: Our intuition may not be right about basic 'mathematical' problems (such as it is impossible that an infinite summation does not get over a particular finite number'). Also ESPECIALLY we may not be able to solve 'complicated' 'mathematical' problems. But not only this time-universe problem does not seem to be a complicated mathematical problem, I have a feeling that it may not be even a mathematical problem, and therefore it may show that there is something 'fundamental' we don't know (such as an unknown dimension!?). - (And even if it is a mathematical problem, I think discovering its answer will be revolutionary.) - This is not the only thing that seems strange that I faced: See for example my argument that 'we can create an INDETERMINISTIC physical motion'.
Is a finite or infinite past about time or universe a sign of 'violation of logic' or magic?