1. There's no conclusive cosmological evidence that time began with the big bang about 15 billion years ago, and it is a matter of some debate whether we live in an oscillating universe (big bang/big crunch/big bang, etc).  However, the issue of the temporal infinity of the world is far from new.  Aristotle believed that the world is eternal, and the eternity of the world was a hot topic in the 13th century between St. Thomas and St. Bonaventure.  Of course, both believed that the world began (the scriptures tell us that); but while for the latter thought he could prove that the world cannot be eternal, the former thought reason was unable to determine the issue and that one has to recur to revelation.

 
2. The past is infinite iff (if and only if) there is an infinite number of same length intervals,  e.g., years, before the present one, e.g.: 0 (present year), -1, -2, -3 (year),....
NOTE: in order to avoid confusion, it's important to keep in mind the following points:

• The qualification “of same length” is somewhat more restrictive than necessary.  What must be ruled out are intervals which became small fast enough to prevent going back to infinity.  For example, if the first interval is 1/2 year, the second 1/4 year, etc., one cannot go back past one year ago.
• Since there's no negative infinite number, there's no infinitely past year or earliest year.
• Each year is separated from any other by a finite number of years (remember that there's no first year).
• There never was a time when the past became infinite because no set can become infinite by adding any finite number of members.  So, if the past is infinite, then it has always been infinite.

• If the past is infinite, then there's no room, as it were, for adding years to the past because all negative numbers have been used already to number the years which are already past.  Time, then, would stop because no year (indeed no second) could become past.  Worse, since if the past is infinite it has always been infinite, time would have stopped a long time ago, which is patently false.  Differently put: if the past is infinite, then we have run out of time (there's no time left).

Reply: Renumber the past years using negative even numbers and add the new year by using negative odd numbers, e.g., -1 (present year), -3 (them next year), etc. We are not going to run out of negative numbers simply because there are as many negative odd numbers as there are negative even numbers as there are negative numbers.
• Any existing set, actually infinite or not, can be increased.  But an infinite library with all the natural numbers on the spines of books cannot be increased because there's no number left.  Hence, existing actual infinity impossible.  Consequently, there cannot be an actual infinite past (the idea here is to prevent renumbering as in the answer to the previous criticism)

Reply: Put a new number, e.g. a rational number, on the spine of the new book.  The point here is that there are as many natural numbers (1, 2, 3,...) as there are rational numbers (the fractions of two natural numbers) as there are natural numbers plus rational numbers.
• Time is given not as a set (i.e. simultaneously in one thought, as a complete whole), but as a succession (i.e. one moment after another).  But how can an infinite succession be formed by successive additions, given that at any stage in the succession one has produced only a finite length of time?

Reply.:
1. it can be formed by an infinity of successive additions.
2. Even if it cannot, all it follows is that the number of years has always been infinite, which presents no problem.  There's no starting point by successively adding to which we are supposed to perform the impossible task of getting to infinity.  Notice, however, how this answer doesn't explain how an infinite number of years could be given to start with.
• If an infinity of years had to pass before today, then today would never had arrived.

Reply: this presupposes a starting year by adding to which we get to today. But there is no such starting year, and from any year in the past one can get to the present year in a finite number of steps.
• Tristram Shandy records one day of his life in one year.  Since there are more days than years in his life, he cannot finish his autobiography (that is, having as many years, i.e. entries, as the days lived, i.e., the subjects of the entries, is a condition for finishing the autobiography).  Worse, the more time goes by, the furtherbehind he is in the autobiography.  In other words, his entries cannot catch up with his life. But supppose now that Shandy were to live forever.  Then, after an infinity of years has elapsed, his entries could catch up with his life (notice, not after any finite number of years, no matter how large) because there is a biunivocal correspondence between the set of years and the set of days (both of them have the cardinality of of N, the set of natural numbers).  But this is absurd, because the more time goes by, the farther behind he falls. Hence, the notion of an infinite time (be it past or future) is incoherent.

Replies:
1. The objection confuses ideas of 'infinitely many' with that of 'all', and of 'more' in a cardinal sense and in an inclusion sense. Biunivocal correspondence is a necessary but not a sufficient condition for finishing the autobiography because it only guarantees an infinity of entries.  But infinitely many entries are not all the entries needed, much in the same way in which there are as many odd numbers as natural numbers, but there are natural numbers which are not odd.

Duplication: It's true that an infinite set A can be properly included in another set B and yet have as many members as A, so that there would be members of B which are not in A (think of the set of even numbers and that of natural numbers).  But why is this relevant?  It looks as if the only reason Shandy's entries  could not catch up with his life if he lived only a finite number of years is that there were more days to write about than entries about them.  This only obstacle is removed once he lives forever.
2. The principle that the more time goes by the more Shandy falls behind, might not be expandable to cover the case of infinite time. (This was Russell's view).

• If there is enough matter, the universe will end in a big crunch, and that perhaps will bring about the end of time.  If there is an oscillating universe or no big crunch will occur because there is not enough matter, then perhaps an infinite time is possible.

NOTE: if infinite past and future, then ordering ...,-2, -1, 0, 1, 2,..., i.e., omega* 0 omega.
• If the universe comes to rest after infinite expansion, then there is a possibility for the future to have the ordering (omega + omega), i.e., 0 (present), 1, 3, 5,... (end of motion) 2, 4, 6,...

NOTES:
1. Then, the years in first series are separated from those in second series (and from present) by an infinite number of years.
2. This hypothesis makes sense only if one thinks that time can exist without motion or change, in a dead universe where nothing happens.  This is equivalent to assuming a substantival theory of time.

Aquinas on the eternity of the world

The XIII century saw a big controversy on whether it could be proved, and not merely accepted by faith, that the world began.  Saint Bonaventure held that it can be proved that the world began, while Saint Thomas Aquinas held that it cannot. Here we look at some of the arguments of Saint Thomas.
1. Since Aquinas adopted the Aristotelean theory of demonstration, according to which science deals with what is necessary, he held that it's impossible to prove that the world began because:

• The essence of things is independent of any temporal modality.  For example, it's impossible to derive from the essence of man whether man is sempiternal or not.
• The divine will concerning creation is not necessary (e.g. God didn't create out of necessity).

1. Nothing can be equal to God in any respect.  But if the world had no beginning, it would be equal to God with respect to infinite duration.  Hence, the world has a beginning.

Reply: Divine duration is not successive.
2. If an infinity of days had to pass before today, then today would never had arrived because it's impossible to traverse the infinite.

Reply: this presupposes a starting day by adding to which we get to today. But there is no such starting day, and from any day in the past one can get to the present one in a finite number of steps.
3. If the world were eternal, then any man would have been begotten of a previous one in an infinite series.  But the father is the efficient cause of the son, and an infinity of efficient causes is impossible.

Reply: Aquinas draws a distinction between two types of series of efficient causes:
• efficient causes which are required per se to bring about a certain effect (think of a big clockwork in which cogwheel B can move cogwheel C only insofar as it's being moved by cogwheel A).  In this case, an infinity of causes is impossible (there must be a spring moving the first cogwheel)
• efficient causes which are not required per se to bring about a certain effect. A man B generates another man C not as a son of a previous man A, but as a man (A's generating power needn't be around for B to generate C).