Infinity of past and future time
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.
3. Criticisms of the possibility that the past is 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.:

it can be formed by an infinity of successive additions.

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:

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.

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).
4. Can there be an infinite future?
The cosmology is unclear:

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:

Then, the years in first series are separated from those in second series
(and from present) by an infinite number of years.

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).
2. Aquinas considers eight objections to his view and answers them.
Here we'll consider only three:

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.

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.

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).