Today, the position that time began is held by big bang cosmologists; however, it is a view which has a long history and was held by many theologians, Augustine and Aquinas to mention just two.  Here, first we shall consider what it means to say that time has a beginning, and then some objections which have been raised against the view that time began.

1. What does it mean to say that “time begins” ?(Let's call the sentence "time has a beginning" "B")
A general answer is that B means that there is a first interval of time of some length, e.g., a first second, or minute, or whatever.
A more detailed answer depends on what view of the structure of time one accepts.  Smith and Oaklander consider two options (time contains no intervals of length zero, and time contains intervals of length zero, i.e. instants), with  the first option itself divided into two suboptions.

A. Time contains intervals of time but no intervals of length zero (instants). There are two interesting sub-options here:

1. Time contains a shortest interval of time which cannot further be divided, a time-atom.  Time,  then, would look like a string of beads in a necklace, with each bead representing an atom of time.  The time series, then would be isomorphic (structurally identical) to the series (or a part of it) of natural numbers, 1, 2, 3, 4, etc.

NOTE:
• Given two moments of time (two time-atoms) it makes sense to ask whether one is next to the other (for example, if there's a first time-atom, one might wonder whether there's a second time atom, etc.)
• The atomic view of time is quite old; it has been revived by quantum mechanics which could be taken to entail that 10 elevated to -43 sec. is the shortest time.  Later we'll look a arguments for it.
• Then, B means that there is a first time, i.e., the first atom of time (the first number or the first bead).
 
2. Time contains no shortest interval of time (there are no time atoms) because any interval, no matter how small, can be divided in half. 

NOTE:
• Time, then, is dense, i.e. between any two times t1 and t2 there is a third time t3 (e.g., take the interval t1-t2 and divide it in half; the midpoint is t3)
• notice that if one allows infinitesimals or one denies actual infinite division, then density doesn't entail the existence of instants.
• Given a moment t1, then, there is no other moment t2 next to it.  In fact, take the interval t1-t2; then, there's a time t3 between t1 and t2; hence, t2 cannot be next to t1.
• There is no first time M.  In fact, suppose a first time M existed. Since M is an interval of non-zero length, it can be divided in half.  Hence, the first half of M is earlier than M, and therefore M is not the first interval of time.  (This doesn't entail that time extends infinitely in the past because time could be a (semi)open interval).
• Then, B means “there is a first interval of some length”, e.g., a first second, or a first century, etc.

B. Time contains intervals of length zero (i.e., instants).
The interesting case, here, is when time is continuous, that is, is isomorphic to the set of real numbers. 
NOTE:
Time is dense, and so it has infinitely divisible intervals; however, it also has instants, i.e., time quantities of zero duration. Then we have a choice with respect to what B means:

• B can be taken to mean “there is a first instant”.
• B can be taken to mean “there is a first interval of some length” if one rejects the idea that there is a first instant.  In this case, time would be a semi-open interval in the direction of the past. 

NOTE: The reason for adopting this alternative is that in big bang cosmology at the first instant the universe would have zero size, which, one might argue, cannot be.

• “X begins” entails that there is a time t at which X does not exist.  But “there is a time t at which time does not exist” is contradictory.  Hence, time cannot begin.

Objection: That time has a beginning doesn't entail that there was a time at which there was no time; rather, it entails that “before” time there was nothing (or no time), i.e., that given any time t, time extends only for a finite distance before t.
 
• Aristotle's argument: a moment is a boundary between past and future.  Hence, any moment must have a successive and a previous one.  Therefore, there cannot be a first moment.  So, time cannot have a beginning.

NOTE:
Evidence that a moment is a terminus (a boundary) between past and future comes from the consideration of the extension of the now, the present.  Is this minute the present? Well, no, since its first part is past and its last future.  Is, then, the present, the 'now' identical to this second?  The same reasoning shows it isn't.  So, the present, the now, the instant is merely a boundary between past and future, with no temporal extension.
Objections:
• At most, the argument shows that the 'now' has zero duration (or perhaps, that it is not a quantity), not that it is a boundary between past and future
• The argument fails to establish its alleged conclusion. Time could have a beginning without having a first moment (e.g., by being dense and being an open or semi-open interval).
 
• Quinton's argument: since the time relation “before than” is transitive and asymmetrical, given any moment, one can coherently describe a previous one.  Hence, there's no first moment.

Objection: One may repeat the second objection advanced against Aristotle's point: time could have a beginning without having a first instant.  However, Quinton might repropose his point by substituting “interval” for “moment,” thus rendering the objection irrelevant, if the intervals don't become too small too fast.
Duplication: Possibility of description doesn't entail existence.

Some general issues.

1. A topology of time.
Let time be a set of moments ordered by the relation T, “being before than.”  Then the standard topology is given by the following axioms, where x, y, z are moments:

1. (x) -Txx  (irriflexivity, i.e., no moment is before itself)
2. (x)(y)(Txy->-Tyx) (asymmetry, i.e., if x is before y, then y is not before x)
3. (x)(y)(z)((Txy & Tyz)->Txz) (transitivity, i.e., if x is before y and y before z, then x is before z)
4. (x)(y)(Txy v Tyx) (connectedness, i.e., every moment is before or after any other moment)
5. (x)(y)(Ez)(Txy->(Txz & Tzy) (density, i.e., between any two moments there is a third one)

1. (x)Ey(Tyx) (no first moment)
2. (x)Ey(Txy) (no last moment).

NOTE:

• Irriflexivity and/or asymmetry rule out a closed time, that is, a time that loops on itself.
• Connectedness rules out fission (a time series that splits into two time series) and fusion (two time series that join into one time series). However, it allows for parallel time series as long as they stay separate.

1. it is possible to traverse an infinite number of intervals of time.
2. there are temporal atoms, and consequently there's no infinite number of intervals beween two times t1 and t2.
3. things exist at discontinuous temporal positions, by leaps, as it were (perhaps by temporal transcreation) and exists only instantaneously.

3. Another philosophical argument for the existence of time atoms. 
Start with a paradox of measurement applied to time:

1. Any time interval T of non-zero duration is not only infinitely divisible, but actually infinitely divided.
2. There are no time infinitesimals.
3. Hence, actual infinite division leads to instants.
4. Instants are magnitudes (quantities of zero duration)
5. Ultra-additivity: if a magnitude is partitioned into classes of parts, then the size of the whole original equals the sum of the sizes of all the parts which are members of the classes.

NOTE: that is, if a quantity Q can be (in principle) divided into parts A, B, and C, which do not overlap and which, taken together, amount to Q, then the size of the whole A equals the sum of the sizes of the members of A, B, and C.  For example, take a string Q of 10 beads, each 1 inch long.  Q can be divided into class A (the first 3 beads), class B(beads 4-8), and class C (beads 9-10).  Since A, B and C don't overlap (they have no common members), and together they make up Q, the size of Q is equal to the sum of the sizes of the beads, that is 10 inches.
This sounds like an obvious truth, perhps it isn't true in all cases.
6. T is partitioned into classes of instants.
7. Hence, T has zero duration.

• Deny (1):
• One could, with Aristotle, deny that any actual infinite exists.  Then, there would be no complete or actually infinite division (only potentially infinite division would be allowed).
• There are time atoms.
• Deny (2): this alternative is certainly possible, but apparently has not been much explored.
• Deny (4):

One could hold that an instant is a terminus (a bounday and not a quantity)between two time segments, like the now is between past and future.
Problem: This, however, would leave unexplained how a quantity could be partitioned into classes of non-quantities.
• Deny (5): this is the view of contemporary analysis, which rejects ultra-additivity in the case of continuous magnitudes.  A line-segment can be partitioned into classes of points (which have no length), and yet it has a positive magnitude.