There are various arguments that have to do with not being able to reach the present if the past were in fact infinite. One of them involves the axiom of not being able to form an actual infinite through successive finite addition. We can again justify this using the counting example. Consider again the story of an immortal person named Count Int who is attempting to write down all the natural numbers and reach infinity with his trusty pen and never-ending supply of paper, taking him exactly one second to write down each number. He starts with one and successively adds one each second (1, 2, 3, 4...). Will he ever reach a point in time where he can honestly say, “I’m done, I’ve reached infinity”? No, the number will just get progressively larger and larger without limit. He can never reach infinity anymore than he can reach the greatest possible number. There will always be a bigger yet finite number in the next second. The Count will never lay down his pen because there is an infinite quantity of those numbers.
Since we’re using numbers, this example can be instantiated to reality whether it be distance (1 meter, 2 meters, 3...) time (1 year, 2 years, 3...) or whatever. In any case, it seems clear that a complete traversal of an infinite region (when done via successive steps of finite size) cannot be done. For instance, suppose a person named Joe Walker tries to completely traverse Infinity Avenue walking at the rate of one meter per second. Infinity Avenue is infinitely long and begins at Hilbert's Hotel. Starting from the hotel, he walks 1 meter, 2 meters, 3, 4... and though he walks tirelessly (just as Count Int is tireless) Joe Walker cannot succeed. The infinite distance is impossible to traverse via successive finite steps, with the only “barrier” to its successful traversal being the sheer magnitude of this impenetrable distance.
However, an infinite past requires that an infinite number of years be traversed before the present is reached, and thus the universe could never have reached the current point in time. And time is certainly traversed via successive finite steps (year after year). So while an actual infinite might be valid mathematically, in reality it just doesn’t work with an infinite past. An infinite past requires an impossible traversal.
The set of all finite numbers is infinite. When counting (1, 2, 3, 4...) each number will be reached eventually. Recall the counting example (starting with 1, adding one each second; 1, 2, 3, 4...), in which our job was to count all finite numbers. Now if we can’t traverse all the finite numbers, which finite numbers won’t we get to? There aren’t any of course. Each one will eventually be gotten to. We thus traverse them all (1, 2, 3, 4...).
In the counting example (1, 2, 3, 4...), it is true that any finite number we can name will eventually be gotten to (whether it be a million, a trillion, or whatever). But will the Count ever manage to count all finite numbers? Ironically, no. The Count will never lay down his pen. This is because no matter how far he gets, there will always be more numbers to count (he can eventually get to a million, but there’s still more numbers to count; he can eventually reach a trillion, but there’s still more to count etc.). The Count never lays down his pen. The Count being able to reach any finite number only implies that he can traverse any finite distance; it doesn’t imply that he can traverse an infinite one.
If the Count eventually reaches infinity, where does he cross the threshold between a finite number and an infinite one? What is the last finite number he writes with his pen before writing “infinity”? Remember that the Count always gets his next value by adding one to the previous value (1, 2, 3, 4...e.g. when he’s at 2 he adds one to get 3, and when he’s at 3 he adds one to get 4). If he eventually reaches infinity, what finite number does he add 1 to in order to get infinity? If all natural numbers could be counted, then along the path of successive finite addition (1, 2, 3, 4…) where does he traverse all those numbers? After what natural number does the Count lay down his pen? Nowhere. Instead, there will always be more numbers to count regardless of what number he has reached so far (a million, a trillion, or whatever), and thus his job (counting all those numbers) is never done. In fact, it can be proven via mathematical induction that the Count cannot reach infinity.
A false analogy is one where there are relevant differences that outweigh the similarities. In this case, the story of the Count is a false analogy because the only reason why the Count cannot reach infinity is because there is no end. With a beginningless traversal there is a difference: it does have an end (the present point in time). Also, the story of Count Int has a beginning, whereas an infinite past does not. Therefore, a beginningless traversal of an infinite region is possible even though the Count’s traversal is not.
It is true there is an endpoint in the beginningless traversal, and it is true there is no beginning in a beginningless traversal, but these are not relevant differences. The bottom line is that any traversal of an infinite region at a finite pace is impossible.
As mentioned earlier, there are actual infinites in mathematics. For instance, the mathematical infinity that comes “right after” natural numbers is symbolized by ω. But that doesn’t mean it can be reached via successive finite addition (the Count would still never reach it for instance). And so it isn’t clear why an endpoint would make an infinite traversal via successive finite steps possible. Suppose Joe Walker tries to reach a point infinitely far away. To do this he must completely traverse the infinite region. He walks 1 meter, 2 meters, 3, 4...and yet Joe can never get there. It isn’t that the traversal will take a really long time, it’s that the infinite traversal is impossible. Note also that the only “barrier” for Joe’s traversal to that point is the sheer magnitude of this impenetrable distance.
Compared to a traversal with a beginning, one view (e.g. philosopher J.P. Moreland) is that the absence of a beginning point actually makes things worse for an infinite traversal, not better. It seems clear that going from zero to infinity (as Joe trying to reach a point infinitely far away) is impossible, but traversing an infinite number of years to reach the present is even worse because it “cannot even get started. It is like trying to jump from a bottomless pit.” [1]
Another sort of infinite past is possible. Traversing an infinite region at a finite speed might be impossible, but this is not so with an infinitely fast traversal. Suppose for instance at the first “moment” of the universe (t = 0) there was an infinite regress of simultaneous causes. There would be no first cause of the universe and no actual beginning.
One of the problems of treating infinity as a number (of things) in physical reality is you can get incoherent—even to the extent of being contradictory—results. The same sort of thing applies here (more later).
Consider this example of an absurdity via an infinite number of things existing in reality (one used by Craig). Suppose we have an infinite number of marbles. Now let’s do some subtraction. We take away all the marbles; infinity minus infinity equals zero. But suppose we take away all the marbles except ninety; infinity minus infinity equals ninety. Note that identical quantities were subtracted each time and we get contradictory answers. Suppose we take every other marble away (e.g. all the odd numbered ones); infinity minus infinity equals infinity. Another contradictory answer. As a result, mathematics has an entirely arbitrary set of rules against this sort of thing to prevent contradictions from happening. Contradictions in mathematics screws up the whole system, e.g. infinity + 1 = infinity + 2, now let’s take infinity away from both sides and get 1 = 2.
In physical reality, while a mathematician might shake his finger, there would be nothing to prevent me from allowing such absurdities when I start taking away marbles and producing contradictory answers. A number of philosophers such as William Lane Craig (and some mathematicians) have argued on the basis of such absurdities that an actual infinite cannot plausibly exist in reality.
Similarly, we get incoherent results with this objection. Traversing an infinite region at a finite pace is impossible. This argument says you can traverse an infinite region if you fast forward through it at infinite speed. But suppose we hit the rewind button on the universe and backtrack the infinite traversal at infinite speed. We do we reach? The beginning? Not only do we get a beginning, we get a contradiction. A beginning was precisely the sort of thing the objection was to avoid. At least in this case, traversing an infinite region at infinite speed is incoherent and ultimately doesn’t work.
