The formula ∀x∃y(y ≺ x) is admissible in temporal logic, and defines a no-beginning sequence. The problem is less that such a sequence is definitively impossible, and more that it is not well-founded. But then this is so much of a problem only if one insists that all infinite sequences must be well-founded; otherwise, the formula from temporal logic doesn't seem as amiss. (C.f. the sections on well-foundedness in the SEP article on metaphysical grounding and on fundamentality.)
But so another issue is that, even if an actually infinite past time elapsing to now is impossible, so is it impossible that there has been only a finite amount of past time:
For let it be granted that it has a beginning. A beginning is an existence which is preceded by a time in which the thing does not exist. On the above supposition, it follows that there must have been a time in which the world did not exist, that is, a void time. But in a void time the origination of a thing is impossible; because no part of any such time contains a distinctive condition of being, in preference to that of non-being (whether the supposed thing originate of itself, or by means of some other cause). Consequently, many series of things may have a beginning in the world, but the world itself cannot have a beginning, and is, therefore, in relation to past time, infinite.
So perhaps either time overall is contradictory (the suggestion of a dialethic solution to the antinomy of past time) or it is not so absolute a reality as we would have to take it to be, to say that it was definitively finite or infinite as to its past extent.
Disjunction of the above options (attempted):
It is epistemically possible that (time is metaphysically possibly finite and metaphysically possibly infinite) or it is epistemically possible that (time is not metaphysically possibly finite and not metaphysically possibly infinite) or it is epistemically possible that (time is both actually finite and actually infinite). Alternatively put: either time is finite or infinite, or time is neither finite nor infinite, or it's both. I'm not of a mind to countenance the dialethic option that much (I tend to interpret the dialethic theory of truth as involving, at best, something like a self-dual truth-value), so I would settle on (using a bit of Gödel-speak):
- If it is relatively consistent that time is finite, then it is relatively consistent that time is infinite; so if it's inconsistent that time is finite, then it's inconsistent that time is infinite, too.
A comprehensive theory of time might very well be amenable to the amount of arithmetic that causes this relative-consistency stuff to pop up (see about e.g. Brouwer's temporal intuitionism).