I don't think this makes sense. When you ask "What are the odds that we happen to be in a finite set?", what probability distribution are you talking about? It is entirely possible to describe a probability distribution over a finite set, and is, in fact, easier to do so than to describe a probability distribution over an infinite set.
But, you said "happen to be in a finite set", which, seems to suggest that you are talking about a probability distribution over "what set we are in", where some of the sets are finite and some of them are infinite?
If so, there is still the question of "what distribution are you talking about over sets?".
If you are hoping for some kind of uniform distribution over sets, and arguing that "most" sets are infinite, then this is still fraught. Assuming ZF, there is an obvious correspondence between "Sets" and "Sets with exactly one element". Namely, [some set A] |-> [the set {A}] .
You would have to define something like a measure or something?

Now, I don't mean to say that you can't define senses of talking about "most sets" or "how common are sets with a given property". I believe some people working with category theory stuff came up with a sense in which the "amount" of finite sets (up to bijection?) is Euler's constant, e. So, I don't mean that you can't come up with a sense of "most sets" in which it is true that "most sets are infinite" or even a sense in which "almost all sets are infinite".

But you would have to both describe the sense in which you mean that, and also argue for why we should consider that to apply to our intuitive notion of what is "likely" to be true of the universe in which we live, rather than some other sense.

Also, the universe having existed for a finite amount of time, doesn't really mean that the, what, set of points in time (in a given reference frame) is finite. Like, maybe time is discrete, but afaiu, we have never observed anything that strongly suggests that it is, and everything we've seen is compatible with using, and we usually do use, mathematical description of which are based on infinitely divisible time (though, there are some things about the planck time and whatnot.) . So, if time is infinitely divisible, the amount of points in time would be infinite, not finite.
So, even if one successfully argued that it be "unlikely" that the set of points in time so far be finite, it doesn't seem clear that the amount of time so far shouldn't be finite, because the amount of points in time could be infinite in cardinality while being finite in measure.

Even if you wanted to argue that the cardinality of amount of points in time should be at least [any cardinality you want], I still don't think that would be a problem, because the class of Surreal Numbers have ordered subfields of arbitrarily large cardinality (as well as having arbitrarily large cardinality in any interval), and I don't think we could do anything to really distinguish between points in time being infinitely divisible and like the real numbers, vs infinitely divisible and like one of the larger subfields of the surreal numbers.

You would have to argue that it be unlikely that something like the measure of the collection of points in time so far be finite, not just something like the cardinality of the points in time.

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> When you look at the axioms in first order logic the requirement of negation is suggestive of the idea that “nothingness” (i.e. something preceding a finite set) is a logical fallacy.

I cannot get any reasonable meaning out of this. First order logic doesn't even have a concept of "finite"/"infinite" or "finite set". I'm pretty sure this is just nonsense.
Having the universe of discourse be empty is a totally valid model of first order logic.

Also, I personally prefer to reserve the "fallacy" to mean "a step in an argument which is not valid", and therefore would not apply it to a concept.

Also, "something preceding a finite set" does not, in my mind, at all correspond to "nothingness", so I don't understand why you equated the two.