Why the size of the universe must be infinite

If you would buy a ticket in a lottery with 100 tickets numbered 1-100, the probability that you got a ticket with number 10 or less would be one in ten.

If you would buy a ticket in a lottery with an infinite number of tickets numbered 1-..., then what would be the probability that you got a ticket with number

10 or less? Infinitesimal.

100 or less? Infinitesimal.

1,000 or less? Infinitesimal.

And so on. One could say that:

The probability that the number on the ticket is no larger than x is at most infinitesimal, for any given finite value of x.

Now, the possible sizes of the universe range from the size of the observable universe (about 93 billion light years across, according to some estimates; the exact number is not important here) and endlessly upward through endless infinities light years across. There is nothing to say that any one of these possible universe sizes, any more likely than any other one of them, is the true size of the universe. Thus, the unknown true size of the universe can be thought of, and treated, as a randomly drawn ticket in a lottery with an infinite number of tickets numbered from 93 billion (again, the exact number is not important) and endlessly upward through endless infinities. One can therefore say something about the probability that the universe is at most 10 times the size of the observable universe, the probability that the universe is at most 100 times the size of the observable universe, the probability that the universe is at most 1,000 times that size, and so on, like in the example above - namely that every one of these probabilities is infinitesimal. Thus, like in the example above, and for similar reasons, we can conclude that: 

The probability that the universe is no larger than x light years across is infinitesimal, for any given finite value of x.

Conclusion:

The above shows that the probability that the size of the universe is finite is infinitesimal. So, the size of the universe must be infinite.

Some questions about the premise:

"Suppose you buy a ticket in a lottery with an infinite amount of tickets, numbered 1, 2, 3, ..."

Is it clear what is meant, in that premise, by "an infinite amount of tickets"? Does it implicitly somehow specify any particular infinity (namely, the set of natural numbers)? Does it implicitly specify whether said infinity is to be treated as a cardinal or as an ordinal?

There are plenty of differently large infinite cardinals: aleph(0), aleph(1), aleph(2) etc. If the amount of tickets would be (supposing that it could be) aleph(1), the amount of tickets with infinite numbers on them would be infinitely larger than the amount of tickets with finite numbers on them, which would mean that the chances of drawing a ticket with an infinite number on it would be practically 1 (to be precise: 1 minus something infinitesimal). If, however, the amount of tickets is aleph(0), then would it follow that only one of the tickets would have an infinite number on it, namely the one numbered aleph(0), meaning that the chances of drawing it would be infinitesimal? Or would there be several tickets numbered with a number of which the cardinality is aleph(0), tickets which would however consist of different series of digits and thereby "have different numbers on them"? Is it correct to say that the countably infinite ticket number 121212121212... is not the same digit sequence, and thereby not the same number, as the equally countably infinite ticket number 232323232323..., although both these numbers have the same cardinality: aleph(0)? If this is so, it seems to suggest that in said lottery there would be aleph(0) different finite ticket numbers, and also aleph(0) different countably infinite ticket numbers. Then the chances of drawing a ticket with an infinite number in said lottery would seem to be 50%.

But suppose instead that the infinity of tickets is to be treated as an ordinal. 

The smallest infinite ordinal is w and is equivalent to aleph(0) in size. However, whereas aleph(0)+1 is not larger than aleph(0), w+1 is larger than w, and w+2 is larger than w+1, and so on. If the lottery contains w*w tickets, each of which has its own unique digit sequence on it, making the tickets have numbers on them ranging from 1, 2, 3... to w*w, then there would be w times more tickets with an infinite number on them than there would be tickets with a finite number on them. Then the chances of drawing a ticket with an infinite number on it would be practically 1 (to be precise:1 minus 1/w).

Moreover, (given that the infinity of tickets may be treated as an ordinal) there could just as well be, for example, w^w^w tickets, or BB(w^w^w) tickets (for definition of BB(w^w^w), go to this article and read about "Busy Beaver number"), or any other infinite amount of tickets grotesquely larger than w. If one has no clue as to what ordinal is referred to by the "infinite amount of tickets" that the lottery is supposed to contain, then what is the best strategy, given sound decision theory, for determining the chances of one randomly drawn ticket having an infinite number on it? Presumably the best such strategy would be to treat the amount of tickets as though its ordinal were something like "the average of all infinite ordinals" (as it is not likely to be any one of the ordinals more likely than any other one of them). But however large such an average one suggests and expresses, it will be way too small to be the average of all infinite ordinals. This seems to further support the idea that the probability of drawing a ticket with an infinite number on it, in a lottery with an infinite amount of tickets numbered 1, 2, 3..., is as close to 1 as anything can possibly be without being exactly 1.

Now consider the fact that the universe may very well be, for example, something like (if its size is expressed as an ordinal) BB(w^w^w) light years across. Then if the chances that the size of the universe is infinite can be determined with the lottery analogy described in the beginning of this text, it seems that the size of the universe must be infinite (or infinite with a probability as close to 1 as anything can be without being exactly 1). But this is if the size (for example expressed as the diameter in light years) of the universe is to be treated as an ordinal. Is it to be treated as an ordinal? If not, why not?