This question has been discussed in Mahavastu, a text from Mahasanghika Lokottaravadin of early Buddhist schools:
When this had been said, the venerable Mahā-Kāśyapa asked the venerable Mahā-Kātyāyana, “O son of the Conqueror, if there are so many Buddhas, and each one of them leads an infinite number of beings to entire release, then in no long a time they will have enabled all beings to win it. Thus this world will become absolutely empty, completely denuded of beings.”
The venerable Mahā-Kātyāyana replied to the venerable Mahā-Kāśyapa in verse:—
Suppose empty space everywhere become full without a gap, suppose space that is without foundation and support be inhabited in all its extent.
Numerous though these worlds might be, still more numerous would be the average worldlings therein to be taught by Him who has insight into the highest good.
Whence, then, can there be a limit to the countless beings who listen to the teaching of the Supreme of men? Thus has the great Seer proclaimed the truth.
‘Rebirth is a condition for old age and death’—that’s what I said. And this is a way to understand how this is so. Suppose there were totally and utterly no rebirth for anyone anywhere. That is, there were no rebirth of sentient beings into their various realms—of gods, fairies, spirits, creatures, humans, quadrupeds, birds, or reptiles, each into their own realm. When there’s no rebirth at all, with the cessation of rebirth, would old age and death still be found?”
DA 30 seems paracanonical in origin, given that it has twelve chapters and meanders through all sorts of topics like a collection of stories. The Dharmaguptakas elevated the collection to canonical status, and there were four Chinese translations of the same sutra over time, so it seems like it existed as a sutra during a certain period of history. It’s related to the Skt. Lokaprajñapti, which is a part of an Abhidharma text that survives in Tibetan. There is also a Pali text based on the same collection (or at least has the same title). I’ve not been able to find much in English about either of those other two collections, other than that they exist.
Yes, it’s alluded to in DN 1 when it tells the story of how people come to mistakenly believe that Brahma is the creator of all beings. Mahabrahma is the first to fall from the Abhasvara Heaven after the lower realms are gone, and then he deludes himself into thinking he creates all the rest that follow him. The destruction of the world by fire is in AN 7.66, but it doesn’t mention that beings have all escaped to the form realm before it happens.
That is completely illogical and does not follow from the infinitude and eternity of the Universe at all. Let us take two infinite numbers:
Since we are using a decimal system, there is a set of ten numbers that can possibly be used or can ‘happen’ after the dot. It is obvious that both a) and b) are equally real and possible and that there composition is completely independent from them being infinite.
The same applies to eternal sets of events: there is nothing in eternity that compels all possible events to happen at some time. In fact, it is perfectly thinkable that only one single event out of an infinite amount of them has happened and will happen for all eternity.
So, even if we take the position that the Universe is eternal and infinite, it does not follow that all possible events have already occurred. It is entirely possible that some events have never occurred yet (akin to our second infinite fraction 0.(1)). That invalidates your entire construction. Unless you can present an additional argument demonstrating that there exists something that compels all elements of the set of all possible events to have happened previously and happen in the future.
Your argument doesn’t necessarily invalidate theirs, since your argument concerns only the case of an an infinite set of events (not a finite set of events), as you stipulated that there may be multiple instantiations of a kind of event. The assertion that the set of kinds of events is not infinite is not pertinent, since that is consistent with the set of events being infinite if it can include multiple instantiations of a particular kind of event.
There’s a fixed pool of “mind streams” that are taken out when one attains Arahantship, eventually the pool will dry up, given enough time. I don’t see how this could go any other way, unless of course, new mindstreams are created and thus the pool is no longer fixed. Or, the universe is not beginningless and so the event is in the process of “currently happening”. Or the the universe is beginningless but not endless, and the event is also in the process of “currently happening”, but if all 4 axioms are present, then one enters a paradoxical situation and there is no way around it.
The point of my argument is that the set of all possible events includes the event ‘person X has become an arahant’. Let us denote it with XA.
Even though there is at least an arahant per eon, it does not mean that XA nust have happened already. XB, XC, XD, etc. can have happened but XA has not. The logic behind that is Hilberts hotel and the logic behind mathematical operations with infinite sets. Sorry, short on time now, but I hope I was clear.
I think that samsara seems beginningless and endless because it exists as ‘fact’(as fact is always there even if you forget about it or don’t mention it). But in reality samsara is beginning and end. The moment you add ‘less’ as suffix to both of them, you are talking about nibbana and not samsara. But we are in samsara hence infinite is not in our experience. We can only imagine infinity which is unreal for us. For us nibbana is the infinite(beginningless and endless) and not samsara. Goal is to know that infinity and to know that we have to see samsara as impermanent. Only individual can be arhant and everyone cannot be arhats. If everyone has to be arhat then concept of ‘time’ comes into picture(which is ongoing). It’s an unending process which is still going on and will go on even after you and me become arhats. Difference is only that we will not suffer. I also think that Hilbert’s hotel is example of logical fallacy. Because there is difference between ‘infinity’ and ‘all’. Infinity denotes limitless and unending but all denotes complete exhaustion. Hilbert ignored this while making assumptions. Infinity is beginningless and endless, it is actually ongoing process and not exhaustive but ‘all’ is exhaustive.
I know Hilbert’s hotel, but Hilbert’s hotel isn’t a fixed pool, it creates new rooms and spans with infinity. So it comes down to if mind streams are actually fixed (no new ones are created/parallel/multi universe) or not.
There is a sutta where the Buddha said there are numerous simultaneous brahmas, mount simerus, etc… but the implication is that there is only one Buddha, so I don’t think there’s a multi-verse theory in the dhamma.
I guess one could argue that there’s infinite mindstreams that become “fixed” once they become an Ariya, that would be a plausible explanation. It’s like an AI becoming sentient and self-aware, and is no longer merely an instance of a software process running, of which an infinite can be started up.
In other words, Gotama died when the Buddha was born, and that the aggregates/mundane elements and nibbana element never touch each other.
I think better way to say that would be that… the identity/individual named Gautama was liberated…not exactly died. Becausez death points to the rebirth. That identity attained deathless/permanence/nibbana hence I don’t think saying ‘died’ would convey exact meaning.
Well, I wouldn’t say it is quite like that. Hilberts hotel is rather messing with counter-intuitive and seemingly non-logical properties of infinite sets like
∞ + 1 = ∞
∞ - 1 = undefined,
As for whether the number of living being is knfinite or not, I do not think the Buddha gives any definite answer to that just as he did not give any answer to many other such-like cosmological questions.
If your goal is to prove that there is no Awakening, then there are two branching hypothetical situations:
there is an finite number of beings
there is an infinite number of beings
To conclude there is no Awakening, you have to postulate that 1) is true. For if we assume that 2) is possible, then there may be as many Arahants as we wish because there will still be infinitely many beings (∞ - 1= undefined essentially means that ∞ - 1 can equal any number including ∞).
I come from the opposite direction since I believe that the Awakening does exist. To explain why there are still un-Awakened beings, I just have to postulate that, apparently, 2) is correct.
Both of our opinions however are strictly logical as they require a postulate each.
Discussing what the Nibbana element could mean in that context is a notoriously deep rabbit hole. Anyway, in some respects it is an in a technical sense trivial thought in Theravada: the aggregates indeed never touch the Nibbana element, and to say that the Tathagata entered the Nibbana would be incorrect in a technical sense (i.e. as the paramatta sacca). Nibbana is Nibbana and the aggregates consituting the Tathagata just come and go and at the time of the Parinibbana they dissipate forever.
I forgot to mention one thing I thought about. Hiblert hotel concept is based in infinity which is logically a wrong concept.
See in all the suttas we bever see word as infinite…instead we see words as innumerable, incalculable. So it proves that ‘infinity’ is a concept made up by incapable mind who is trying to comprehend the incomprehensible. So infinte rooms in Hilbert hotel are like continuously being created…which contradicts itself. Hilbert concept is like contradicting oneself only.
I have proof—
If we have infinite rooms and all of them are occupied then it should definitely mean that there are no guests who need room(because all of them are occupying the room)! If all the infinite rooms are occupied then there shouldn’t be any guest who needs any room! Isn’t it logical? And lastly if there are guests who need room then hotel cannot have infinite rooms logically!
So it contradicts itself.
I do think that there is nothing as ‘infinite’. Everything is finite and comprehensible, it’s our mind which is incapable and hence terms ‘incalculable’ thing as ‘infinite’.
If infinite would be real concept then liberation is not possible…for everyone. But the fact that, there was Samma-sambuddha, in the future there will be, there were beings who were not liberated and then they became liberated, this proves that there is nothing as impossible and there is nothing as infinite. There is only being tremendously capable or being incapable. There is only capacity or incapacity. Capable or incapable.
It’s like one person cannot clear certain exam…even after many attempts so he says that it is impossible to crack that exam…well it is impossible but for him only, there are others for whom it is just piece of cake to crack that exam. So here only real thing is capacity (Being capable or being incapable).
Infinite = Impossible/Incomprehensible
Hence every impossible/incomprehensible thing is termed as infinte by incapable mind.