Infinity Problem

For a few months now, I have a problem that I just can’t solve, even though it’s very simple. When we all have existed forever and not got enlightened yet, then statistically enlightenment must have the probability of 1 to infinity per life. Because if the probability was 1 to whatever number then the probability of not having achieved it in infinite lifetimes would also be 1 to infinity. So if there is the slightest possibility that the Buddha didn’t achieve true freedom from rebirth, for example the likelyhood of 1 to 9.999.999.999 then that would still be much more likely than the chance that he did. I just cannot understand how: 1. The Buddha never said anything about this. 2. If the Buddha was right than there must be some explanation, why I am wrong or there is some solution that the likelyhood is not infinitely low, because to trust that the likelyhood of 1 to infinity happened to me in this life does seem irrational for me to believe in especially when we cannot be 100% sure to know that the Buddha was right.

I thought about this from many angles but found no solution that did not came out to be nonsense after reflecting more.


This is how i understand it.

A set of numbers greater than 1 and lower than 2 contains an infinite (uncountable) amount of numbers. Another set of numbers between 1 and 3 is twice as big if you measure it but also contains an uncountable amount of numbers.

Just because both sets are infinite as to the number of elements, the second set is still measurably larger and has elements not contained in the first set.

A being’s transmigation is immeasurable & uncountable as to one’s past and is potentially immeasurable as to one’s future lest he becomes awakened, having become awakened the future becomes measurable & finite or determined.

The immeasurability & uncountability of the past is akin to the immeasurability & uncoubtability of the infinite amount of numbers lower lower than 0. This dataset is immeasurable & uncountable but it still doesn’t contain any number >0.

Likewise even if one’s past is immeasurable & uncountable it still doesn’t contain all possible eventuality and a living person’s transmigation is measurably larger compared to another which has attained parinibbana.

Yes, but this has nothing to do with my problem.
(Sorry, I don’t want to be rude, it’s just that as I understood your post, it seems to be correct (except the part at the end, I believe it is possible that a person does never achieve nibbana, so it is also immeasurably longer than someone who does) but I don’t see any connection to my problem except that it is also about infinity)

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I will try to address specifics.

I think it is wrong to assert that each life has a probability of awakening. A cow’s life has zero probability of awakening. A human’s life may or may not have a probability of awakening, it’s not determined lest one is a Bodhisatta.

Also asserting that the probability is 1/infinity doesn’t translate to a value, the expression ‘divide by infinity’ is dividing by an undefined value so you aren’t actually dividing, it doesn’t give you a value of a probability because infinity is not a number.

To me it seems absurd to speak of a probility as 1:∞ because the difinition of a probability is arrived at by dividing one number by another number, not by dividing a number by a symbol representing a concept of an incalculable set.

Maybe a mathematician can explain this better.

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Thought more about it,

If you divide by a number you get a value, if you divide by a higher number the resultant value is lower, if you divide by an infinitely high number then result is zero.


It’s not a probability, it is an impossibility.

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Should be said

When we all have existed forever and not got enlightened yet, than statistically enlightenment must have been impossible in the past.

It is impossible only under those conditions and may still be possible under different conditions because the past doesn’t include the future.

Hence one can’t determine the possibility [probability] of enlightenment based on a past dataset which doesn’t contain it.

I think i understood your problem now and that you are asking about a dataset which does include awakening.

As in a Buddha’s life being one of an uncountable amount of past lives and it being one of an infinitude.

I think the idea of a probability still doesn’t apply because it was never a probability in the past.

It is not like a dice is rolled each time one takes birth as one’s birth is determined by previous development.

Another way to think about it is that the measured ‘end or limit’ of a set containing infinite numbers isn’t itself a number within the set. Yet the measured end of a set is a truth even tho it’s not a number, therefore one can’t calculate it’s probability.


Yeah, I struggled with this problem for a while myself! But, at the end of the day there’s a subtle math error there.

Are you familiar with the two envelopes paradox? Or selecting random irrational values?

There are certain situations where modeling an arbitrary event as a random event or assuming that P=0 implies impossibility is a misuse of statistics and gives wrong answers. Especially when dealing with infinite distributions, you simply can’t sum things up and divide like that. Here are a couple math videos that address this problem from different angles:

Hope they help!


Thanks both of you, I will need some time to be able to grasp your answers as I am not very learned in these things.

At some time in the past I thought the answer is simply that there are things that are contradictory but still true, and that I might not get a solution to this other than that it is paradoxically not extremely unlikely even though it has never happened in an infinite past.
Now my question is: I have read in some places in the internet that infinity always leads to contradictions and no one has yet been able to solve these. Is that true? What are these contradictions? Is there a list of all contradictions of the concept of infinity somewhere?

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Not really. Like Zeno’s Paradoxes of Motion which can be understood with Calculus, problems involving infinity can be explained, they just require somewhat advanced maths.


Can you try to explain this a bit simpler?

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Did you watch the videos? I think the dartboard illustration is about as simple as it can get :sweat_smile:

You throw a dart at a dartboard. What is the probability that it lands exactly, precisely in that one spot? P=0 right? Cause there are an infinite number of other spots where it could have landed. Yet it has to land somewhere!

It’s the same with Nibbāna. There are an infinite number of lives where the Dhamma could land in our heart and wake us up, but it only lands once.

Does that help?

The problem is that P=1/6 is for things like dice rolls, where there are a finite number of options to pick from. Said another way, P=1/∞ isn’t quite the same thing as P=0

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Have you read my first post?
I am not saying it is impossible, I am saying 1 to infinity is infinitely unlikely. And If there is the slightest chance that the Buddha was wrong that would be infinitely more likely, hence my problem.

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As I said before, this is an inappropriate use of statistics. The Buddha is either right or wrong in fact and that fact-about-the-world cannot be modeled as a random variable. See the (classical) two envelope paradox above.

The other way to reason your way out of this box (if you want to) is to remember that Nibbāna has infinite value. So even an infinitesimally small chance is worth it, because the reward is so great.

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We cannot know for a 100% that the Buddha was correct. And for that reason we can say that there is a likelyhood of him being correct or not. And that likelyhood is not infinitely high because there is nothing there, that can assure us he is that likely correct.

And I have to disagree with the last part:
I don’t think the likelyhood of one to infinity is worth it, because the result is so great, instead the likelyhood of staying forever in samsara being infinitely likely is a hellish nightmare.

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If anyone has a solution to this problem, I invite you to share it with me, as I am suffering a crisis of losing faith while still being aware of the suffering of samsara.

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Think about this problem differently.

Suppose you see a man build a house in 30 days, the house is finished on the 30th day.
Is the probability of him finishing the building of a house 1/30?

This is why you can’t, as Bhante pinned it, ‘model an arbitrary event as a random event’.

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Thanks, as I see it it doesn’t work that way.
As far as I know we don’t all work towards nibbana from beginningless time, instead we wander around aimlessly. And even the Buddha tought, as far as I know, that the appearance of a Buddha is a sheer coincidence. So we can come to the conclusion that it is impossible that the likelyhood is higher than 1 to infinity because otherwise it would have happened almost surely.

It is as if an unskilled builder was to build a house.

Being unskilled he is making mistakes and as a result of those mistakes the house keeps breaking down but he doesn’t give up.

Sometimes he learns, sometimes he doesn’t learn, sometimes he forgets what he’s learned, sometimes he doesn’t forget, this or that time maybe he will overcome and maybe he won’t.

How long will this building go on?

It can go on and on and he can in theory come up with endless amount of excuses for why it went wrong this time or that time.

It doesn’t mean that he will never finish but there is no guarantee that he will.

However if he was to finish then you wouldn’t model the finishing as a random event would you?

It doesn’t matter how long it takes him to finish it, 30 days, 300 days or 3000 days, or even infinity, you still don’t calculate the probability of finishing as a random event in the manner of 1/30, 1/300, 1/3000 or 1/infinity because it is an arbitrary event and not a random one.

Important Sutta here is Ahara Sutta

"Monks, I will teach you the feeding & starving of the five hindrances & of the seven factors for Awakening. Listen & pay close attention. I will speak…

I won’t post the entire discourse but it points out that there is a on & off development of good and bad qualities and there is no guarantee that the good ones will culminate given enough time.

As Bhante pinned it

Even if a man builds a house in 30 days, the day he finishes it is one out of 30 days or 1/30 but it is not a statement about probability, there was no chance that it would be finished on day 2 3…29, it took 30 days and finishing the build wasn’t a random event.

Likewise if one attains Nibbana after beginningless transmigation and we express it as P=1/∞ or P=0 we aren’t making a statement about a probability we are just saying that something occurred out of an infinity.

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All beings are working to overcome delusion, all are working for happiness.

Nibbana is a removal of taints.

No beings want to be unhappy and are working for the highest happiness they can conceive of.

Less delusion allows them to conceive of a greater happiness.

Nibbana is the highest happiness.

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I’m not sure how much detail there is in EBTs, but the traditional Buddhist presentation is that liberation is impossible while a being is in the lower realms below the human world or in the higher realms in the heavens. So, the actually number of lives when there’s a possibility of liberation is a subset that’s less than infinity, if we really want to take it to this extreme of mathematically calculating it. (Is there some mathematics for infinite subsets of infinity? I have no idea myself.) There’s also the later idea that a person can liberate themselves without encountering a Buddha by realized the truth of dependent origination in other ways (i.e., become a pratyeka-buddha), which would raise the chances a bit more.

Interestingly, in later Buddhist thought, it was asserted that it’s a certainty that each being will eventually be liberated because suffering is the primary cause for developing wisdom and seeking liberation from it. The only uncertainty is the duration of time spent in samsara.

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