The infinity problem in Buddhism is the following. “Each of us” has been in the rebirth process for an infinite time. Thus, “each of us” has been exposed to Buddha Dhamma countless times, i.e., made infinite attempts to attain *Nibbāna*. How is it possible that all of “us” have not attained *Nibbāna*?

**July 15, 2017; Revised February 5, 2018; Re-written March 23, 2021; re-written September 24, 2022**

##### Introduction – “Infinite Monkey Problem”

1. The question is based on the following statement in several *suttas* in* Anamatagga Saṃyutta *starting with the “*Tiṇakaṭṭha Sutta* (SN 15.1)“: “There is no discernible beginning to the rebirth process.” In other words, we have had an infinite time to attain *Nibbāna *(because an infinite number of Buddhas must have been born too.) So, why have we all not attained *Nibbāna* yet?”

- This issue has been discussed in discussion forums without a conclusion. See “The problem of infinity in Buddhism” at Dhamma Wheel and “The infinity problem in Buddhism” at the Sutta Central forum in 2017.
- This question seems to have its origin in the “infinite monkey theorem,” which states that a monkey hitting keys at random on a typewriter keyboard for an infinite amount of time will almost surely type any given text, such as the complete works of William Shakespeare.
- By the way, this infinite monkey theorem is another evidence of how unimaginably large infinity is: “Infinity – How Big Is It?“.

##### Detailed Analysis of the Infinite Monkey Theorem

2. First, the monkey theorem is based on a monkey typing on a keyboard and generating random letters. It is assumed that **the same monkey** will keep typing on the keyboard non-stop for an infinite time.

- For example, the probability that the monkey will come up with the word “banana” would be
**less than 1 in 15 billion, but not zero.**Thus it is a theoretically**possible**outcome, even though improbable. - But the enormity comes to view when you realize that you have to get a WHOLE TEXT correctly without making too many errors at a stretch (in one continuous period.) For example, to get TWO words correctly, like “yellow banana,” has a probability that is the multiplication of the probabilities of getting each word right. The probability that a monkey gets those two words correctly is less than 1 in a billion-billion (1 in 10^18). That is extremely small.

3. The above Wikipedia article says: “..the probability that **monkeys filling the entire observable universe** would type a single complete work, such as Shakespeare’s *Hamlet*, is **so tiny that the chance of it occurring during a period hundreds of thousands of orders of magnitude longer than the age of the universe** is *extremely* low (but technically not zero).”

- That is because scientists estimate the age of our current universe to be only about 14 billion years. Infinity cannot be assigned a number. Any large number you can think about CAN NOT BE the largest number because you can just add 1 to that number to make it bigger. There is no ending! Thus, given an infinite time, it is theoretically possible that a monkey could type out the complete works of William Shakespeare.
- However, as we see below, the rebirth process involving a “lifestream” can not be compared to the
**same monkey**typing on a keyboard for an eternity.

##### Two Relevant Issues

4. We will discuss **TWO aspects of this issue.**

**First**, we will show that the infinite monkey theorem’s MECHANISM does not apply to the rebirth process.**The rebirth process DOES NOT involve a “person/soul/**traveling the*ātman*“*Saṁsāra***(like a single monkey typing for an infinite time.)**Nothing like a soul “moves” from this life to the next. Even the next moment in life arises based on causes and conditions based on the present moment, i.e., via the*Paṭicca Samuppāda*process.**Second**, we will show that even if an infinite number of living beings has attained*Nibbāna,*there will still be an infinite number left!

## First Issue

##### There Is No “Person” Traveling the Rebirth Process

5. During the rebirth process, various forms of life manifest. Even though we live human lives now, we have lived in most of the 31 realms described by the Buddha. **We have been born a Deva, a Brahma, an animal, a hell-being, etc. countless times!**

- The concept of a “lifestream” moving from life in one realm to another (rather than a soul incarnating or appearing in different forms) is what the Buddha described. See “What Reincarnates? – Concept of a Lifestream.”
**No “unchanging/permanent essence” like a soul moves from one life to the next.**If that were the case, stopping the rebirth process and attaining*Nibbāna*would be impossible! How can a permanent entity cease to exist?- What is taken to the next life is
*anusaya*/*gati*/*saṁyojana*. None of those is permanent. Each one can change even momentarily! - Furthermore, all those entities are associated with suffering. Elimination of
*anusaya*/*gati*/*saṁyojana*is the end of suffering, not the end of an entity like a soul or an*ātman.*See “Yamaka Sutta (SN 22.85) – Arahanthood Is Not Annihilation but End of Suffering.”

6. When one understands that by comprehending “Paṭicca Samuppāda, Tilakkhana, Four Noble Truths,” one becomes a *Sotapanna* by removing significant parts of *anusaya*/*gati*/*saṁyojana. ***That is getting rid of the wrong view of a “soul/ ātman” or “sakkāya diṭṭhi.”** See “Sakkāya Diṭṭhi and Tilakkhana.”

- As that
*Sotapanna*progresses on the Noble Eightfold Path, more of the*anusaya*/*gati*/*saṁyojana*are removed. See**“**Kilesa – Relationship to Akusala, Kusala, and Puñña Kamma**”**and**“**Anusaya, Gati, Bhava – Connection to Manomaya Kāya (Gandhabba).**”** - With the attainment of the
*Arahant*stage, ALL*anusaya*/*gati*/*saṁyojana*will be lost from the mind. At the death of the*Arahant*, there is nothing left that can initiate another birth, and thus, the rebirth process stops.

##### Path to *Nibbāna* Is Not a Mechanical Process

7. Therefore, reaching *Nibbāna* CAN NOT be considered a mechanical process and, thus, CAN NOT be compared to a monkey hitting arbitrary keys on a keyboard to generate Shakespeare’s *Hamlet*.

- We can consider a monkey typing a single letter to a living-being hearing the correct Buddha Dhamma. Therefore, we can call that a “single-shot” at
*Nibbāna*, corresponding to a single keystroke by a monkey. - However, a mathematician could still say that there will be an infinite number of such “single-shots” by a living being over an INFINITE time.

8. But the key issue is that it is NOT **the same person** who heard the correct Buddha Dhamma infinite times!

- Therefore, the problem is with the question itself.
**The infinity problem in Buddhism is phrased as follows:****“Each of us”**has been in the rebirth process for an infinite time. How is it possible that all of us have not attained*Nibbāna*? **That question assumes that a****fixed person/soul/***ātman*is repeatedly reborn!*Paṭicca Samuppāda*explains that there is no such permanent entity being reborn. See “What Reincarnates? – Concept of a Lifestream.”- We went through the above discussion to show that the two processes cannot be equated. Now, we address the SECOND issue mentioned in #4 above.

## Second Issue

##### Infinite Number May Have Attained *Nibbāna*

9. It is indeed possible that an infinite number of living beings HAVE ATTAINED *Nibbāna* in the past.

- Not only that,
**an infinite number of living beings may have attained**Of course, attaining the*Buddhahood*in the past.*Buddhahood*is infinitely more difficult than attaining*Arahanthood*. - Therefore, the infinite set of living beings who have attained
*Nibbāna*is “much larger” than the infinite set of living beings who have attained*Buddhahood*. **The key to this puzzle is to realize that “many levels of infinity” exist.**It has been revealed by mathematicians within the past hundred years, thanks to the pioneering work of the mathematician George Cantor. See “George Cantor – The Man Who Founded Set Theory.” The following video provides good insights too.

##### A Nice Visualization of Infinity Within Infinity – The Infinite Hotel Paradox

10. The following video explains why there can be “smaller infinities” types within infinity. In particular, the set of positive integers is a “smaller infinity.” Those who have attained *Nibbāna* fall under that category. Regardless of how many have attained *Nibbāna*, more could attain *Nibbāna*.

- The following video discusses an infinite number of buses filled with an infinite number of guests arriving at an infinite hotel. It is shown that the infinite hotel can accommodate all of them and more!

- I have set the video to stop around 2:10 minutes. It is enough to see that the hotel can accommodate an infinite number of guests at any time. During the presence of a
*Buddha Sāsana*, only a finite number of living beings (humans,*Devas*, and*Brahmas*) attain*Nibbāna*. - The rest of the video is more mathematical and shows that even an infinite number of buses with an infinite number of passengers in each bus can be accommodated! You can watch the whole video by clicking “watch on Youtube.”

##### Other Related Issues

11. Of course, several other questions now arise: Where do all these infinite numbers of living beings live? Do they all live in our Solar system? It will take many more future posts to explain these fully, but we can summarize them as follows.

- Brief answers to those questions are as follows: According to the Buddha, an uncountable number of planetary systems are populated with living beings. While an uncountable number of living beings live in our Solar system, there are an uncountable number of such planetary systems (
*cakkavāla*) in the world. Each cluster of 10,000 such*cakkavāla*can have a Buddha appearing periodically. Thus, there could have been an infinite number of Buddhās. - Such details are in
*suttas*in the*Tipiṭaka*, mostly in the*Digha Nikāya*. I briefly discussed one*sutta*: “Buddhism and Evolution – Aggañña Sutta (DN 27).”

##### Summary/Additional Resource

12. Infinity is a very complex concept. There are many levels of infinity. Infinity minus infinity can still be infinity.

- Therefore, even as there have been an infinite number of Buddhās and an even higher infinity of those who attained
*Arahanthood*, there are still an infinite number of living beings (including us) who have not yet attained*Nibbāna*. - Even more importantly, there is no contradiction per the “infinite monkey theorem” because the premise/assumption of that theorem does not hold for the rebirth process.
**Thus, there is no contradiction in either case.**