Are There Different Sizes of Infinity?

Shyamasundara Dasa

Copyright © 2014-2016

In the Purusha Sukta1.3-4 from Rig Veda 10.90 [1] it states that the material creation of the Lord is only one quarter of the total creation while the remaining three quarters is the spiritual sky.

etavanasya mahima-ato jyayash-ca purushah
pado-asya vishva bhutani tri-pad-asya-amrtam divi

tri-pad-urdhva udait-purussah pado-asye’ha-abhavat-punah
tato vishvang vya’kramat-sashana-anashane abhi

“The past, present, and future universes are manifestations of the Lord's powers, but the Lord Himself is much greater. All the living beings of the universe are but one-quarter portion, and the eternal nature in the spiritual sky exists in three-quarters portion. The three-quarters portion of the Lord transcends the material portion. The Lord in the one-quarter portion manifested the universe again, as He had done repeatedly before. The Lord of the one-quarter portion began the work of creation, by going all around, taking the form of all animate and inanimate objects.”

The material and spiritual sky

"The Maha-Vishnu, into whom all the innumerable universes enter and from whom they come forth again simply by His breathing process, is a plenary expansion of Krishna. Therefore I worship Govinda, Krishna, the cause of all causes." Brahma-samhita 5.48
Artwork courtesy of The Bhaktivedanta Book Trust International, Inc. www.krishna.com Used with permission.

This reference to the creation of the Lord being one part material and three parts spiritual is found in many places in the Vedic literature. We do not propose to catalog all such references, as that is not the point of this short essay.

Though not stated directly in the verse but from the context it is also to be understood that both the material world and spiritual world are infinite in measure. Here are some references indicating the infinite nature of the material and spiritual sky:

“Every universe is covered by seven layers -- earth, water, fire, air, sky, the total energy and false ego -- each ten times greater than the previous one. There are innumerable universes besides this one, and although they are unlimitedly large, they move about like atoms in You. Therefore You are called unlimited [ananta].” Srimad Bhagavatam 6.16.37

If the material creation is unlimitedly large what then of the spiritual sky? The following verse indicates that the spiritual sky is one of the Lord’s unlimited immeasurable opulences.

“Let me offer my respectful obeisances unto Him who is the associate of the members of the Yadu dynasty and who is always a problem for the nondevotees. He is the supreme enjoyer of both the material and spiritual worlds, yet He enjoys His own abode in the spiritual sky. There is no one equal to Him because His transcendental opulence is immeasurable.” Srimad Bhagavatam 2.4.14

And, this verse indicates the same.

“Above Brahmaloka is the planet of the cows, which is protected by the Sadhyas. O Krsna, that great planet is infinitely expansive, pervading the unlimited spiritual sky.” Brhad Bhagavatamrta 3.7.81

For a sweet yet detailed explanation of how the material and spiritual worlds are infinitely large see Caitanya Caritamrta 2.21.

The question now arises and which has puzzled many persons is how could it be possible that if both the spiritual and material world are infinite how then can the spiritual world be three times bigger than the material world? How can one infinity be three times bigger than another infinity? The concept of infinity itself boggles the mind what to speak of different sizes of infinity.

Some devotees take this statement of sastra as fact simply because they have implicit faith in sastra. While others have difficulty in accepting it and this creates a seed of doubt that burrows deep into their consciousness and undermines their spiritual life because they doubt the veracity of sastra.

“But ignorant and faithless persons who doubt the revealed scriptures do not attain God consciousness; they fall down. For the doubting soul there is happiness neither in this world nor in the next.” Bhagavad-gita 4.40

In order to get a glimmer of an understanding of how there can be different sizes of infinity we shall use as an analogy some ideas from the mathematician Georg Cantor’s theory of transfinite numbers. When I was studying real analysis before becoming a devotee I recall my professor commenting that Cantor’s “naive set theory” of the late 19th century had paradoxes and that until the Zermelo-Fraenkle axiomatic set theory was published mathematicians who studied infinite set theory took refuge in theology or risked insanity. It has been suggested that Cantor believed his theory of transfinite numbers had been communicated to him by God.[2] And, now to the analogy:

bijection mapping of x to y

Let us define some terms.

A set is called “countably infinite” if it has a one-to-one correspondence with the natural number set, N, where N = {1, 2, 3, ...} (“…” means it continues indefinitely.)

“x → y” means “x corresponds to y”.

Now to indicate how the infinite spiritual sky is three times bigger than the infinite material sky.

Suppose we had an infinite set of numbers TC= {1, 2, 3, ...}.

If we took every fourth number and put it into a separate set “MS” it would consist of MS = {4, 8, 12, …}.

The set of remaining numbers, let us call it SS = {1, 2, 3, 5, 6, 7, 9, 10, 11,…}

We note that there is a one to one correspondence between N, the set of Natural numbers and MS, because 1 → 4, 2 → 8, 3 → 12, 4 → 16, … to infinity. Therefore set MS is countably infinite.

We also note that there is a one to one correspondence between N, the set of Natural numbers and SS because 1 → 1, 2 → 2, 3 → 3, 4 → 5, 5 → 6 … to infinity. Therefore according to the definition set SS is also countably infinite.

Thus both sets MS and SS are countably infinite, yet SS is 3 times larger than MS. Thus by analogy with transfinite numbers we have an idea of how Krsna’s infinite Total Creation (TC) can be one part Material Sky (MS) and three parts Spiritual Sky (SS), and that both material and spiritual sky are infinite.

Larger infinities

In mathematical analysis “countably infinite” is one size (cardinality) of infinity expressed as ℵ0 (Aleph null) the smallest size of infinity. However there are bigger sizes of infinity for example the set of all the real numbers R between 0 and 1, that is, R = {0 < x < 1}. This set R is not “countably infinite” as per definition, because there is no “one to one correspondence” with the natural set N.

To see why that is consider that you can have an infinite sequences approaching 1, for example .9, .99, .999, .9999, ... and never get to 1. But also consider that between .9 and .99 you can create another infinite sequence .9, .98, .989, .9899, .98999, … and never get to .99. And, between .9 and .98 you can create another continuous infinite sequence .9, .97, .979, .9799, .97999 … that would be less than .98. In fact between any two real numbers in the set R you can create an infinite sequence, and thus can create an infinite number of nested infinite sequences. This set of uncountably infinite numbers is designated as ℵ1 (Aleph one).

(Cantor was able to prove that the real numbers were bigger than the natural numers using the "diagonal proof." An explanation of this proof would greatly increase the size of this article and would be off the topic. Those who are interested will find this article very accessible -- Here's The Simple Proof That There Must Be Multiple Levels Of Infinity.)

A detailed discussion of the topic of Aleph numbers and infinite set theory is beyond the scope of this short essay, interested persons can visit the links at the end if they want to know more.

But, to get an idea of what it means to be infinite to an infinite degree consider Lord Brahma’s description of Govinda:

“He is an undifferentiated entity as there is no distinction between potency and the possessor thereof. In His work of creation of millions of worlds, His potency remains inseparable. All the universes exist in Him and He is present in His fullness in every one of the atoms that are scattered throughout the universe, at one and the same time. Such is the primeval Lord whom I adore.” Brahma Samhita 5.35

Lord Brahma indicates that each unlimited universe is springing from Lord Krsna, and that each universe is composed of infinite atoms, yet in each atom exists the same unlimitedly potent Govinda from whom more unlimited universes could expand. (Sadaputa Prabhu once suggested the mathematical analogy of an N dimensional Hilbert space, where N = ∞, for Krsna’s ability of being connected to infinite atomic particles in an infinite number of dimensions.)

This phenomena that Lord Brahma describes was experienced by Mother Yashoda (Srimad Bhagavatam 10.8) on the occasion of her looking into Krsna’s mouth to see if He had been eating dirt and saw in His mouth the whole cosmic manifestation including her looking into His mouth.

And, by Markandeya Rsi (Srimad Bhagavatam 12.9 and Mahabharata, Vana Parva: Markandeya-Samasya Parva, 187-188) who after asking for the benediction of experiencing the Lord’s illusory energy was floating in the pralaya ocean for millions of years and then saw a small child sucking its toe under a banyan tree. The child inhaled Markandeya Rsi into Himself wherein Markandeya Rsi found himself inside another universe where everything was normal. And, after trying unsuccessfully for millions of years to find the limits of this new universe was exhaled back into the pitch darkness of the pralaya ocean into the presence of the baby (Krsna) sucking His toe under the banyan tree. (The Mahabharata version provides complimentary details not found in the other rendition.)

I hasten to state that the mathematical analogies I have described are the product of finite (and fallible) human imagination. The atma is only one ten thousandth of the tip of a hair in size, and it is thus absurd to suggest that we could ever truly comprehend Lord Krsna’s unlimited potencies. The finite cannot comprehend the infinite, unless the infinite gives its blessings (Bhagavad-gita 10.10). But, the analogies are somewhat useful in appreciating the concept of infinity. Krsna can only be understood through sastra-“sastra yonit vat” - Vedanta Sutra 1.1.3, not by mental speculation.

To conclude we have demonstrated via the analogy of transfinite number theory that it is indeed true that while the material world is infinite it is only one part of the Lord’s creation, the other part the spiritual sky is three times the size of the infinite material sky.

Notes:

[1] ^ There are various redactions of the Purusa Sukta. The hymn finds place in texts such as the Atharvaveda (19.6), the Samaveda (6.4), the Yajurveda (VS 31.1-6), the Taittiriya Aranyaka (3.12,13) and it is commented upon in the Shatapatha Brahmana, the Taittiriya Brahmana, the Shvetashvatara Upanishad and the Mudgala Upanishad. The Purusha Sukta is also mentioned with explanations and interpretations in the Vajasaneyi Samhita (31.1-6), the Sama Veda Samhita (6.4), and the Atharva Veda Samhita (19.6). Among Puranic texts, the sukta has been elaborated in the Bhagavata Purana (2.5.35 to 2.6.1-29) and in the Mahabharata, Mokshadharma Parva 351 - 352.

[2] ^Dauben, Joseph W. (June 1983), Georg Cantor and the Origins of Transfinite Set Theory, Scientific American 248 (6): 122–131

Further reading:

Aleph number

Countable set

Naive set theory

Zermelo-Fraenkle set theory with the axiom of choice

Real Numbers

Gödel's incompleteness theorems

Different kinds of ∞, and the א numbers

Ordinal number

Hilbert space

Dedekind Cut