Has Cantor Proved the Muslim Theologians Wrong?

Answered by Shaykh Faraz Khan

Question: 1. The universe can never be infinite because otherwise we would never exist. But how come God is infinite if the universe can’t be infinite?

2. Do Zeno’s paradox and Cantor’s work in mathematics show that actual infinite does exist in reality?

Answer: Assalamu alaikum wa rahmatullah,

I pray this finds you in the best of health and states.

Please pardon me for the undue delay in answering your questions.

The Universe as Finite, and God as Eternal

Regarding your first question, the universe cannot possibly be infinite because it is bound in space and time, and therefore occurs as discrete and separate events. Distinct events can never amount to an actual infinity, since an actual infinity cannot possibly be ‘traversed’ (since there is always more to traverse!), let alone completed by successive addition of separate events (since you can always add one).

The only infinity that exists in the universe is ‘potential infinity’, which is purely conceptual, such as the endless divisibility of a finite measure that can take place in the mind; or such as the possibility of always ‘adding one’ to any finite number or measure. So endless future events (like the everlasting abodes of Paradise and Hell in the afterlife) are potential infinities, not actual: they shall last forever, by God’s creative act, yet at any one moment their elapsed time period is always finite.

God the Exalted, however, is not bound in space and time. He is transcendent above any contingency and temporality, since His existence is metaphysically necessary and not merely possible. Therefore His eternality does not entail the fallacies of an actual infinity in space-time. The universe necessarily had a beginning, while its Creator is necessarily eternal and timeless.

Zeno’s Paradox

Regarding your second question, neither Zeno’s Paradox nor Cantor’s set theory prove the possibility of an actual infinity in space-time.

Zeno’s Paradox has many types (Achilles Paradox, Dichotomy Paradoxes, etc.) which all attempt at showing that certain temporal events cannot possibly occur since they entail infinities. Yet all are based on the common fallacy of treating the potential infinity as actual, and are therefore non-paradoxical. A finite distance does not actually contain an infinite number of points. As stated above, finite distances have only potential infinities, by way of unending conceptual division.

Cantor’s Set Theory

Cantorian set theory, which treats the actual infinite as a determinate totality, pertains only to the mathematical realm of concepts and as such has no bearing on extra-mental existence (objective reality). As stated above, actual infinities cannot exist in reality, as demonstrated by the obvious contradictions entailed otherwise, such as the impossibility of further addition to an actual infinity. In real time or space though, any number, however large, can validly be added to.

Another paradox of set theory untenable in the real world is the equivalence of a whole to its part: it is self-evident (a ‘first principle’) that in reality, a whole is always greater than its part. But in the Cantorian realm of actual infinites, a infinite set of all integers (1, 2, 3, …) is shown by one-to-one correspondence to be equivalent to an infinite set of even integers (2, 4, 6, …).

Applied to the real world, the position of a beginningless universe would entail no difference between (a) past infinity until the present moment (the whole), and (b) past infinity until an event a thousand years ago (its part), which is clearly absurd. (In the classical kalam literature, this reductio ad absurdum argument is termed burhan al-tatbiq, or the ‘demonstration by way of one-to-one correspondence’.)

Indeed, several mathematicians acknowledge that modern set theory is merely an abstract model of intellectual discourse, completely divorced from any implications in the physical universe. Bernard Bolzano for example, an early pioneer of modern set theory, admits that actually infinite sets exist only in the “realm of things which do not claim actuality, and do not even claim possibility.”

Moreover, several antinomies were identified that over time made set theory less and less tenable. Specifically, Zermelo and Russell discovered a contradiction in set theory that caused its abandonment by even former proponents like Dedekind and Frege. And David Hilbert’s analysis of set theory illustrates that the existence of an actual infinity in reality (real time and space) would not only entail numerous paradoxes, but would in fact undermine the axioms of finite numbers and hence mathematical reasoning in general. As such, Hilbert concludes that actual infinities are not only logically absurd but also mathematically inadmissible in reality.

[Spitzer, New Proofs for the Existence of God; Craig, Kalam Cosmological Argument]

An Infinity in the Singularity?

According to the ‘Big Bang Theory’, which is pretty much accepted as factual in the scientific community, the universe began about 13.7 billion years ago from a singularity, or point of infinitely dense matter. Aside from the problematic name of the theory, it does serve as empirical confirmation of what Muslims believe that the universe had a beginning, although such confirmation is not necessary since that theological position is proven metaphysically (by logical demonstration, or burhan). And if for some reason the theory changes, the rational demonstration of the universe’s beginning still remains undeniable.

Yet based on the Big Bang Theory, the singularity by no means shows the occurrence of an actual infinity in space-time, since the very first moment after the singularity was the beginning of space-time, and the infinite density of matter of the singularity basically means there was literally ‘nothing’ in existence. So even science confirms what Muslim scholastics (scholars of kalam) proved by logic, that the cosmos was created from nothing, or ex nihilo.

The question then is, what caused the Big Bang? Metaphysically, there must be a cause, as the causal principle itself is known a priori; it is self-evident that every occurrence must have an efficient cause. Yet the cause of space-time itself must be transcendent above space-time, lest it too need an efficient cause, resulting in either infinite regress (an impossibility!) or circular reasoning (also impossible!). Therefore one is logically forced to affirm an efficient cause that is outside space-time, or eternal. Its existence is therefore necessary, not merely possible, since the possibly existent would be a temporal occurrence.

The Necessarily Existent (or Abidingly Real; in Arabic, al-Haqq) must have neither beginning nor conceivable end; be completely dissimilar from all things contingent and temporal; and be wholly self-sufficient, as need for another entails contingency and hence temporality.

The Necessarily Existent must also be unique and one, without partner or co-sharer, as composition or partnership would entail temporality along with other logical absurdities. And in light of creation, the Necessarily Existent must possess the attributes of life, knowledge, will, and power; while in light of His absolute perfection, He must possess hearing, sight, and speech.

This Ultimate Reality is what is referred to by Muslims as Allah, or God. And out of His pure largesse and mercy, He has chosen to reveal Himself through scripture sent to messengers, whose evidence was the miracles they performed that were inimitable.

The last of God’s messengers is Prophet Muhammad (peace and blessings be upon him and all prophets), whose miracle is both the revelation given to him, the Qur’an, as well as hundreds of other miracles narrated through sound chains of transmission, coupled with his pristine and cosmic character. And this final revelation, the Qur’an, confirms every reality of the Necessarily Existent mentioned above that is rationally necessary.

And Allah knows best.



Checked & Approved by Shaykh Faraz Rabbani