The Infinity of God: In What Sense Is God Infinite?

The infinity of God is the unlimited fullness of His being: nothing outside Him bounds Him and nothing in Him falls short, so His infinity is a perfection, not an enormous quantity. Since Georg Cantor proved there is no largest infinity, a God whose infinity was a size would sit on a ladder with no top. Below are the classical doctrine, its Scripture, the objection at full strength, four rival answers (Aquinas, Craig, process theism and Spinoza), Frame-Indexed Classical Theism's answer, and five tested objections.

Key takeaways

• God's infinity is of essence, not magnitude. Aquinas: "It is one thing to be infinite in essence, and another to be infinite in magnitude." His is wholly the first.
• God is not the top rung of Cantor's ladder. Every transfinite number has a greater, and Cantor held that the true Absolute, "which is in God, admits no kind of determination."
• Each rival keeps part of the truth. William Lane Craig denies the infinity is quantitative, process theism keeps God related to the world, and Spinoza takes "unlimited" at full strength.
• Frame-Indexed Classical Theism (FICT) separates two senses of "infinite." Where Craig makes infinity an umbrella term and Spinoza lets it swallow the creature, FICT holds that "infinite" said of God's being and of our counting differ in kind.

What is the infinity of God?

Infinite means without limit. Said of God, it means that no boundary of any kind applies to His being: He is not limited from outside, since nothing outside Him gives Him His existence, and He is not limited from within, since He lacks nothing that being could have. Thomas Aquinas devoted a whole question of the Summa Theologiae to the point and concluded that "it is clear that God Himself is infinite and perfect." He then drew the line on which everything below depends: "It is one thing to be infinite in essence, and another to be infinite in magnitude." His infinity is of the first kind. He is not an endless amount of anything.

The Reformed tradition put the same doctrine into its catechisms. The Westminster Shorter Catechism answers the question "What is God?" in these words: "God is a spirit, infinite, eternal, and unchangeable, in his being, wisdom, power, holiness, justice, goodness and truth." Infinity there is not one item on a list. It qualifies every item. God's power is power without limit, which is omnipotence; His knowledge is knowledge without limit, which is omniscience. That is why the tradition counts infinity among the incommunicable attributes that no creature shares, and why the hub on the attributes of God treats it as the attribute most often misread as a size.

The misreading is where the modern objections begin. By divine simplicity, His infinity is no part He has. It is His very being, considered as unlimited, and the classical doctrine of God never meant it as a number.

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Is God infinite? What Scripture says

Scripture says that He is, and it says it of His understanding and His greatness long before any philosopher defined the word. The psalmist joins two claims in a single breath:

"He determines the number of the stars and calls them each by name. Great is our Lord and mighty in power; his understanding has no limit." (Psalm 147:4-5, NIV)

The Hebrew behind "has no limit" is 'ên mispār, which the interlinear renders "there is no numbering." The God who numbers the stars is numbered by nothing.

Solomon, dedicating the temple he had spent years building, confessed that no building could hold the God he was praying to: "But will God really dwell on earth? The heavens, even the highest heaven, cannot contain you. How much less this temple I have built!" (1 Kings 8:27, NIV). Isaiah, answering a people who complained that their way was hidden from God (Isaiah 40:27), wrote: "Do you not know? Have you not heard? The LORD is the everlasting God, the Creator of the ends of the earth. He will not grow tired or weary, and his understanding no one can fathom" (Isaiah 40:28, NIV). The creation has ends, and He made them. He has none. "Great is the LORD and most worthy of praise; his greatness no one can fathom" (Psalm 145:3, NIV).

These texts teach that no measure reaches God. They do not teach a theory of sets, and they should not be pressed into one. Paul's response is the right one: "Oh, the depth of the riches of the wisdom and knowledge of God! How unsearchable his judgments, and his paths beyond tracing out!" (Romans 11:33, NIV).

The series carries the same question into time, change and mathematics.

Why an infinite God looks incoherent: the question at its strongest

The puzzle is old, and modern mathematics has sharpened it. Aristotle distinguished two concepts of infinity, the potential and the actual. A potential infinite is a process that can always go on, as counting can. An actual infinite would be a completed infinite whole, an idea that, as the Internet Encyclopedia of Philosophy records, was for Aristotle "close to a contradiction in terms." Aquinas agreed where multitudes were concerned: "it is impossible for an actually infinite multitude to exist, even accidentally."

Then Cantor changed the mathematics. He showed that infinite collections can be compared and ranked, that the real numbers outnumber the natural numbers, and that every infinite set is outrun by the set of its subsets. The result is a ladder of transfinite numbers that climbs without end, with no largest rung. That creates a dilemma for anyone who calls God infinite. If the divine infinity is a quantity, either it is one of Cantor's infinities, in which case a greater one exists and God is not maximal, or it is the totality of all of them. And the totality is exactly what Cantor, in a letter to Dedekind in 1899, called "an inconsistent, absolutely infinite multiplicity": a collection that cannot be gathered into one thing without contradiction.

The Christian apologist faces a second version of the same problem. The kalam cosmological argument holds that an actual infinite cannot exist in reality, so the past cannot be infinite and the universe must have begun. The reply arrives at once, and a reader once put it to William Lane Craig in nine words: "how is God infinite if actual infinites cannot exist?"

Suppose, then, that His infinity is not a quantity at all. The worry changes form but does not go away. The word "infinite" always invites the question infinite in what?, and once number is ruled out it is hard to say what is left. And the opposite reading is no safer. If God has no boundary in any respect, then the existence of a single sparrow seems to mark a place where God is not, and the believer is left asking how there can be room for anything but God. Here is the question at full strength. Divine infinity looks like a paradox if it is a quantity, an empty word if it is not, and a threat to every creature if it is taken without qualification.

How have theologians answered? Four accounts at their strongest

Classical theism: infinity of essence. Aquinas's answer, which the Thomist tradition has kept, is that the divine infinity belongs to form and act, not to quantity. Matter limits form, and quantity belongs to matter, so a being that is pure act, with no matter and no unrealized potential, is unlimited in a sense no quantity can be. He adds that "Things other than God can be relatively infinite, but not absolutely infinite." Created things can be infinite only in a qualified way: matter, for instance, can take on endlessly many forms, yet it is limited by the form it has. The strength of the view is that it escaped Cantor's paradox centuries before Cantor stated it, because it never placed God among the quantities that can be ranked. Its strain is intelligibility. Critics such as R. T. Mullins argue that a life complete without succession is beyond our power to conceive, a charge the attributes hub takes up.

Craig: infinity as qualitative. Craig defends the kalam, so the objection strikes him first, and he answers it directly. In a 2009 reply he wrote that "when theologians speak of the infinity of God, they are not using the word in a mathematical sense to refer to an aggregate of an infinite number of elements." In his words, "God's infinity is, as it were, qualitative, not quantitative." Omniscience need not mean an infinite stock of propositions, since "God has a single undivided intuition of reality, which we finite knowers break up into individual bits of information called propositions."

The answer lets Craig hold two things together: actual infinites cannot exist in the created world, so the past had a beginning, and God is nonetheless infinite, because His infinity was never a count. That is exactly right, and any adequate account must keep it. The strain is in what Craig concedes next. "Really 'infinity' is just a sort of umbrella term used to cover all of God's superlative attributes," he writes, and "If you abstract away all of those attributes, there really isn't any distinct attribute called 'infinity' left over." If that is so, the doctrine becomes a summary of other perfections, and the word the theologian shares with the mathematician becomes an accident of language.

Process theism: a God both infinite and finite. Process theism, in the form Charles Hartshorne gave it, divides God in two. In the Stanford Encyclopedia's account, this God "is dipolar in having an enduring character embodied in successive states," and His infinity lies chiefly in "the unlimited capacity to influence, know, and care." The view's strength is that God is really related to a changing world, affected by what His creatures do and suffer, which many find closer to the God of the prophets than a God unmoved by anything. Its cost is that God's actual, concrete being becomes finite and growing, and the classical God who does not change His mind is surrendered.

Spinoza: the absolutely infinite substance. Baruch Spinoza accepted the third horn. "By God, I mean a being absolutely infinite," he defined, and he drew the conclusion: "Besides God no substance can be granted or conceived," and "Whatsoever is, is in God, and without God nothing can be, or be conceived." No account takes "unlimited" more seriously. But no creature now stands outside God as a real other, and creation from nothing gives way to a world that follows from God's nature by necessity.

Volume Five, still in production, works through the mathematics of infinity; follow its progress.

How does Frame-Indexed Classical Theism answer the infinity of God?

Frame-Indexed Classical Theism (FICT) is a research program in analytic theology that resolves classical theological dilemmas by diagnosing them as artifacts of a single error: one term carried across two frames as though it meant the same thing in each. On infinity, Craig keeps the classical God by thinning the word to a summary, process theism keeps the word by dividing God, and Spinoza keeps both by abolishing the creature. FICT keeps the classical God, the full word and the creature, because it finds two senses where the dilemma assumed one.

In God's own frame, the infinite is the completed fullness of the divine nature, which Volume Five defines as "not an endless amount of anything, but the positive absence of all limit in the One whose being is underived." In our finite frame, the infinite is the endlessness of an operation: whatever number we reach, a further one is available. Both are real. They differ in kind, as the fullness of a being differs from the openness of a process, and the difference follows from the difference between God's nature and ours, since a being's nature fixes the way it knows.

Infinity belongs to God the way His goodness does: essentially and positively. Against Craig, then, something is left over when the other attributes are named. What is left is the fullness itself, the plenitude of which power, knowledge and goodness are aspects.

This is why the two claims never contradict each other. A contradiction needs one proposition affirmed and denied in the same sense and at the same time. "The infinite is actual" is said of God's being; "the infinite is potential" is said of our counting. The senses differ, and they share no moment, since God's eternal present is not a position on the creature's timeline.

Cantor's ladder now has its place. Each transfinite number is a real infinite, gathered by a finite mind into a single unit it can work with, and every such unit has a next. God is not a unit and has no next. He is not on the ladder at all, not because He stands too high on it but because He was never one of its rungs, any more than the sun is one of the colors a prism throws on the wall. Cantor saw this himself. He walled off the Absolute from his transfinite numbers and wrote that "The absolute can only be recognized, but never known, not even approximately known."

The endless ladder is how the one Infinite appears to a finite mind that receives it a piece at a time. That does not demote mathematics. The mathematician keeps every theorem Cantor proved, and his unending climb is a real, God-sustained engagement with what exceeds him. As Volume Five puts it, "That God is the completed Infinite does not falsify the mathematician's endless progressions; that our progressions never complete does not embarrass God's plenitude."

The same distinction explains why there is only one actual infinite. A completed infinite would owe its completeness to nothing outside itself, and what owes its wholeness to nothing outside itself is self-existent. Aseity admits no second, so the actual infinite is God alone.

The third horn falls with the first two. God's fullness of being is not total extent, as though He occupied all the room and left none. Creatures are not parts of the Infinite. They are held in being by it: "For in him we live and move and have our being" (Acts 17:28, NIV). He is the ground of being, not the largest thing there is.

The method behind this answer is laid out in Volume One, which you can download free.

Does the infinity of God survive its hardest objections?

We have run thousands of simulated attacks against Frame-Indexed Classical Theism, drawn from academic and peer-reviewed sources. Five bear directly on the doctrine, and it holds under each. Aquinas met the size objection before Cantor stated it, and Craig met it by calling the infinity qualitative. FICT meets all five below with the two senses set out above.

Does an infinite God fall into Cantor's paradox?

Here's one of those attacks and how the Many Beings framework would address it.

A set theorist presses Cantor's own result. Cantor held that the Absolute is not a transfinite quantity, and that no consistent quantity can be "the infinite"; any attempt to treat the totality as one finished thing leads to contradiction. So when a theologian says that God's eternity is an infinite content too great for any finite mind to hold, he seems to treat that infinity as a magnitude, a size beyond every size. A maximal quantity is exactly what Cantor's paradox forbids.

The defense grants the mathematics and denies the reading. His infinity, as Volume Three states it, is "the formal, qualitative infinity of pure act," following Aquinas. It is the perfection of a being with no unrealized potential, not a quantity at the far end of the quantities. No comparison of sizes is made, so none can fail. Cantor's paradox shows that no magnitude can be the infinite, and the classical doctrine never said God was one. The objection is sound against a theology that makes God the largest number; it does not touch a God who is pure act.

If an actual infinite is impossible, how can God be infinite?

Here's one of those attacks and how the Many Beings framework would address it.

A constructivist mathematician, who accepts only the infinities that a procedure can actually build, presses the question Craig's reader asked. The same series that denies an infinite past, on the ground that it would be an actual infinite, affirms that God's eternity is a completed, simultaneous whole. Why is one completed infinite incoherent while another is the very model of coherence? The asymmetry looks like special pleading.

The defense is that "completed" is being used in two senses. The worry about an infinite past is a worry about traversal: a beginningless series would have to be run through by adding one moment to another until the present arrived. God's eternity is not a series at all. It has no before and after within it, so nothing in it was ever run through step by step. Volume Three adds a further safeguard. It rests the beginning of the world on its dependence on God, and treats the traversal argument as contested corroboration rather than as the proof. The eternity of God and Volume Three take this up in full.

Doesn't every stretch of time already hold infinitely many instants?

Here's one of those attacks and how the Many Beings framework would address it.

A set theorist who treats Cantor's infinities as fully real points to the standard mathematics of the continuum. Any stretch of time, however short, contains as many instants as there are real numbers, which is uncountably many. So if "the only actual infinite is the Creator," a single hour already refutes the claim. The thesis seems to prove too much.

The defense concedes the mathematics and shows that the argument never rested on it. The uniqueness of the divine infinity was never a claim about counting. It is a claim about self-existence: the created order exists only as God sustains it, so it cannot be a self-standing, beginningless whole, and Volume Three calls this contingency "the primary and decisive ground." A bounded stretch of time with uncountably many instants is still wholly dependent on God. What the series denies to creation is completion of itself, a whole that owes its completeness to nothing outside itself. Whether the instants within a bounded stretch count as an actual infinite is a question about how many instants there are, and Volume Five, the volume on the mathematics of infinity, is where that question belongs.

Can an infinite mind know every truth without contradiction?

Here's one of those attacks and how the Many Beings framework would address it.

Patrick Grim's best-known argument, that there is no set of all truths, is answered in the article on whether God is omniscient. Grim has a second argument that needs no set at all. It builds on the Knower paradox of David Kaplan and Richard Montague (1960). Add to ordinary arithmetic a symbol meaning "known by an omniscient being," require that whatever it covers is true and that it covers every truth of its own language, and a self-referring sentence produces a contradiction. If even the claim "for every truth, God knows it" generates the paradox one truth at a time, then an infinite understanding looks impossible.

The defense accepts the result and locates it. What the Knower proves is that God's knowing cannot be installed as a symbol inside the creature's own formal language. The troublesome sentence speaks of that symbol, not of God; it records what the symbol cannot do, and God knows that truth as He knows every other. Within any one system we may assert the claim level by level. The unrestricted reach belongs to God's one act of knowing, which no sentence pattern of ours contains.

Would an infinite God rule out the Trinity?

Here's one of those attacks and how the Many Beings framework would address it.

Aquinas argued that God is one "from the infinity of His perfection": "If then many gods existed, they would necessarily differ from each other," and each would lack what the other had. The same reasoning appears in the claim that two self-existent infinite beings would bound each other. A Trinitarian reviewer turns it inward. The church confesses three Persons, each fully God. Does the argument for a single infinite being exclude the Trinity?

The defense is that the argument excludes a second self-existent being, a second reality standing of itself outside God, and the Persons are not three beings. The church's confession is one substance in three persons, distinguished by their relations of origin and not by division, each Person the one divine essence and not something added to it. One essence has one aseity, so the Trinity introduces no second infinite. The position offers no doctrine of the Trinity of its own; it receives the doctrine from Scripture and the councils, where John states it in a single sentence: "In the beginning was the Word, and the Word was with God, and the Word was God" (John 1:1, NIV). Volume Four, Resolving Immutability & Interaction, states the confession the position receives.

Each volume takes one of these tests in full; see the series.

Frequently asked questions

Is God infinite or finite?

Christianity teaches that God is infinite, and historic confessions such as the Westminster Shorter Catechism say so. Some modern views make God partly finite: process theism holds that God's concrete experience grows and is limited at each moment. On the classical view, God has no limit in being or in any perfection, and His infinity is fullness of being rather than an immense amount of anything.

What is the difference between God's infinity and His eternity?

Infinity is the broader idea: God's being has no limit of any kind. The classical tradition then names that infinity by what it is set against. With respect to time, it is His eternity, the life without beginning, end or succession treated in the eternity of God. With respect to space, it is His immensity, and His omnipresence is His presence to everything He has made.

If God is infinite, how can anything else exist?

Because His infinity is not a quantity of room. He is not the largest thing, filling space so that nothing else fits. He is unlimited being, and creatures exist as beings He gives and sustains, not as rivals for space within Him. Acts 17:28 states the relation: "in him we live and move and have our being." Spinoza's conclusion follows only if infinity means total extent.

Can the human mind understand a God without limits?

We can know Him truly but never comprehend Him, which is the difference between a true grasp and a complete one. Scripture reveals God in words we understand, and the tradition of apophatic theology insists that every concept still falls short of Him. A finite mind receives the infinite a piece at a time, as Romans 11:33 confesses: "his paths beyond tracing out."

Does divine infinity prove that God exists?

No. The doctrine says what God is, not that He is. The arguments for His existence are a separate inquiry, surveyed in the post on evidence for God. One of them, the kalam, argues from the impossibility of an infinite past, and Volume Five treats it in full. What the doctrine does accomplish is defensive: it shows that infinity in God is not a contradiction, which removes a common objection to the arguments that conclude He exists.

Where to go from here

The ladder of infinities that never ends is no problem for the doctrine. It is what His infinity looks like from below, to a mind that can only take the unbounded one unit at a time and always finds another. Cantor, Aquinas and the psalmist agreed on the essential point. The God who numbers the stars is numbered by nothing, and no rung of any ladder reaches Him.

Volume One sets out the method that separates the two senses of "infinite" and applies it from first premises, and it is yours free. The series carries it into evil, time, change and mathematics.