Limit and infinity are not two neighboring notions but one, seen from two ends. A sequence (a_n) has limit L precisely when a process that could be extended infinitely — step after step, without stopping — nevertheless acquires a precise target: however close you wish to approach L, there is a point beyond which all terms remain within that closeness. The limit does not deny the infinity of steps; it gathers it into a single point. This is exactly what calculus showed against Zeno’s paradox: Achilles can never overtake the tortoise, Zeno said, because between them lies an infinite sequence of ever-smaller distances still to cross — ancient reason was paralyzed before this infinity. Only the concept of limit showed that a sum with infinitely many terms (1/2 + 1/4 + 1/8 + …) can still have a finite, exact value: 1. The infinity of the steps and the finality of the limit do not exclude each other — infinity, passed through the structure of the limit, becomes calculable.
But exactly where the limit seems to have tamed infinity, infinity splits in two, and the limit no longer reaches both halves alike. Aristotle allowed only the potential infinite — a sequence to which one more term can always be added, without there ever being a completed collection. This is the infinity the limit knows how to hold in check. But Cantor dared the actual infinite, treated as a completed, finished magnitude — and found a hierarchy of infinities, some strictly „larger” than others. Here the limit no longer helps: there is no sequence of steps that *tends toward* the set of real numbers starting from the set of natural numbers — the difference between them is not a threshold crossed by approach, but a change of order for which the limit, as a tool, was never designed. The limit tames the infinite that lets itself be gathered; there is an infinite that does not.
The same pair returns in philosophy, but now the limit belongs to the mind itself, and infinity is what the mind cannot contain without fracturing. Socrates named the mind’s contour „knowing that one does not know” — limit as the shape of honesty. Kant went further: the mind knows the phenomenon but cannot reach the thing-in-itself, and when that same mind tries to think the infinity of the world — whether space and time have or lack a boundary — it arrives at an antinomy: it can prove, with equal logical force, both that the world is finite and that it is infinite. The mind’s limit and the infinity of its object collide here directly — not as in mathematics, where the limit gathers infinity into a result, but as a rupture reason suffers within itself the moment it tries to think infinity as an object.
Levinas overturned the pair. For him, infinity is no longer what the mind cannot encompass out of weakness, but what exceeds every limit of the concept precisely through excess of meaning, not of magnitude: the face of the other person is a small infinity, present in the midst of the finite, which breaks the limit of any totalization — not because it is too large for the mind, as with Kant, but because it can never be fully grasped without being betrayed. Here the mind’s limit is no longer a failure but exactly the place where infinity makes itself known, asking to be received, not calculated — a structure patristic thought will take up again below, on different ground.
Contemporary psychology lives out the same pair, but inverted: here it is not the limit struggling against a desired infinity, but infinity that pretends to be something it is not — a false infinite, dressed in the clothes of endless multiplication, put in place of a limit the person does not wish to accept. Hegel called „bad infinity” (schlechte Unendlichkeit) that repetition which stays on the same plane — a sequence of „and yet more,” „and yet more,” with no change of nature, only a quantity growing without limit without ever becoming something else. This is the exact structure of endless scrolling on a screen: every threshold of saturation reached is immediately pushed further, not through deepening but through repeating the same gesture — a horizontal infinity that approaches nothing in particular, only keeps extending, exactly like the asymptote from the first section, but here lived as craving, not as theorem.
Augustine had already named the phenomenon, many centuries before Hegel: the restless heart seeking rest in finite things, however many are gathered — and therefore remaining unsatisfied, because the sum of finite quantities, however long, never produces the quality of a true infinite, just as an infinite sum of ever-smaller terms can converge, mathematically, to a finite result, yet no sequence of finite pleasures converges toward the infinite the heart demands. Psychology can measure very precisely the physiological limit of saturation and the pattern by which desire always circumvents it, asking for more — but it cannot say why this circumvention always takes the shape of an infinite, false or true; it can describe the mechanism, not distinguish the imitation from the original.
Patristic
Here the limit and the infinite, held apart or in conflict in all three points above, find the only place where they no longer fight but fulfill each other. St. Anthony the Great crossed the limit of his fallen nature not by prolonging his own efforts, but through an intervention from outside the system: grace, which does not continue nature’s trajectory but meets it from a plane nature could not, by itself, reach. The limit here is not overcome by the infinite addition of one’s own steps, as in calculus, but lifted all at once, by gift.
And St. Gregory of Nyssa shows what happens to infinity once the limit has thus been crossed: because God is infinite, the soul that enters communion with Him never reaches an end to its ascent — but, unlike all the infinities above, this never-reaching-an-end is neither a logical paradox (as with Kant), nor a bottomless craving (as with Hegel’s bad infinity), nor even an ever-larger magnitude to be added up (as with Cantor), but the very structure of blessedness itself: epektasis, the ceaseless stretching-forth toward what cannot be exhausted, because to reach an end would mean God has a boundary — and would therefore no longer be God. The limit of each step the soul reaches does not stop the ascent but becomes the starting point for the next — exactly as the mathematical limit gathers the infinity of steps into a single point, except that here the point itself is infinite, and therefore closes nothing, but keeps opening further.
So, at last, the four threads meet: mathematics shows that one infinity can be gathered by a limit, but another refuses to be gathered; philosophy shows that the mind fractures before an infinity it tries to seize by force, but can also be met by an infinity that asks only to be received; psychology shows a craving that takes the shape of infinity without having its aim; and patristic thought shows the one infinity that lets itself be neither gathered by a calculation, nor grasped by a concept, nor sated by a craving — but only loved, step by step, without end, by a person becoming ever more gathered, not more scattered, at every threshold which the Infinite One comes to cross from His side toward hers.
