Infinity used to be the place where mathematics ran out of road, a hazy horizon rather than a mapped landscape. Now a pair of new, exotic infinities is forcing researchers to redraw that map and confront the possibility that some of the field’s most trusted tools behave very differently at the outer limits of size. What began as a technical investigation into enormous numbers has turned into a challenge to the way I, and many others, were taught to think about the infinite.
Instead of a single boundless “beyond”, mathematicians are uncovering a layered hierarchy of infinities with their own internal rules, personalities and even paradoxes. The latest work suggests that two freshly defined kinds of infinity do not just extend that hierarchy, they twist it, hinting that the arithmetic of the infinite might need a serious rewrite.
How two strange infinities crashed the party
The new work centers on two colossal number systems that live so far beyond everyday counting that even familiar infinite sets, like the natural numbers, look tiny by comparison. Researchers from the Vienna University of Technology in Austria and the University of Barcelona set out to understand how these gigantic quantities behave when you try to add, multiply or compare them, and in the process they stumbled on two distinct “sizes” of infinity that had never been formally isolated before. Their analysis shows that these entities are not just bigger versions of what came before but occupy qualitatively new rungs in the hierarchy of the infinite, with algebraic behavior that breaks expectations drawn from more familiar infinite sets.
In technical terms, the team was probing the structure of extremely large ordinals and cardinals, the abstract numbers that measure order and size in set theory. By carefully tracking how these objects grow, they identified two thresholds where the usual patterns of comparison and combination suddenly shift, creating what amount to new phases of infinity. Reporting on the work describes how these thresholds behave like “rule breakers” inside the transfinite universe, suggesting that the landscape of infinite sizes is more fractured and surprising than the standard textbook picture of a smooth ladder of ever larger sets, an insight that is unpacked in detail in coverage of the new infinities.
From Cantor’s ladder to a maze of infinities
To grasp why these discoveries matter, I have to go back to the late nineteenth century, when Georg Cantor first showed that infinity comes in different sizes. Cantor proved that the set of real numbers is strictly larger than the set of natural numbers, even though both are infinite, and he built an ascending tower of infinite cardinals to capture that idea. Following Cantor, generations of mathematicians learned to think of infinity as a well ordered ladder: start with the countable, climb to the uncountable, then keep going to ever more enormous but structurally similar levels, each one a bigger version of the last.
That picture was already mind bending, but it still suggested a kind of regularity, as if the infinite world could be wallpapered with blueprints of itself. Recent reporting on the new work stresses that, following Cantor, mathematicians soon realised they could construct ever larger infinite sets, yet the new constructions show that some of these levels behave in ways that do not mirror the earlier ones at all. The latest research on a mind blowing new kind of infinity argues that Cantor’s ladder is less like a straight staircase and more like a branching maze, with unexpected turns where the rules of comparison and arithmetic change, a shift that is highlighted in analyses of how, following Cantor, the theory evolved.
Meet the “exacting” and “ultra-exacting” infinities
At the heart of the new story are two particularly extreme kinds of infinite size that some commentators have nicknamed the “exacting” and “ultra-exacting” infinities. These are not marketing labels but attempts to capture how unforgiving these numbers are about structure: to even define them, you need sets so large that they encode detailed blueprints of themselves, and then you demand that this self reference holds in a brutally precise way. The first threshold, the exacting infinity, marks the point where an infinite set can be reconstructed from a compressed description that still somehow contains every detail of its own layout.
The second threshold, the ultra-exacting infinity, pushes that idea further, to sizes where the set can be “wallpapered” with copies of its own blueprint in a way that defies the usual tricks mathematicians use to tame large cardinals. A lively discussion among specialists and philosophically minded readers has framed these as “Two New Infinities Discovered”, with one contributor describing numbers “so large that one could wallpaper the universe with blueprints of itself”, a phrase that appears in a thread that also notes the figure 39 in the context of the debate. That conversation, while informal, reflects how radically different these new infinities feel compared with the already vast sizes that came before.
Why these infinities “break the rules”
What makes the exacting and ultra-exacting levels so disruptive is not just their size but the way they scramble familiar relationships between infinite sets. In standard set theory, once you climb high enough up the ladder of cardinals, many structural patterns repeat, and large cardinals often satisfy elegant reflection principles that let mathematicians transfer properties from the whole universe of sets down to smaller fragments. At the new thresholds, some of those reflection principles fail, and operations that usually preserve order or size begin to behave in unexpected ways, which is why researchers describe these infinities as breaking the rules of math rather than simply extending them.
One striking feature is how these sizes interact with the idea that there is an “infinity of infinities”. Earlier work already showed that there is no largest infinite cardinal, so in that sense there is an endless supply of ever bigger infinities. The new results refine that picture by carving out two specific points in this endless ascent where the combinatorial behavior of sets changes character. Reporting on how there is an infinity of infinities, and how researchers just found two new ones that challenge mathematical order, emphasizes that these thresholds are not arbitrary curiosities but natural fault lines in the transfinite landscape, places where the usual hierarchy of sizes becomes more intricate, as explored in coverage of how two new strange infinities challenge that order.
Inside the Vienna–Barcelona collaboration
Behind these abstract ideas is a concrete collaboration between specific institutions and people. The core results come from mathematicians at the Vienna University of Technology in Austria and the University of Barcelona, who combined expertise in set theory, logic and the theory of large cardinals. Their work, which has circulated as a detailed preprint, uses sophisticated tools from modern set theory to isolate the exacting and ultra-exacting thresholds and to prove that they really do behave differently from previously known infinite sizes, rather than being disguised versions of familiar cardinals.
Accounts of the project describe how the team’s analysis grew out of earlier investigations into enormous ordinals and the combinatorics of the infinite, and how the collaboration drew on a broader community of researchers whose citation records can be traced through profiles such as a leading set theorist’s scholar citations. One widely shared summary notes that, recently, mathematicians from the Vienna University of Technology in Austria and the University of Barcelona presented their findings on a peer reviewed preprint server, highlighting how the work moved from specialist seminars into a broader mathematical conversation, a trajectory captured in reporting on how recently, mathematicians from the Vienna University of Technology in Austria and the University of Barcelona shared their results.
Infinity contains everything, including new kinds of infinity
One of the most counterintuitive lessons of modern set theory is that infinity is not a single destination but a universe in its own right, with room for endlessly many different infinite sizes. Commentators on the new work have leaned into that idea, pointing out that infinity contains everything, including multiple kinds of infinity, and that the discovery of two more is less an exception than a continuation of a long running pattern. From this perspective, the exacting and ultra-exacting levels are part of an infinite set of infinities, each one defined by ever stricter structural demands on how sets can encode information about themselves.
That framing helps explain why the new results feel both shocking and oddly inevitable. If infinity already contains an infinite set of infinities, then it is natural to expect that some of those levels will behave in ways that surprise us, just as exotic phases of matter emerge when physicists push materials to extremes of temperature or pressure. Coverage that describes how “Infinity contains everything, including multiple kinds of infinity” and notes that there is “an infinite set of infinities” uses the new discoveries as a vivid example of that principle in action, a theme that is developed in reporting on how Infinity contains everything, including these new sizes.
What this means for the foundations of mathematics
For working mathematicians, the arrival of new infinite sizes is not just a curiosity, it is a potential shift in the foundations on which many theories rest. Large cardinals already play a central role in understanding the consistency of set theory, the structure of the real line and the limits of what can be proved within standard axiomatic systems. The exacting and ultra-exacting infinities raise fresh questions about which axioms we should accept, how far the hierarchy of sets extends and whether some long standing conjectures about the continuum and related objects might behave differently once these new levels are taken into account.
Some analysts have suggested that the new work could eventually feed into debates about whether the usual axioms of set theory are sufficient, or whether we should adopt stronger principles that assert the existence of ever larger infinities as a way to settle undecidable questions. Reporting that asks how two new strange infinities challenge mathematical order notes that these sizes might lurk at the core of deep open problems, hinting that they could become part of the standard toolkit for future generations of logicians. One detailed explainer on how two new strange infinities challenge mathematical order points out that the discoveries invite mathematicians to rethink what counts as a “natural” axiom, a theme that surfaces in coverage of how two new strange infinities might lurk at the core of set theory.
Why non mathematicians should care
It is tempting to treat all this as an esoteric game played with symbols on a blackboard, far removed from anything in daily life. Yet the way we understand infinity filters into everything from computer science to physics to philosophy. Concepts like countability, uncountability and the size of the continuum underpin the theory of computation, the design of algorithms and the analysis of what problems can be solved in principle. When new infinities appear that disrupt the usual hierarchy, they can change how we think about the limits of calculation, the structure of possible universes and even the meaning of “all possible outcomes” in probabilistic models.
There is also a cultural dimension. Infinity has long been a playground for paradoxes and thought experiments, from Hilbert’s hotel to Zeno’s arrow, and the discovery of new infinite sizes gives philosophers and educators fresh material to work with. A widely shared explainer on how two new types of infinity challenge mathematical order uses an Illustration by Midjourney to convey the strangeness of these concepts visually, and frames the story with the simple question “What is this? Is there an …”, inviting readers to confront just how alien these new sizes are compared with the numbers they use every day. That blend of rigorous theory and imaginative presentation is captured in coverage of How two new strange infinities are being communicated to a broader audience.
The debate: do infinities really come in different sizes?
Not everyone is comfortable with the idea that infinity can be sliced into distinct magnitudes at all. While Cantor’s work is widely accepted within mainstream mathematics, there is a persistent undercurrent of philosophical resistance that questions whether talk of larger and smaller infinities reflects genuine mathematical reality or just artifacts of a particular formalism. The arrival of two more exotic infinite sizes has reignited some of those debates, with critics arguing that piling new cardinals on top of old ones risks drifting away from any intuitive notion of quantity.
One online discussion titled “Proof that infinity does not come in different sizes” captures this tension vividly, with participants challenging the standard hierarchy and others defending it by pointing to the concrete combinatorial properties that distinguish one infinite set from another. In that thread, the phrase “Two New Infinities Discovered. Here’s an introduction to the exacting infinity and the ultra-exacting i…” is used both as a hook and as a provocation, prompting questions about whether these constructions reveal something deep about mathematics or simply stretch the language of size beyond its breaking point. The presence of detailed arguments and the specific reference to “Two New Infinities Discovered, Here’s an introduction” in that forum underscores how the new work is already shaping philosophical as well as technical conversations, a dynamic reflected in the discussion linked through Two New Infinities Discovered, Here.
Where the search for larger infinities goes next
For the researchers involved, the discovery of the exacting and ultra-exacting levels is less an endpoint than a starting signal. Once two such thresholds have been identified, it is natural to ask whether there are more, perhaps an entire sequence of ever more demanding self encoding conditions that carve the infinite landscape into finer and finer strata. Each new level would bring its own algebraic quirks and foundational questions, and mapping them could become a major program in set theory over the coming years, much as the study of measurable, supercompact and huge cardinals shaped the field in the late twentieth century.
There is also the question of how these ideas might connect to other parts of mathematics and theoretical computer science. Some large cardinal axioms have surprising links to determinacy in infinite games, regularity properties of sets of real numbers and the behavior of algorithms on transfinite inputs. If the exacting and ultra-exacting infinities turn out to have similar connections, they could influence areas as diverse as descriptive set theory, model theory and the study of proof complexity. Popular accounts that describe how mathematicians casually discovered two new infinities hint at this broader horizon, suggesting that what looks like a rarefied advance in pure logic could eventually ripple outward into other disciplines, a possibility that is sketched in explainers on Mathematicians Casually Discovered New Infinities.
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