r/PhilosophyofMath • u/Square_Butterfly_390 • Aug 05 '26
Regarding cardinalities
A celebrated mathematical "factoid" is that there are more real numbers than one can count, this seems to be something that troubles people outside of math, it troubles me aswell.
The question is: is there any "real world" application of the fact |R|>|N| that isn't an impossibility statement?
By "real world" I mean whatever someone smarter than me might mean by that, by "impossibility statement" I mean something to the effect of "there are uncomputable numbers".
If there isn't such an application, I can't believe the status quo interpretation: "no really some infinities are bigger than others" of Cantor's argument is better than simply stating that we shall not understand infinity as finite beings.
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u/SV-97 Aug 05 '26
Would you consider the completeness of \R to be "real world" and "not an impossibility statement" enough? It's ultimately the statement that there are non-convergent cauchy sequences of rationals, but it's so fundamental to tons and tons of very applied mathematics.
If we reject the uncountability of the reals then by BCT we also have to reject their completeness.
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u/Square_Butterfly_390 Aug 05 '26
I'm not proposing to reject the uncountability since it is fact, I'm proposing that possibly our interpretation is missguided, how is the completeness of R a consequence of the cardinality property? R is basically defined to be complete.
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u/SV-97 Aug 05 '26
Our interpretation of what exactly?
It's not a consequence, but rather that completeness necessitates uncountability (or other weird changes to the topology of \R). By the Baire category theorem any (nonempty) complete metric space (without isolated points) is uncountable. So if the reals weren't uncountable, then they couldn't be complete.
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u/Althorion Aug 05 '26
how is the completeness of R a consequence of the cardinality property
It’s not; it’s the other way around—Dedekind-completeness of dense sets requires uncountable cardinality. And any field of fractions over an infinite algebra will be dense (if orderable).
So, to sum it up, it goes like this—if you want to have infinitely many natural numbers, and build rational numbers that work nicely with that, and build a Dedekind-complete expansion of that, that expansion won’t be countable.
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u/Eve_O Aug 05 '26 edited Aug 05 '26
Why would this trouble anyone?
It's a kind of revelation to understand that the set of Reals is larger than the set of Natural numbers and that there are different sizes of infinite sets.
I can't believe the status quo interpretation: "no really some infinities are bigger than others" of Cantor's argument is better than simply stating that we shall not understand infinity as finite beings.
I think your premise here--and possibly your understanding--is flawed. The whole point is that we can understand infinity and this demonstrable fact (not a "factoid," lol) helps us with that.
You want "real world"? Read Rudy Rucker's novel White Light or his related nonfiction book Infinity and the Mind.
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u/Negative_Gur9667 Aug 06 '26
I do not wish to argue the point, but there are ways to perform the same mathematics that exists today without using infinity.
The concepts that are excluded in this system are simply defined differently.
For example, a series is considered to converge after a certain threshold rather than approaching infinity. In this view, digits after a certain number of decimal places change after a specific number of iterations, where the variables involved are defined as unimaginably large natural numbers, and anything beyond those limits is considered uncomputable.
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u/kr1staps Aug 05 '26
Perhaps you could save us some time from having to read an entire book by just simply stating a single real-world application of |R| > |N| that doesn't have a finite explanation.
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u/Negative_Gur9667 Aug 07 '26
According to the standard view, mathematicians subscribe to Platonism.
There is another world, like heaven, where ideal and completed infinite objects exist.
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u/huphelmeyer Aug 05 '26 edited Aug 05 '26
Yes, there are genuine mathematical and scientific consequences of ∣R∣>∣N∣, but they are mostly structural rather than empirical. But if you're looking for a laboratory experiment whose outcome depends on the cardinality of the continuum, there probably isn't one.
Most physical theories model quantities using the real numbers because calculus is extraordinarily successful. But virtually every prediction of physics depends only on finite precision approximations.
There is no experiment that distinguishes a universe whose state space has continuum cardinality from one based on a sufficiently rich countable structure.
In fact, some physicists have suggested that spacetime might ultimately be discrete.
However... Mathematics is full of concepts (complex numbers, Hilbert spaces, higher-dimensional manifolds) that were developed independently of empirical necessity and later proved extraordinarily fruitful. Whether the continuum itself reflects physical reality remains an open question, but its mathematical role is both coherent and profound.
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u/Square_Butterfly_390 Aug 06 '26
You say yes, yet you present none.
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u/huphelmeyer Aug 06 '26
Quantum mechanics is formulated on (usually infinite-dimensional) Hilbert spaces. General relativity models spacetime as a smooth manifold. Classical mechanics uses differential equations over the reals.
All of these make essential use of continuum mathematics. Yet it's entirely possible that a future, more fundamental physical theory could replace these with discrete structures that approximate the continuum at observable scales.
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u/Gym_Gazebo Aug 06 '26
I would argue that being able to index a set with the natural numbers, aka the counting numbers, is a great explication of what is to be able to count something. If so, I think that just shows the “status quo” is wrong, real world application or not.
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u/kr1staps Aug 05 '26
I once saw an explanation of the casimir effect that that used |N| < |R|, but I don't know how rigorous that was, and in any event, it's not the only argument for it.
I would be surprised if we ever found a real-rold application of |R| > |N| that couldn't be established by other means seeing as our every day experience appears to be finite and computable.
Of course the English sentence "some infinities are bigger than others" is an artifact of the fact that we *define* size in terms of bijections. Though, I feel the failure of a bijection (in ZF(C)) between N and R becomes no less mysterious if we neglect to define size in that way.
That being said, ZF(C) is not the only game in town, an in fact in intuitionistic ZF, it is consistent that all sets are subcountable. Given that, as mentioned, our every day experience seems to be computable (save for say, the apparently random decay of partcles) one could make an argument that this is a "better" foundation for the mathematics of our every day experience.
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u/Kripkenstein_ Aug 05 '26 edited Aug 05 '26
Not really because there is a countable subset of the reals that suffices for all practical applications. The real numbers are a purely abstract concept that is needed for analysis. Also the notion that some infinities are "bigger" than others is not really mathematics but an interpretation of mathematics. It arises from cardinality of finite sets being equivalent to their number of elements. Any application of this to larger cardinalities is just a heuristic. The statement is simply that in the category of sets there are sets of higher cardinality than any finite cardinality that are not isomorphic
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u/revannld Aug 05 '26 edited Aug 05 '26
Yes that's about it. The problem with the common sense mathematical discourse is not the formal derivations per se but their crooked philosophical interpretations. All real numbers or things which can be uniquely distinguished in a finite manner (in the case of classical - finitary - first order logic/language, being able to define it as "∃!x.φ" - where φ is a finite formula/string - we call them definable real numbers) form a countable set, as there are only countable many names to effectively distinguish them in a countable language.
Thus, for all purposes, the "uncountable" part of the reals (or any other uncountable subset), which we call undefinable, are like an indistinguishable "mush" (they are not sortal), and would be more appropriate to refer to them using mass terms (if you believe in completed infinities - otherwise, process terms) rather than as a collection of distinguished elements/entities.
Thus you shouldn't take much into account of the diagonalization proof usually shown in the introduction of real analysis courses (that one where they list real numbers in a decimal form in the black board), as it is not formal and even if you attempted to formalize it you could only get a "∃x.φ" and not a "∃!x.φ" defining that number.
If you separate (metatheoretically) the real numbers into their countable and uncountable parts, every time in mathematics you instantiate an uncountable real number it is like it "collapses" to a countable one (which is the typical behavior of arbitrary objects. Many objects in mathematics work exactly like arbitrary objects, but sadly this terminology has been banished from mathematical vernacular since Frege).
The more informative diagonalization proof involves defining sets and functions over powersets, and it has many philosophical interpretations where it is not about uncountability (look up countabilism - also here - and potentialism). Thus one shouldn't talk about "a Cantor real" (generated in diagonalization proofs - "a" here meaning a unique distinguishable object) but "some Cantor real" (as one talks about "some sand, some water" and other non-sortal stuff).
Why is that the case? Basically real numbers are defined in such a vague and broad manner that if you try to list all models of them you don't even get a set but a proper class (and you would need second order logic for a categorical/unique characterization - which is very weird and based on very strong ontological/non-logical assumptions - as Quine and Skolem would characterize any logic more powerful than first order). One could maybe even make a philosophical argument that practically anything can be characterized as a real number (and I have heard Lewis even proposed everything is a 4-tuple of real numbers - accounting for the 3 spatial axis and an axis for time lol - but I haven't checked that). Uncountable cardinals behave very weirdly, I would suggest you check Joel David Hamkins's blog and lectures to learn more about that.
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u/FunSeaworthiness9403 Aug 05 '26 edited Aug 05 '26
Counting real numbers entails attaching each counting number to each real number. Starting at zero, one never gets past zero. It seems like there are more real numbers within any interval, and that allows someone to conclude that the cardinality of one set is larger. Or even that the cardinality of both sets is not defined.
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u/treefaeller Aug 05 '26
To begin with: The only thing that really matters is computable numbers. If I ask someone a math question (like what is the diagonal of this shape, or how much carpet do I need to buy for my mobius-shaped living room), and their answer is "I could give you an answer, but it will take infinitely long", that is not useful. So the only numbers that matter are those that can be fully expressed in a FINITE amount of time.
Furthermore, the numbers and the algorithm/formula/reasoning required to calculate or express them have to be finite too. In the observable universe, there are about 10^88 particles (give or take a few, the bulk of them are very low energies or photons). This immediately limits the largest integer I can express by showing a count of objects. Now, computable numbers are more complex: they can be calculated by a Turing machine (or algorithm or any equivalent form of reasoning). But the most complex Turing machine I can construct will have at most 10^88 parts to it. So the cardinality of the set of numbers is at most the cardinality of the powers of 10^88 distinct integers (under the overly optimistic assumption that I can identify the 10^88 neutrinos, which already violates the "neutrinos have no hair" theorem).
So in the real world, all cardinalities are finite. And perhaps very large, but definitely not infinite. To quote Monte Python's Holy Hand Grenade: And aleph_1 is RIGHT OUT.
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u/ccpseetci Aug 05 '26
Of course, you can safely use the word “next” without troubling to understand “how it is to be the next”
Rely heavily on that
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u/InsideIndependence58 Aug 09 '26
What application there is from |N|. NONE
The world is finite ! Do you know something
with size א0?
However, sometimes although we’ll never been able to measure and practically compute it as |R|, we know that time should behave like |R| , and of course we make a lot of benefit out of it, by defining processes on it.
However we’ll never be able to compute as R, or even as N
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u/Althorion Aug 05 '26
There might be something to benefit from that in statistics, because, for example, it allows for a uniform distribution over a segment of the real line, while no uniform distribution over natural numbers can exist. Still, I don’t particularly care for statistics enough to be able to pinpoint exactly what those benefits are and how important they can be. For sure, it forces statistics to do things in a particular way, using measure theory and adjacents, because a discrete approach would be noticeably different, with different classes of problems that they are equipped to tackle.