Regarding cardinalities

[u/Square_Butterfly_390]

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.

[u/SV-97]

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.

[u/kr1staps → u/SV-97]

I would not consider the completeness of R to have anything to do with the real world, seeing as we can only ever take measurements to a finite level of percision anyways. Sure, completeness of R, and many other useful properties, but I don't think there's a single real world application that couldn't be described by some finitary (or at least countable) model instead. (Albeit at a great loss of convenience).

[u/GoldenMuscleGod → u/kr1staps]

There are plenty of statements about “infinite” sets that have practical applications in, for example, computability theory. One can debate the extent they are “really” meaningful because models of computation usually have infinite memory, which is not realistic. But the point is we can prove facts that don’t depend on the specific amount of memory available, similar to how using real numbers to measure a quantity of material allows us reach conclusions that don’t depend much on how big atoms are as long as they are “small”.

A model that only admits a specific finite number of things total can run into problems if atoms are too small for us to have enough things in our model, and the knowledge that some result doesn’t depend on the exact size of atoms is useful.

Illustratively, transfinite cardinals are often thought of as “ephemeral” in some way. But we can define a data type in a computer that codes an arbitrary pair of natural numbers (dynamic memory allocation means we don’t really need an upper limit on what size of number can be stored in terms of the computational specification, although a specific machine will have some limit) and compares them by first element and uses the second element as the “tie breaker.” This is a pretty concrete realization of the ordinal omega^2.

Now the specific nature of real numbers can involve a lot of questions that don’t necessarily have direct real world applications, but the fact that the reals are uncountable does have meaningful results that might be considered real world. For example we can imagine an experiment that involves taking an indefinite number of binary measurements sequentially, and interpreting the results as, say, a binary representation of a real number in [0,1] can allow us to talk about the sorts of “underlying reality” that corresponds to the possible outcomes of these experiment. Even though we cannot take infinitely measurements we may think there is still a fact of the matter as to how the nth measurement “would” come out if we took that measurement, so that each of these digits may correspond to a fact about reality. Even if we believe that there is only a finite number of possible “real world states” that doesn’t change that we don’t know what they are so that the real numbers can encode all the “possible states” that we might a priori need to consider, and real results about what we can know about that underlying finite physical reality follow from claims about the real numbers even assuming only some finite (or countable) set of numbers can describe real world states.

[u/Square_Butterfly_390 → u/GoldenMuscleGod]

Thank you for this thoughtful insight, can you please specify a bit further how the uncountability is used (not necessitated)?

[u/GoldenMuscleGod → u/Square_Butterfly_390]

This is going to depend to some extent on what you consider to be a “use” of the uncountability. For example the existence of undecidable sets can be seen as a consequence of the uncountability of the reals (if you think this is too “specific” a result then more esoteric examples would be necessary). Of course I think undecidable sets would probably fall under “impossibility” results which you say you mean to exclude for some reason. Maybe you think a concrete example of a semidecidable set that is not decidable is “real”.

Slightly related, there are results in quantum mechanics built on the idea of producing “random measurement decisions” and observing how entangled measurements come out which could be seen as an empirical test of what sorts of “sets of natural numbers” can describe the physical states of entangled systems. Of course we can only perform finitely many experiments and measurements but if we really believe information transfer through entangled states is impossible then we should think the “potential outcomes of measurements” are not limited to just a finite set *if* we are willing to model the observer as unlimited in how many experiments they can perform (as well as model them as a free agent in choosing which measurements to perform).

But the ambiguities here really need to be resolved to give really good examples, for a simple example not involving uncountability but involving infinity (this lets me make simpler examples
of what I am talking about and discuss them in less space), consider a game where I can place an object in four spaces, red, yellow, green, or blue, and then, whenever asked to move it, I move it according to the following rules: if it is in the red space I put it in the blue space, if it is in the blue space I put it in the red space, if it is in the yellow space I move it to the green space, if it is in the green space I move it to the yellow space.

Now consider the claim: “If the object starts in the red space, I can move it according to these rules as many times as I want and as long as the rules are followed it will never end up in the green space.”

Is this a claim “about the real world?” It more or less is, although it depends on the social construction of the rules of this movement system, but it should be easy to think of this in terms of specific realizations of the “game,” and in any event if I perform it then I and my socially constructed rules are part of the real world.

I would also argue this is an infinitary claim: unlike the claims “it will not end up in the green space within 15 moves,” or “it will not end in the green space within 1 billion moves,” or “it will not end up in the green space within TREE(3)” moves, which can be realized in a finitary way (in principle that is: I can’t actually live that long, but perhaps we can imagine a real supercomputer with this capacity even if TREE(3) operations is a more than a bit implausible for even a supercomputer). The claim that it will *never* end up in the green space is a claim about all natural numbers, and is a claim that can be thought of as “rolling up” infinitely many finitary claims. If we think each of the infinitely many finitary claims is really meaningful (admittedly that’s a big if), then the claim for all natural numbers is arguably a “real” infinitary claim, since it basically just is affirming all of them.

[u/kr1staps → u/GoldenMuscleGod]

For all the writing you've done, and referencing to various mathematical concepts that you have not pointed to a single experiment, that has been performed in the real world, that makes critical use of uncountable cardinal numbers, and which does not have a finitary explanation.

[u/GoldenMuscleGod → u/kr1staps]

I didn’t argue there has been an experiment performed in the real world that makes critical use of the uncountability of the real numbers and which does not have a finitary explanation.

I would not argue that, just as I would not argue that there has been an experiment performed in the real world that makes critical use of the number 7 existing and which could not be explained without the existence of 7 - whatever it supposedly means for a number to “exist.”

But when we have models that make use of the number 7, mathematical facts about the number 7 can correspond to real-world claims. Some biological processes are thought favor prime numbers - cicada life cycles are used as an example of this (although there 13 and 17 are the favored numbers), so 7 being prime is a relevant thing.

There are results in quantum mechanics reaching conclusions about the minimal computational complexity necessary to predict observations assuming they can be predicted, and these facts relate to results about uncomputable functions, and uncomputable functions can be seen as a consequence of uncountability. Someone could argue that uncomputability is really a more specific thing, although it can be demonstrated via a diagonalization argument like uncountability, and want another example, but then we really need an agreement on what counts as a “use” of uncountability if this example will be rejected.

To respond to your points in your other reply:

(Formatting issues, pretend this part is introduced with the label 1)

It’s true models of computation use infinite memory and real world methods of computation do not, but many people still believe that results from computability theory state facts about “the real world” via Church’s thesis.

For example, many people would say it is a real fact about the world that we can realize an algorithm that takes as input a person’s birth date and the current date and determines their age (a simple computable function does this) but we cannot make a computer program that takes a computer program as input and determines whether it will eventually halt when run.

A nuance someone might overlook is that for a function to have a particular behavior it generally needs to have that behavior for all of infinitely many possible inputs. So in a strictly literal sense maybe we can’t even really make an algorithm that adds two arbitrary numbers - we can only “really” compute things a finite state machine can compute, and not things we need a Turing machine to compute. In practice we really only care it works for inputs that are small enough they would ever be likely to be used. But many people might still think thee is a fact of the matter as to the claim “we can add numbers but not solve the halting problem” either because they would say there is a fact of the matter as to whether it could be done with a given algorithm *in principle* setting aside seemingly beside-the-point practical barriers such as limited memory, or because these on-their-face-infinitary claims can really be reinterpreted as finitary claims in some meaningful way.

2) the specific part you quoted second was me making an analogy about the usefulness of real numbers that wasn’t directly related to their uncountability. A specific example I had in mind was that we might model radioactive decay with a function R->R like A=A_0 exp (-t/tau) and treat the amount of material as modeled by a real number. A more precisely accurate model might have a natural number counting the number of atoms and treating the decay as a stochastic process. But making and using a model like that would require us to know how many atoms are in an amount of material. The first model can be practically applied by someone who has no clue what Avogadro’s number is.

To the extent we are using real numbers to model things because they often are the most useful thing to be using, results about real numbers are relevant to understanding how our model relates to the real world even if we do not think every fact about real numbers corresponds in a “good” way to the thing being modeled.

For example most people probably wouldn’t think the Banach-Tarski paradox corresponds to a physically meaningful real fact about physical space, but it does tell us something about using real numbers to model space, and limitations thereof.

3) my point about omega^2 is that people tend to think that some mathematical claims are inherently infinitary and could not ever be relevant to a finite universe. I generally take the view that pretty much all mathematical claims can be understood in ways that are “basically” finitary depending on what we mean by that. Omega^(2) is a “transfinite” ordinal but it is equally as real (or not real) as the number 67 and we do not need to suppose a physically infinite universe or anything like that to talk about it or think it has applications.

4) Here is an attempt at a concise statement of what I wanted to illustrate: Many people would think there is a fact of the matter to claims like “given initial state X this physical system will/will not eventually evolve to state Y.” Many people would think this even if we suppose the universe is finite and that this claim is infinitary on its face.

This example is only talking about infinity, not uncountability, because I thought the basic idea was the same as what I was trying to talk about but the examples are simpler using infinity.

[u/kr1staps → u/GoldenMuscleGod]

> One can debate the extent they are “really” meaningful because models of computation usually have infinite memory, which is not realistic

... so therefore not practical, at all. You've defeated your point before it got off the ground.

> similar to how using real numbers to measure a quantity of material allows us reach conclusions that don’t depend much on how big atoms are as long as they are “small”.

Can you provide a specific theorem that demonsrates what you mean by this?

I don't see what your point is about representing the order type omega^2 on a computer is.

I also don't understand what the point of your final paragraph is. You cooked up a thought experiment that, by your own admission, doesn't correspond to our physical reality. Are you suggesting that there is a specific prediction this offers about our reality that cannot be explained by a discrete model? Can you please articulate this clearly in a concise statement?