Peano arithmetic is enough, because Peano arithmetic encodes computation
(math.stackexchange.com)239 points by btilly 3 days ago
239 points by btilly 3 days ago
After putting on my boots and wading through all of that I think that you have one edit to make.
In the "Why Lisp?" section there is a bit of basic logic defined. The first of those functions appears to have unbalanced parentheses.
>(defun not (x) > (if x > false > true)
I have a compulsion that I can't control when someone starts using parentheses. I have to scan to see who does what with who.
You later say in the same section
>But it is really easy to program a computer to look for balanced parentheses
Okay. This is pretty funny. Thanks for the laugh. I realize that you weren't doing that but it still is funny that you pointed out being able to do it.
This later comment in the "Basic Number Theory" section was also funny.
>; After a while, you stop noticing that stack of closing parens.
I really enjoyed reading this post. Great job. Though it has been a long time since I did anything with Lisp I was able to walk through and get the gist again.
It was a great read. I enjoyed how you laid it all out. It reminded me of some of my upper level math coursework. Easy to follow if you take it step by step and stop to consider the implications. Some things become obvious to the most casual observer.
I found all the jokes funny as well. Thanks for the blog post. Extremely nice read. I love the approach.
For myself the issue goes back to my college mathematics courses, especially differential equations. I worked those homework problems by hand on a large format tablet, roughly 24" x 36", carefully laying them out step by step so that I could walk through them in the future and make sense of the solution process. Counting and matching parentheses was pretty critical since a missed parenthesis may not pop out at you like it would in a compiler error or by walking through code.
I automatically count and match even today, 40 years later.
Python block indentation is an example of nested structure that's at least easier to visually scan. You don't need to count opening/closing parens, just look down a column of text - assuming nobody mixed tabs and spaces. (but I wouldn't go as far as saying you don't need to scan it)
This is fascinating! I haven't read much past the intro yet, but I find the whole premise that you can prove all specific instances of Goodstein sequences terminate at 0 within PA, but not that all sequences terminate (it's a trivial result but still interesting).
I also find it super weird that Peano axioms are enough to encode computation. Again, this might be trivial if you think about it, but that's one self-referential layer more than I've thought about before.
One question for you btilly - oddly enough, I just recently decided to learn more Set Theory, and actually worked on an Intro to Set Theory textbook up to Goodstein sequences just last week. I'm a bit past that.
Do you have a good recommendation for a second, advanced Set Theory textbook? Also, any recommendation for a textbook that digs into Peano arithmetic? (My mini goal since learning the proof for Goodstein sequences is to work up to understanding the proof that Peano isn't strong enough to prove Goodstein's theorem, though I'll happily take other paths instead if they're strongly recommended.)
My apologies for having missed this request.
I don't have good suggestions for a good set theory textbook. Grad school was 30 years ago, and I didn't specialize in logic.
The best set theory book that I read was https://www.amazon.com/Naive-Theory-Undergraduate-Texts-Math.... But that one is aimed at people who want to go into math but do not wish to specialize in set theory, and not at people who want to actually learn set theory.
Thanks! I read through a bunch of that textbook when I first decided to learn more maths a few years ago, but I relatively quickly started dual-reading a different textbook ("Introduction to Set Theory" by Hrbacek and Jeck), which is a more in-depth, non-naive introduction. That's the book I'm continuing to work through now.
Thanks for this. In another strange internet coincidence, I was asking ChatGPT to break down the fundamentals of the Peano axioms just yesterday and now I see this. Thumbs up!
I suspect in the near future (if not already) ChatGPT data will be sold to data brokers and bough by Amazon such that writing a prompt will end up polluting Alexa product recommendations within a few minutes to hours.
> teaching kids set theory as a foundation for math
Very reminiscent of the New Math pedagogy of the 1960s. Built up arithmetic entirely from sets. Crashed and burned for various reasons but I always had a soft spot for it. It also served as my introduction to binary arithmetic and topology.
I've noticed this too. I will be researching a topic elsewhere and then it seems to pop up in HN. Am I just looking for patterns where there are none, or is there some trickery happening where HN tracks those activities and mixes in posts more relevant to my interests with the others?
It may also be that there's some common cause for (1) a topic being on HN's front page and (2) someone researching it elsewhere. E.g., maybe a couple of days ago there was an interesting social-media post related to it, and (1) one person saw it and wrote up something interesting enough to get on HN and (2) another person saw it, started digging around to find out more, and then read HN.
(I am not suggesting that that very specific sequence of events happened in this case. It's just an example of how "I was looking this up for other reasons and then I saw something about it on HN" could genuinely happen more often than by chance, without any need for weird tracking-and-advertising shenanigans.)
This proves that ChatGPT sells your data to HN which then decides which posts to put on the front page.
Boot sector Lisp bootstraps itself.
https://justine.lol/sectorlisp2/
Also, lots of Lisp from https://t3x.org implement numerals (and the rest of stuff) from cons cells and apply/eval:
'John McCarthy discovered an elegant self-defining way to compute the above steps, more commonly known as the metacircular evaluator. Alan Kay once described this code as the "Maxwell's equations of software". Here are those equations as implemented by SectorLISP:
ASSOC EVAL EVCON APPLY EVLIS PAIRLIS '
Ditto with some Forths.
Back to T3X, he author has Zenlisp where the meta-circular evaluation it's basically how to define eval/apply and how to they ared called between themselves in a recursive way.
Hey! This is fantastic and actually ties in some very high disparate parts of math. Basically, reorient & reformulate all of math/epistomology around discrete sampling the continuum. Invert our notions of Aleph/Beth/Betti numbers as some sort of triadic Grothendieck topoi that encode our human brain's sensory instruments that nucleate discrete samples of continuum of reality (ontology)
Then every modal logic becomes some mapping of 2^(N) to some set of statements. The only thing that matters is how predictive they are with some sort of objective function/metric/measure but you can always induce an "ultra metric" around notions of cognitive complexity classes i.e. your brain is finite and can compute finite thoughts/second. Thus for all cognition models that compute some meta-logic around some objective F, we can motivate that less complex models are "better". There comes the ultra measure to tie disparate logic systems. So I can take your Peano Axioms and induce a ternary logic (True, False, Maybe) or an indefinite-definite logic (True or something else entirely). I can even induce bayesian logics by doing power sets of T/F. So a 2x2 bayesian inference logic: (True Positive, True Negative, False Positive, False Negative)
Fun stuff!
Edit: The technical tldr that I left out is unification all math imho: algebraic topology + differential geometry + tropical geometry + algebraic analysis. D-modules and Microlocal Calculus from Kashiwara and the Yoneda lemma encode all of epistemology as relational: either between objects or the interaction between objects defined as collision less Planck hyper volumes.
basically encodes the particle-wave duality as discrete-continuum and all of epistemology is Grothendieck topoi + derived categories + functorial spaces between isometry of those dual spaces whether algebras/coalgebra (discrete modality) or homologies/cohomologies (continuous actions)
Edit 2: The thing that ties everything together is Noether's symmetry/conserved quantities which (my own wild ass hunch) are best encoded as "modular forms", arithmetic's final mystery. The continuous symmetry I think makes it easy to think about diffeomorphisms from different topoi by extracting homeomorphisms from gauge invariant symmetries (in the discrete case it's a lattice, but in the continuous we'd have to formalize some notion of liquid or fluid bases? I think Kashiwara's crystal bases has some utility there but this is so beyond my understanding )
> Invert our notions of Aleph/Beth/Betti numbers as some sort of triadic Grothendieck topoi that encode our human brain's sensory instruments that nucleate discrete samples of continuum of reality (ontology)
There’s probably ten+ years of math education encoded in this single sentence?
My apologies to ikrima for being critical, but I think anyone who thinks "aleph/beth/Betti numbers" is a coherent set of things to put together is just very confused.
Aleph and beth numbers are related things, in the field of set theory. (Two sequences[1] of infinite cardinal numbers. The alephs are all the infinite cardinals, if the axiom of choice holds. The beth numbers are the specific ones you get by repeatedly taking powersets. They're only all the cardinals if the "generalized continuum hypothesis" holds, a much stronger condition.)
[1] It's not clear that this is quite the right word, but no matter.
Betti numbers are something totally different. (If you have a topological space, you can compute a sequence[2] of numbers called Betti numbers that describe some of its features. (They are the ranks of its homology groups. The usual handwavy thing to say is that they describe how many d-dimensional "holes" the space has, for each d.)
[2] This time in exactly the usual sense.
It's not quite true that there is no connection between these things, because there are connections between any two things in pure mathematics and that's one of its delights. But so far as I can see the only connections are very indirect. (Aleph and beth numbers have to do with set theory. Betti numbers have to do with topology. There is a thing called topos theory that connects set theory and topology in interesting ways. But so far as I know this relationship doesn't produce any particular connection between infinite cardinals and the homology groups of topological spaces.)
I think ikrima's sentence is mathematically-flavoured word salad. (I think "Betti" comes after "beth" mostly because they sound similar.) You could probably take ten years to get familiar with all the individual ideas it alludes to, but having done so you wouldn't understand that sentence because there isn't anything there to understand.
BUT I am not myself a topos theorist, nor an expert in "our human brain's sensory instruments". Maybe there's more "there" there than it looks like to me and I'm just too stupid to understand. My guess would be not, but you doubtless already worked that out.
[EDITED to add:] On reflection, "word salad" is a bit much. E.g., it's reasonable to suggest that our senses are doing something like discrete sampling of a continuous world. (Or something like bandwidth-limited sampling, which is kinda only a Fourier transform away from being discrete.) But I continue to think the details look more like buzzword-slinging than like actual insight, and that "aleph/beth/Betti" thing really rings alarm bells.
you know what, I nerd sniped myself, here's a more fleshed out sketch of the [Discrete Continuum Bridge
https://github.com/ikrima/topos.noether/blob/master/discrete...
well, you're in luck because I'm about to make a fool of myself in trying to tease Terence Tao over at https://mathstodon.xyz/@mathemagical
Wish me luck!
You know what, since you put in all that work, here's my version using p-adic geometry to generalize the concept of time as a local relativistic "motive" (from category theory) notion of ordering (i.e. analogous to Grothendieck's generalization of functions as being point samples along curves of a basis of distributions to generalize notions of derivatives):
https://github.com/ikrima/topos.noether/blob/aeb55d403213089...
I was pretty amazed by how expressive Peano Arithmetic is when I first started using it. At first it just seems like a basic system, but once you realize that computation itself can be encoded inside PA, and that it can simulate many kinds of computation, things that felt complicated start to click.
Does anyone have recommendations for beginner-friendly resources that explain these encoding techniques?
Anything by https://en.wikipedia.org/wiki/Gregory_Chaitin, for example https://www.amazon.com/Exploring-Randomness-Discrete-Mathema....
The interesting part to me (I have a background in both math+programming) isn't so much the encoding of computation but that one can work around the independence of goodstein's theorem in this self-referential way. I think this implies that PA+"PA is omega-consistent" can prove goodstein's theorem, and perhaps can more generally do transfinite induction up to epsilon_0? EDIT: I think just PA+"PA is consistent" is enough?
Asker of the SO question here: I edited the question to link to a few other answers on this kind of thing. Essentially, "PA is consistent" is not enough, but a "uniform reflection principle" that says "if PA proves something, it's true" is enough. I'm not 100% certain that this principle is equivalent to omega-consistency, but if I'm reading this correctly, it should be:
https://en.wikipedia.org/wiki/%CE%A9-consistent_theory#Relat...
The Wikipedia article says T is omega-consistent if "T + RFN_T + the set of all true sentences is consistent", which should mean the same thing as "T + RFN_T is true".
Unfortunately not, and apparently no other purely universally quantified formulas will do either (so this is a more general thing, not specific to Con(PA)): https://math.stackexchange.com/questions/5003237/can-goodste...
On the first question, how do you encode omega-consistency as a formula of PA? Just curious, it's not at all obvious to me.
> On the first question, how do you encode omega-consistency as a formula of PA? Just curious, it's not at all obvious to me.
I was also wondering about that, but going by the Wikipedia definition, it doesn't seem too complicated: you say, "For all encoded propositions P(x): If for every x there exists an encoded proof of P(x), then there does not exist an encoded proof of 'there exists an x such that ¬P(x)'." That is, if you can prove a proposition for each integer in the metatheory, then quantifiers within the target theory must respect that.
Thanks, I see, so you pick some Gödel numbering and then quantifying over propositions is actually just quantifying over elements of the domain (and using your encodings of Sub(...) and Proves(...) and such). I see why that might have a chance of working, because it's now much higher up the arithmetic hierarchy.
I also like the recursion. In essence, you're making a meta-proof about what PA proves, and given that you trust PA, you also trust this meta-proof.
> I think just PA+"PA is consistent" is enough?
It's not clear to me how. I believe PA+"PA is consistent" would allow a model where Goodstein's theorem is true for the standard natural numbers, but that also contains some nonstandard integer N for which Goodstein's theorem is false. I think that's exactly the case that's ruled out by the stronger statement of ω-consistency.
I think so, the math exchange post mentions that the PA + transfinite induction works on epsilon_0 proves PA. It seems likely to me that PA + PA is consistent would be able to prove transfinite induction on epsilon_0.
This is very like Boyer-Moore theory[1], which builds up mathematics from the Peano axiom level.
Boyer and Moore wrote an automated theorem prover that goes with this theory. I have a working copy for GNU Common LISP at [2].
"It is perhaps easiest to think of our program much as one would think of a reasonably good mathematics student: given the axioms of Peano, he could hardly be expected to prove (much less discover) the prime factorization theorem. However, he could cope quite well if given the axioms of Peano and a list of theorems to prove (e.g., “prove that addition is commutative,” . . . “prove that multiplication distributes over addition,” . . . “prove that the result returned by the GCD function divides both of its arguments,” . . . “prove that if the products over two sequences of primes are identical, then the two sequences are permutations of one another”)." - Boyer and Moore
Your comment to JoJoModding in Math Stackexchange is incorrect.
"That's because there are nonstandard models of PA which contain infinite natural numbers. So PA may be able to prove that it produces a proof, but can't prove that the proof is of finite length. And therefore it might not be a valid proof."
If PA proves "PA proves X" then PA can prove X. This is because the key observation is not that there are nonstandard models, but rather that the standard natural numbers model PA.
Therefore if PA proves "PA proves X", then there is in fact a standard, finite natural number that corresponds to the encoded proof of "PA proves X". That finite natural number can be used then to construct a proof of X in PA.
I haven't had the time to go through your argument in more detail, but it's important to note (because the natural language version of what you've presented is ambiguous) that you haven't shown "PA proves 'Provable(forall n, G(n))'" in which case it would be the case that in fact "PA proves 'forall n, G(n)'", but rather "PA proves 'forall n, Provable(G(n))'".
My logic is very rusty at this point, but if someone could give me an argument that you cannot move the 'Provable' outside the 'forall', I would really appreciate that, without making reference to Goodstein sequences. In other words, that in general for propositions 'P' it is not true that proving "forall n, Provable(P(n))" implies you can prove "Provable(forall n, P(n))".
> If PA proves "PA proves X" then PA can prove X.
Not true.
From PA we can construct a function that can search all possible proofs that can be constructed in PA. In fact I outlined one way to do this at the end of my answer.
With this function, we can construct a function will-return that analyzes whether a given function, with a given input, will return. This is kind of like an attempted solution to the Halting Problem. We know that it doesn't always work. But we also know that it works a lot of the time.
From will-return we can create a function opposite-return that tries to return if a given function with a given input would not, and doesn't return if that function would. This construction is identical to the one in the standard proof of the Halting Problem.
Now we consider (opposite-return opposite-return opposite-return). (Actually you need a step to expand the argument into this recursive form. I've left that out, but that is identical to the one in the standard proof of the Halting problem.)
PA can prove the following:
- PA proves that if PA can prove that opposite-return returns, then it doesn't. - PA proves that if PA can prove that opposite-return doesn't return, then it does. - PA proves that if it can prove everything that it proves that it can prove, then PA must have a proof of one of the two previous statements. - Therefore PA proves that if it can prove everything that it proves that it can prove, then PA is inconsistent.
This is a form of Gödel's second incompleteness theorem.
And, therefore, there must be a distinction to be made between "PA proves" and "PA proves that it proves".
I am not sure I follow and it is possible I may not be on the same wavelength, but here is another thought.
For any sentence S, if first-order PA (or any first-order theory) proves S, then that means that S holds true in every model of that theory, via the completeness (not incompleteness) theorem.
The statement "PA proves x" is equivalent to saying "there exists a natural number N which is a Gödel number encoding a proof of x." The letter "x", here, is a natural number that is assumed to encode some sentence we want to prove, that is, x = #S for some sentence S.
The above is a predicate with one existential quantifier: Provable(x) = there exists N such that IsProof(N, x) holds true, where IsProof says that N is the Gödel number of a valid proof ending in x.
If PA proves "Provable(x)", that means that the predicate "Provable(x)" holds in every model of PA. This means every model of PA would have some natural number N that satisfies IsProof(N, x). Of course, this number can vary from model to model.
However: the standard model, which has only standard natural numbers, is also another model of PA. So if PA proves "Provable(x)," and Provable(x) thus holds in every model, it must also hold in the standard model of PA. This means that there must be a regular, standard, finite natural number N_st that encodes a proof of x.
Since the standard model is an initial segment of every other model of PA, then every other model includes the standard model and all the standard natural numbers. Thus, if PA proves Provable(x), then the standard number N_st must exist in all models.
So we cannot have a situation where PA proves Provable(x) but all proofs of x must be nonstandard and infinite in length. This would mean no such proof exists in the standard model of PA - but then, via completeness, PA would not be able to prove "Provable(x)".
For every n, the function will return a function, that PA proves has a proof in PA.
Suppose that we're in a nonstandard model. For all standard n, there really will be a proof. For nonstandard n, there may or may not actually be a proof.
And so PA cannot prove from the fact that it proved the existence of a proof, that PA actually proves it.
> > If PA proves "PA proves X" then PA can prove X.
If we assume PA to be sound, then indeed everything it proves is true.
> Not true.
Now you're saying PA is unsound.
But your article wasn't about PA proves "PA proves X", it was about "forall n : PA proves G(n)".
For PA not to prove "forall n: G(n)", there is no soundness issue, only a ω-consistency issue.
I think we are saying the same thing.
If PA proves that it proves a statement, PA cannot conclude from that fact that it proves that statement.
If PA proves that it proves a statement, and then fails to prove it, PA is unsound.
There exist collections of statements such that PA proves that it proves each statement, and PA does prove each statement, but PA does not prove the collection of statements.
Our understanding of the last is helped by understanding that "PA proves that PA proves S" is logically not the same statement as, "PA proves S". Even though they always have the same truth value.
> PA proves that if it can prove everything that it proves that it can prove, then PA must have a proof of one of the two previous statements.
I don't believe this is true. I don't know what result you're using here, but I think you're mixing up "provable" and "true".
In particular your line of reasoning violates Lob's Theorem, which is a corollary of the second incompleteness theorem.
The standard model is a model of PA only if PA is consistent, which it cannot prove unless it is inconsistent (Godel's theorem). So your proposed proof doesn't work in PA itself, which is the point of that comment, I believe.
I believe my proof actually works in systems even weaker than PA (but the metatheory of how exactly to do the encoding in a first-order theory is a bit of a headache for me at the moment).
In particular I'm not relying on the consistency of PA. If PA is inconsistent, "If PA proves 'PA proves X' then PA can prove X" also holds (trivially), because then PA can prove anything.
Hmm, earlier you wrote:
> This is because the key observation is not that there are nonstandard models, but rather that the standard natural numbers model PA.
This fact (that the standard natural numbers model PA) implies that PA is consistent, so your argument is implicitly assuming that from the outset, right? If PA is consistent then this is not something that PA can prove, so I don't see how you could run your proof in PA itself. And certainly not how you could do it in a weaker theory. Logic is subtle though... so what am I missing?
As far as I know what you're describing (namely that "Provable(X)" implies "X") is called the uniform reflection principle and added to PA is much stronger than PA alone.
I was talking to someone about inductive data types, and showed them the zero/succ definition of `Nat`, e.g. in Lean or Rocq.
It was interesting because they were asking "is this all you need? What about the Peano axioms? Is there anything more primitive than the inductive data type?"
I bring it up because it's good not to take for granted that the Peano axioms are inherent, instead of just one design among many.
> Is there anything more primitive than the inductive data type?
I believe that the natural numbers are more primitive than inductive data types, since all inductive data types may be constructed from the natural numbers alongside a small number of primitive type formers (e.g. Π, Σ, = and Ω).
You need some source of infinite-ness, otherwise the entire theory can be modelled by finite sets. It can be provided by the natural numbers or W types or inductive types, but the naturals are arguably the most fundamental of the three.
Pure lambda calculus is enough because lambda calculus encodes computation.
Related to PA consistency, it can be proven in PA. https://youtu.be/6pjLmmkZnIA
This definitely needs some context for the non-logicians in the house! Gödel's second incompleteness theorem shows that, if PA can prove its own consistency, then PA is inconsistent (and can therefore prove anything, including false things).
The work linked here doesn't show that PA is inconsistent, however: what it does is to define a new, weaker notion of what it means for PA to “prove its own consistency” and to show that PA can do that weaker thing.
Interesting work for sure, but it won't mean anything to you unless you already know a lot of logic.
But is computation enough? The computable reals are a subset of the reals.
"Reals" (tragically poorly named) can be interpreted as physical ratios.
That is: Real numbers describe real, concrete relations. For e.g., saying that Jones weighs 180.255 pounds means there's a real, physical relationship -- a ratio -- between Jones' weight and the standard pound. Because both weights exist physically, their ratio also exists physically. Thus, from this viewpoint, real numbers can be viewed as ratios.
In contrast, the common philosophical stance on numbers is that they are abstract concepts, detached from the actual physical process of measurement. Numbers are seen as external representations tied to real-world features through human conventions. This "representational" approach, influenced by the idea that numbers are abstract entities, became dominant in the 20th century.
But the 20th century viewpoint is really just one interpretation (you could call it "Platonic"), and, just as it's impossible to measure ratios to infinite precision in the real world, absolutely nothing requires an incomputable continuum of reals.
Physics least of all. In 20th and 21st century physics, things are discrete (quantized) and are very rarely measured to over 50 significant digits. Infinite precision is never allowed, and precision to 2000 significant digits is likewise impossible. The latter not only because quantum mechanics makes it impossible to attain great precision on very small scales. For e.g., imagine measuring the orbits of the planets and moons in the solar system: By the time you get to 50 significant digits, you will need to take into account the gravitational effects of the stars nearest to the sun; before you get to 100 significant digits, you'll need to model the entire Milky Way galaxy; the further you go in search of precision, the exponentially larger your mathematical canvas will need to grow, and at arbitrarily high sub-infinite precision you’d be required to model the whole of the observable universe -- which might itself be futile, as objects and phenomena outside observable space could affect your measurements, etc. So though everything is in principle simulatable, and precision has a set limit in a granular universe that can be described mathematically, measuring anything to arbitrarily high precision is beyond finite human efforts.
As far as we know quantum mechanics does not have a granularity. You can measure any value to arbitrary precision. You must have limits on measuring two things simultaneously.
Granularity is implied by some, but not all, post standard model physics. It's a very open question.
Whether or not nature is discrete is an open question, but it's rather strongly implied, and there's absolutely nothing to suggest that the universe is incomputable.
> You can measure any value to arbitrary precision
Quantum mechanics lets you refine one observable indefinitely if you are willing to sacrifice conjugate observables.
You can measure that one observable to an arbitrarily (not infinitely!) high precision -- if you have an correspondingly arbitrary duration of time and arbitrarily powerful computational resources. That measurement of yours, if exceedingly precise, might require a block of computronium which utilizes the entire energy resources of the universe. As a practical matter, that's not permitted. I'm certainly unaware of any measurement in physics to more than 100 significant digits, let alone "arbitrary precision".
In fact, as it turns out, no experiment has ever resolved structure down to the Planck length. A fundamental spatial resolution is likely to be quite a lot smaller than the Planck length; even the diameter of the electron has been estimated at anywhere from 10^-22m to 10^-81m.
The question I was responding to asked whether reals are "necessary" -- plainly, insofar as reality is concerned, they are not.
We cannot measure anywhere near the Planck length or anything like 100 significant figures.
My concern with the reals goes the other direction. The reals are not closed under common operations. A lot of the work is done in complex numbers, which are. The rationals are not closed under radicals, and radicals seem pretty fundamental.
You can define a finitist model despite this. But it's ugly, and while ugliness is not physically meaningful, it tends to make progress difficult. A usable solution may yet arise; we shall (perhaps) see.
Numbers interpretable as ratios are the Rational numbers, by definition, not the Reals.
This entire discussion is about mathematical concepts, not physical ones!
Sure, yes, in physics you never "need" to go past a certain number of digits, but that has nothing to do with mathematical abstractions such as the types numbers. They're very specifically and strictly defined, starting from certain axioms. Quantum mechanics and the measurability of particles has nothing to do with it!
It's also an open question how much precision the Universe actually has, such as whether things occur at a higher precision than can be practically measured, or whether the ultimate limit of measurement capability is the precision that the Universe "keeps" in its microscopic states.
For example, let's assume that physics occurs with some finite precision -- so not the infinite precision reals -- and that this precision is exactly the maximum possible measurable precision for any conceivable experiment. That is: Information is matter. Okay... which number space is this? Booleans? Integers? Rationals? In what space? A 3D grid? Waves in some phase space? Subdivided... how?
Figure that out, and your Nobel prize awaits!
The set of values of any physical quantity must have an algebraic structure that satisfies a set of axioms that include the axioms of the Archimedean group (which include the requirements that it must be possible to compare, add and subtract the values of that physical quantity).
This requirement is necessary to allow the definition of a division operation, which has as operands a pair of values of that physical quantity, and as result a scalar a.k.a. "real" number. This division operation, as you have noticed, is called "measurement" of that physical quantity. A value of some physical quantity, i.e. the dividend in the measurement operation, is specified by writing the quotient and the divisor of the measurement, e.g. in "6 inches", "6" is the quotient and "inch" is the divisor.
In principle, this kind of division operation, like any division, could have its digit-generating steps executed infinitely, producing an approximation as close as desired for the value of the measured quantity, which is supposed to be an arbitrary scalar, a.k.a. "real" number. Halting the division after a finite number of steps will produce a rational number.
In practice, as you have described, the desire to execute the division in a finite time is not the only thing that limits the precision of the measured values, but there are many more constraints, caused by the noise that could need longer and longer times to be filtered, by external influences that become harder and harder to be suppressed or accounted for, by ever greater cost of the components of the measurement apparatus, by the growing energy required to perform the measurement, and so on.
Nevertheless, despite the fact that the results of all practical measurements are rational numbers of low precision, normally representable as FP32, with only measurements done in a few laboratories around the world, which use extremely expensive equipment, requiring an FP64 or an extended precision representation, it is still preferable to model the set of scalars using the traditional axioms of the continuous straight line, i.e. of the "real" numbers.
The reason is that this mathematical model of a continuous set is actually much simpler than attempting to model the sets of values of physical quantities as discrete sets. An obvious reason why the continuous model is simpler is that you cannot find discretization steps that are good both for the side and for the diagonal of a square, which has stopped the attempts of the Ancient Greeks to describe all quantities as discrete. Already Aristotle was making a clear distinction between discrete quantities and continuous quantities. Working around the Ancient Greek paradox requires lack of isotropy of the space, i.e. discretization also of the angles, which brings a lot of complications, e.g. things like rigid squares or circles cannot exist.
The base continuous dynamical quantities are the space and time, together with a third quantity, which today is really the electric voltage (because of the convenient existence of the Josephson voltage-frequency converters), even if the documents of the International System of Units are written in an obfuscated way that hides this, in an attempt to preserve the illusion that the mass might be a base quantity, like in the older systems of units.
In any theory where some physical quantities that are now modeled as continuous, were modeled as discrete instead, the space and time would also be discrete. There have been many attempts to model the space-time as a discrete lattice, but none of them has produced any useful result. Unless something revolutionary will be discovered, all such attempts appear to be just a big waste of time.
Your whole comment can just be TL;DR Forth and the fixed point philosophy :)
Rationals > irrationals on computing them. You can always approximate irrationals with rationals, even Scheme (Lisp, do'h) has a function to convert a rational to decimal and the reverse. decimal to rational.
Are they?
The idea that uncountable means more comes from a bad metaphor. See https://news.ycombinator.com/item?id=44271589 for my explanation of that.
Accepting that uncountable means more forces us to debatable notions of existence. See https://news.ycombinator.com/item?id=44270383 for a debate over it.
But, finally, there is this. Every chain of reasoning that we can ever come up with, can be represented on a computer. So even if you wish to believe in some extension of ZFC with extremely large sets, PA is capable of proving every possible conclusion from your chosen set of axioms. So yes, PA is enough.
If you're not convinced, I recommend reading https://www.amazon.com/G%C3%B6del-Escher-Bach-Eternal-Golden....
This is an under specified question, until some observable goal is attached to "enough". Enough for what?
I like to think about it like this: while real numbers in general are impossible to compute, write down or do anything else with them, many statements about real numbers can be expressed in a useful, computable form.
It all gets mind-bendingly mind-bending really fast, especially with people like Harvey Friedman ( https://u.osu.edu/friedman.8/foundational-adventures/fom-ema... ) or the author of this post trying to break the math by constructing an untractably large values with the simplest means possible (thus showing that you can encounter problems that do not fit in the universe even when working in a "simple" theory)
(i saw the username and went to check audiomulch website, but it did not resolve :( )
> But is computation enough?
Of course not, but that would invalidate the entire project of some/many here to turn reality into something clockwork they can believe they understand. Reality is much more interesting than that.
Given my former experiences with encoding type-level computation in Haskell and Rust, I'd kinda rephrase that statement in the title, and say "peano arithmetic is enough, if you only ever need to go up to 88." https://upload.wikimedia.org/wikipedia/commons/b/bd/D274.jpg
This is a stack overflow question that I turned into a blog post.
It covers both the limits of what can be proven in the Peano Axioms, and how one would begin bootstrapping Lisp in the Peano Axioms. All of the bad jokes are in the second section.
Corrections and follow-up questions are welcome.