11 ms·
The field of “useful reals” between rational and real numbers (2019)
- klodolph 7y agoNote that like the rational numbers, the field of “useful reals” is not complete. So if you have a sequence of “useful reals” that is Cauchy, it will converge to a real number but it may or may not converge to a “useful real”.
- function_seven 7y agoSorry if I misunderstand you. If I have a Cauchy sequence of "useful reals", wouldn't the convergence be, by definition, a "useful real"? That is, I can write down the Cauchy sequence, so it's now symbolically noted, right? Or are you referring to a Cauchy sequence that exists, but can't be defined using our symbology?
- klodolph 7y agoThere are uncountably many Cauchy sequences of useful reals. You can’t write them all down. So now you have to also restrict yourself to “useful Cauchy sequences of useful reals”. This is a rabbit hole with no end.
- wolfgke 7y ago> There are uncountably many Cauchy sequences of useful reals. You can’t write them all down. This does not hold if you demand that, for example, the map k -> a_k that represents the Cauchy sequence, is a computable function.
- pdonis 7y agoTo prove that a field is complete, your proof must hold for any Cauchy sequence, not just the ones that meet some constraint you impose.
- klodolph 7y agoI am a bit shocked, because that seems like a really dishonest way of doing things. “Surprise! Actually, I am not talking about Cauchy sequences, but only computable Cauchy sequences.” If you change the rules you had better be up front about it. What you are describing is a completely different definition for “complete metric space” than what is commonly accepted by the mathematical community at large. So do not be surprised that by using different definitions, you come to different conclusions.
- avmich 7y agoInteresting that for Cauchy sequences in question they necessarily have to be non-constructive, i.e. one can't name any element in the sequence. Edit: oops, not that. The sequence itself - not the "useful reals" element of the sequence - has to be non-constructive...
- wolfgke 7y ago> I am a bit shocked, because that seems like a really dishonest way of doing things. “Surprise! Actually, I am not talking about Cauchy sequences, but only computable Cauchy sequences.” Rather: If countability is important to you, you should change the rules so that the property that the field is closed w.r.t limits of Cauchy sequences does not make your set uncountable. Redefining the rules if something does not work is how you do mathematics works all the time: - A PDE does not have a solution in a classical sense and you hate this? No problem: You invent the theory of weak solutions and distributions and simply change the concept what is to be considered a solution of the PDE. - The concept of algebraic varieties turns out to be to limiting to obey the rules that you would love them to have? No problem: You define the concept of algebraic schemes and now talk about algebraic schemes instead of varieties (https://en.wikipedia.org/w/index.php?title=Scheme_(mathematics)&oldid=942687770 https://en.wikipedia.org/w/index.php?title=Scheme_(mathemati...). TLDR: Mathematics is often the art of "defining your problems away".
- klodolph 7y ago> Redefining the rules if something does not work is how you do mathematics works all the time: Exploring the consequences of alternative definitions is fantastic, let’s do more of that. Redefining the terms that somebody else uses in a conversation sucks royally. Everybody hates it when people do that. Don’t be that guy.
- ultrafilter 7y agoSet theorists have located the "end" for all practical and most impractical purposes. Let M be the minimal countable transitive model of ZFC. Declare a real to be useful if and only if it is in M.
- OscarCunningham 7y agoThis needs axioms beyond ZFC though. Even assuming ZFC is consistent isn't enough to know that there's a minimal countable transitive model.
- wololongong 7y agoYou cannot write a general Cauchy sequence of useful reals with a finite number of symbols. Hence you cannot in general express its limit with a finite number of symbols.
- ummonk 7y agoName one such sequence of "useful reals" that is Cauchy but doesn't converge to a "useful real". You can't, can you? "Useful" Cauchy sequences of "useful reals" (i.e. those you can define) all converge to a "useful real".
- doomrobo 7y agoA nit: "reals are a field extension of ℚ. They could be considered an algebraic number field..." This is not an algebraic extension. Pi is a "useful real number" and it is not algebraic over Q.
- klodolph 7y agoYes—and to elaborate, the reason why an algebraic field extension of ℚ cannot contain π is because: - If it is a field, it contains π, π², π³, … which are linearly independent. - By definition, an algebraic field extension is finite dimensional.
- arberavdullahu 7y agoYou are wrong! The algebraic field extension ℚ[π] contains π.
- OscarCunningham 7y agoI think people would normally call that a transcendental extension and not an algebraic one.
- klodolph 7y agoℚ[π] is not an algebraic extension of ℚ.
- jopolous 7y agoI guarantee that is not an algebraic extension. It's not even a finite extension
- lonelappde 7y agoThat's not a valid critique, as other commenters explained
- doomrobo 7y ago
- jepler 7y agoNot "between" in the sense of having an intermediate cardinality between rationals and reals, since they are exactly the numbers available from strings in some symbolic system or other. Seems to be a slightly expanded case of algebraic numbers, since additional forms (like infinite definite integrals) are allowed.
- xtacy 7y agoYep, that's right. Its cardinality is the same as rationals, since it's countable.
- perl4ever 7y agoHow disappointing. Unlike most of the time, I read the article first and now that I'm here, that was the question I had - clearly it's smaller than reals, but how and why is this field larger than rational numbers? Guess it's not.
- klodolph 7y agoIt’s larger than the rational numbers in the sense that it is a strict superset. Cardinality is what a lot of people reach for when they are talking about “larger” or “smaller”, but there are lots of other useful concepts which we can translate to “larger” and “smaller”. So when someone says “larger” or “smaller”, your first step might be to try and translate that relationship into a more precise mathematical concept, like cardinality or measure. Casual terminology also leads to weird discussions. Like when someone asks whether some function is “close” to another, and these functions are defined in terms of vector spaces. Unfortunately, “closeness” does not necessarily exist in a vector space. So the answer may be that the question does not make sense.
- effie 7y ago> “closeness” does not necessarily exist in a vector space. The asker will give a definition. For example, two vectors are close if sqrt of dot product of difference of the two vectors is smaller than some number delta.
- emacdona 7y agoThe author claims in the notes that "The useful reals are similar, but not quite equivalent to other ideas in mathematics, such as [...] computable numbers." Is that correct? What is the complement of the Computable Numbers in the Useful Reals? What is the complement of the Useful Reals in the Computable Numbers? I've always thought of Computable Numbers as all numbers able to be represented by a finite string, ie: a computer program that would generate the number to any desired precision. How does that differ from the set of numbers with a finite symbolic representation? Hmmmm... maybe by asking that question I've led myself to the answer. Chaitin's Constant has symbolic representations, one of which being the Wikipedia page that describes it: https://en.wikipedia.org/wiki/Chaitin%27s_constant https://en.wikipedia.org/wiki/Chaitin%27s_constant. Does that mean it's included in the complement of the Computable Numbers in the Useful Reals? Are the Computable numbers a subset of the Useful Reals?
- emacdona 7y agoI say "it" when I mention Chaitin's Constant, but really I believe it's an entire set of constants. Is that set countable? So many questions... :-)
- emacdona 7y agoLooks like the wikipedia page says there's a Chaitin's constant for each Computable Function, so yeah, countable. That's if I'm reading it correctly. Even if the constant differs for every program that computes a given Computable Function... still countable, though (if I'm doing my math right).
- tromp 7y agoYep, it's countable. I have defined a relatively simple one in https://tromp.github.io/cl/Binary_lambda_calculus.html#Halting_probability https://tromp.github.io/cl/Binary_lambda_calculus.html#Halti...
- Sniffnoy 7y agoThe standard term is "definable", not "useful": https://en.wikipedia.org/wiki/Definable_real_number https://en.wikipedia.org/wiki/Definable_real_number But yes, Chaitin's constant is an example of a number that is definable but not computable.
- scarejunba 7y agoDoes this field behave differently from Q in some 'useful' way?
- H8crilA 7y agoIt has sqrt(2), for starters? Not sure what do you mean by useful. It is not "useful" in the sense that reals are most "famous" for: it is not complete. Cauchy sequences can diverge in the useful reals field.
- scarejunba 7y agoAh, I was curious if there are any interesting properties.
- lonelappde 7y agoCompleteness in the "full" reals is a useless feature, though. All is gives you is an emotional crutch to pretend your cauchy sequences can be mapped to regular numbers. But it doesn't give you anything you didn't already have in the cauchy sequences and useful reals.
- steerablesafe 7y agoYou are of course right, reals are isomorphic to equivalence classes of Cauchy sequences on Q. But once you are dealing with equivalence classes of Cauchy sequences on Q you might as well give it a name. Maybe call it R.
- H8crilA 7y agoHis point is different. You cannot (by definition) ever write a "name", a formula, a rule, a lim expression, anything really, for a real that is not in the useful reals.
- deleted 7y ago[deleted]
- PaulHoule 7y agoThis is one of my favorite obscure math topics. I think of the "useful reals" being the "reals that have names". Alan Turing developed the Turing machine to get a handle on the "useful reals" since you can make a Turing machine write them out one digit at a time. Given that, I don't like the term "real numbers" at all because they are phony compared to the "useful reals" -- if you reject the axiom of choice then the construction that Cantor does to construct a real isn't valid. Despite calling for a rebuild of math and science based on computation, Steve Wolfram has yet to take the critical step of rejecting the axiom of choice. I wish he would man up.
- OscarCunningham 7y ago> if you reject the axiom of choice then the construction that Cantor does to construct a real isn't valid Are you talking about Cantor's argument that the reals are uncountable? That doesn't need choice.
- thaumasiotes 7y agoElaborating, the hypothesis that Cantor disproves is "The real numbers are countable -- that is to say, the real numbers can be put into one-to-one correspondence with the natural numbers". You never have to use the axiom of choice, because the hypothesis tells you there is a one-to-one function between the reals and the naturals. You can then order the reals in the order suggested by their image in the naturals: f(0), f(1), f(2), ...
- leni536 7y agoMany sets have a "useful" subset this way. Even the class of all sets have a "useful" subclass.
- arberavdullahu 7y agoThe author claims that this set is countable but not sure if that is true. My argument is based on Cantor's theorem [1], which states that the power set has cardinality strictly greater than the set. In order for the set of symbols to be finite field it must grow therefore since rational is infinitely countable from Cantor it must hold that "useful reals" is uncountable. [1] https://en.wikipedia.org/wiki/Cantor%27s_theorem https://en.wikipedia.org/wiki/Cantor%27s_theorem
- selectionbias 7y agoIf you have a finite set of symbols, then the set of finite sequences of those symbols is countable. The key here is 'finite sequences', if you were to allow for infinite sequences then the set if uncountable.
- Sniffnoy 7y agoNone of this is somehow secret. The standard name for this is "definable"[0]. Although, one has to be really careful with this sort of thing; there are apparently a number of subtle logical issues[1] that come up when talking about these... (Note, by the way, that there's any number of other fields one could put inbetween; such as the field of algebraic reals, or computable reals, or the fraction field of the ring of periods...) [0] https://en.wikipedia.org/wiki/Definable_real_number https://en.wikipedia.org/wiki/Definable_real_number [1] https://mathoverflow.net/a/44129/5583 https://mathoverflow.net/a/44129/5583
- dwohnitmok 7y agoWell that Math Overflow post is excellent. One of the logical issues is that there is a model of ZFC where all reals are definable/useful. I'm guessing that's not what the author of this blog post is going for... If this seems impossible given that the number of definitions is countable, note first that it is possible that a model of ZFC is itself countable (in a larger ambient model), but it cannot witness the countability of sets within itself. So when we say that a set is uncountable in ZFC, it is sometimes useful to make the distinction that it is only uncountable in the implicit model under discussion. Then note that definability, unlike countability, cannot be itself defined in the language of ZFC (due to Tarski's undefinability of truth result). Note that this is different from saying it's independent of ZFC. It cannot even be expressed in ZFC. Hence, unlike countability, there is no "relative" concept of definability, at least not relative to first-order ZFC. Therefore the statement "every element of this model is definable" is more absolute than "every element of this model is countable" (but not absolutely absolute, we still have an ambient model we're working in, just a richer theory for that model). The usual diagonalization argument within our entirely definable model of ZFC to try to construct a definable real number not contained in any countable enumeration of definable real numbers fails because we have no enumeration of definable real numbers. This is not a failure of constructivism (it is ZFC after all, we do have choice), but rather a consequence of the fact that definability cannot be expressed in ZFC so we don't have a way of even talking about the set of all definable real numbers within our model.
- dwohnitmok 7y ago
- OscarCunningham 7y ago> A “useful real” is just a real number that can be precisely described (not just approximated!) by some symbolic notation. Obviously, this definition is loose and depends greatly on your choice of symbols and their definitions. In fact, the definition is necessarily loose. If you could make it precise then you could carry out Cantor's diagonalisation procedure to produce a precise description of a real which couldn't be precisely described, a contradiction.
- solinent 7y agoAnother argument that's not completely non-constructive. The real numbers have to be constructed. Typically, a number is represented by a Cauchy sequence or a Dedekind cut. To determine if a real number is representable symbolically, we simply need a finite sequence of symbols which stands for this Cauchy sequence, lets say. Theroem: The real numbers and definable numbers are the same set. Assume a real number exists but is not definable. This means at the very least we have a mathematical statement saying there exists a number such that some logical predicate is valid (we may not even have a construction in ZFC), which can also be constructed using a Cauchy sequence. This mathematical statement is embedded in ZFC, and since we are humans it must be finite. In fact, you could come up with a binary representation for such a statement using methods from Godel, by mapping each symbol to some binary representation. Therefore, this number can be represented as a sequence of zeros or ones, a contradiction. QED
- OscarCunningham 7y agoI don't understand that argument. But in any case, Cantor's argument is very constructive. It literally gives you the decimal expansion of the new number not in your set.
- solinent 7y agoI didn't do it that much justice because I was discovering it independently, my arguments can easily be made rigorous, but you'd need a background in pure mathematics to understand it. However, there's a section on Wiki: https://en.wikipedia.org/wiki/Definable_real_number#Definability_in_models_of_ZFC https://en.wikipedia.org/wiki/Definable_real_number#Definabi... They start with a stronger definition of a definable number, so they find that they do exist. I think given the argument above there must be a hole in my own argument, I'd have to go beyond ZFC.
- currymj 7y agoit's good to see the "real numbers are fake" crowd out in full force!
- lonelappde 7y agoAhem. Repeat after me, the Creed of Numbers: " The imaginary numbers aren't imaginary. The real numbers aren't real. "
- dfox 7y agoThere is well defined name for "useful reals": Algebraic numbers. Of course the well-definedness necessitates some limit on how the symbolic description looks like (ie. algebraic numbers are roots of polynomials with rational coefficients) because every real number can be described by some arbitrarily complex symbolic notation. Edit: I vaguely remember that there used to be some name for the intersection of algebraic and real numbers, but I neither can remember it nor can find it on wikipedia.
- evanb 7y agoπ and e are not algebraic numbers. So the set of "useful" (read: definable) reals is larger than the algebraic numbers.
- JoshuaDavid 7y ago> every real number can be described by some arbitrarily complex symbolic notation This seems like it would have to be false, because otherwise the reals would be countable (iterate through every possible 1-character string, then every possible 2 character string, then 3 chars, etc and in a finite (but potentially very very large) amount of time you would come across the description of any real number that can be described).
- deleted 7y ago[deleted]
- superjan 7y agothis reminds me of unit testing, where the tests come up with arbitrarily defined numbers, and the function you test tries to come up with a consistent way to count them. If you can change your function each time a test is added, the tester never wins. Isn’t this similar? It seems like cherrypicking to include simple formulas with e and pi in your numbering system.
- stephencanon 7y agoAside from all the other issues people have raised, equality is not decidable for the “useful reals”. While they form a field, they do not form a computably-ordered field, which makes them quite a bit less useful than many other number systems.
- deleted 7y ago[deleted]
- NelsonMinar 7y agoAnother related topic of interest is constructivism in mathematics. Unfortunately the wikipedia article is pretty abstruse, anyone have a more down to earth one? https://en.wikipedia.org/wiki/Constructivism_(philosophy_of_mathematics) https://en.wikipedia.org/wiki/Constructivism_(philosophy_of_... (Note this is different from constructible numbers, which the author mentions. That has to do with classical geometry.)
- btilly 7y agoMy favorite is the collection of essays touching on the topic in the book *The Mathematical Experience". All of the other essays in the same book are also good. :-)
- lonelappde 7y agoSuch a weird perspective. The author thought they discovered something that true and interesting and kind of fundamental but wasn't already published, but didn't think it was worth publishing to the math community?
- joppy 7y agoThinking about maths is fun, and some people do it for leisure and write what they find in innocuous places like blogs. Usually the things you come up with are already well-known by a different name (as was the case here), so one would usually not publish something like this. Think of it just like a random blog post on someone’s thoughts. Just because it contains maths doesn’t mean it needs to be published or not, it can be free to live its own life.
- D_Alex 7y agoIf you look at the integers between say Graham's number (https://en.wikipedia.org/wiki/Graham%27s_number https://en.wikipedia.org/wiki/Graham%27s_number) and TREE3(https://en.wikipedia.org/wiki/Kruskal%27s_tree_theorem https://en.wikipedia.org/wiki/Kruskal%27s_tree_theorem) you can observe that practically all of these integers, while "computable", cannot be defined within the known constraints of this universe. Which raises an interesting question: In what meaningful sense do these numbers exist? They are just out of reach as the non-definable real numbers...
- circlefavshape 7y agoIn what meaningful sense do any numbers exist? This comes up with my kids sometimes ... are numbers real?
- minikites 7y agoI liked this Numberphile video on that topic: https://www.youtube.com/watch?v=1EGDCh75SpQ https://www.youtube.com/watch?v=1EGDCh75SpQ
- voxl 7y agoYou're mixing up "defined" with "defined via a decimal numeral" we can define these numbers without much difficulty via finite formula that compute them. This is a completely valid definition, it is just not a decimal numeral. An interesting idea might be "useful integers" which requires whatever definition we have to allow approximation of any finite subsequence with error converging to zero given more computational power.
- pwdisswordfish2 7y agoGP did not mix up anything. Some of those finite formulas also will be too long to be written within the constraints of this universe. The pigeonhole principle applies just as much to finite formulas as it does to finite strings of decimal digits.
- dchyrdvh 7y agoThe premise of this idea - that anything describable can be written in a binary firm and is thus countable - seems wrong. It's wrong because we easily invent new concepts and put them into a symbolic form. We could invent a new concept, agree on a new symbol for it and add it to our alphabet. The set of ideas isn't countable and so our alphabet isn't countable. This alphabet can't be translated into some binary form either.
- lisper 7y agoWhat makes you think that the set of ideas isn't countable?
- dchyrdvh 7y agoSo long as every real number exists, has properties and so on. Every such number is a separate idea. They exist, no matter whether we know about them or not.
- lisper 7y agoAh. Personally, I distinguish between potential ideas and actual ideas. To be an actual idea, it has to reside in someone's brain (or a computer, or some other data-processing system). The reals correspond to the set of potential ideas, but the set of actual ideas is not only countable, but almost certainly finite.
- dchyrdvh 7y agoWe can call it a materialized idea, like an implemented software algorithm. I'm indeed talking about the world of ideas that's not real, i.e. non material. The proof of the Fermat's theorem has always existed, but only recently it's been discovered by Wales.
- lisper 7y agoProofs must be finite so there can only be countably many of them.