Emile Borel: Real number enthusiast or skeptic? | Sociology and Pure Mathematics | N J Wildberger

[@johngough2958]

I just wanted to say thanks for a great exposition. I first encountered the concept of normal number in a paper by Mark Kac which focussed on probability - a random sequence of coin tosses should be normal base 2, etc. I've just watched with my 13 year old and he picked up your ideas in the video - which is the highest accolade!

[@dsm5d723]

Deriving what is there for everyone to miss is the mark of supposed mathematical genius.

[@dylanjayatilaka8533]

It's nice to hear that I'm not the only one who watched math videos with their kids.

[@NuYiDao]

Love this. Hoping you will cover Clifford at some point.

[@Anders01]

I recently learned that Aristotle described potential infinities and later Georg Cantor 'completed' those infinities into transfinite numbers. One idea that gave me is that infinite can be defined as a single potential, meaning instead of many potential infinities there is only one inexhaustible potential.

[@santerisatama5409]

Aristotle expanded on (and made of mess of) Plato's dialogue 'Sophist', where culminating his study of Forms, Plato concluded based on discussion of Great Kinds (being and non-being, movement and rest, sameness and difference) that the ultimate nature of being is 'dynamis' (which is translated into Latin as potentia). When I first read Sophist, I was already under the influence of David Bohm and went wow: Plato is speaking philosophy of quantum theory! Plato's discussion shows that simplistic bivalent logic does not apply to non-being, which becomes a kind of a being characterized by dynamis. And as the dialogue can be read also as Plato's take between Heraclitus and Parmenides, the conclusion of 'dynamis' goes with process philosophy.

[@jonathancohen2351]

All we need to do is compute the digit of the know it all number that corresponds to the question "are real numbers a valid mathematical concept?"

[@robharwood3538]

Or, like Russell/Goedel: "Does the know-it-all number have a 0 located at the digit whose index represents this question?"

[@alisadiqtcs]

Hi Dr. Wildberger, I am really impressed by your meticulous analysis of real number subject. I have two ideas which you may add further thinking in this domain (i am not aware if this dimension is already explored) 1. Probability of non-pattern-seeking numbers We are trying to find pattern or probability of occurrence. Another angle may be that there are two types of numbers, 1) numbers chasing a pattern, which can be discovered through some formulae or probability, 2) numbers whose normal behaviour is to avoid a pattern. So a probability of avoiding patterns may be an opening slot of this conundrum. 2. Broken number: When a paper is torn into two pieces, without making a neat seam to tear it evenly, the edges of those torn pieces can't have any pattern because of uneven force application in opposite sides. Similarly, all (or some) cuts on the real number line can be thought of such a breakage, which may be part of an undiscovered bunch of complex addition-subtraction of two or more rational numbers which when placed together sit in the hole of the real number line. Although I am not much intelligent but may be such ideas can be explored further to solve through a very ordinary way of thinking. thanks

[@TrevorAndersen0]

Your first point seems like the concept of computable vs uncomputable numbers.

[@SevenThunderful]

I was expecting some kind of Godel like self referential yes or know question about the know it all number itself. One issue I've always had with real numbers is that they actually only have a finite number of informational bits, if one assumes that you are observing the number in noise of finite variance. Is a truncated decimal or binary decimal the best finite representation of this? Any arithmetic on these quantities only increases the noise.

[@jaanuskiipli4647]

"Is the first digit of Borel's know it all number 1?" We find the answer at Borel's number position x (we can calculate x, relatively easily because the question is not too long and there is only a limited amount of questions that come before it, for the sake of example let's say x = 123456), next natural question is "Is the Borel's number value at position x 1?" and so on.

[@robharwood3538]

Looking forward to the next video. Have you considered asking a native French speaker to help with the translation? Perhaps someone with 'no dog in the race', but perhaps interested in math or history or math history? Just a thought! 😊

[@joepike1972]

Two ideas come to me on having watched this video. In response to the normal number. It reminds me of a program I was working on that I implemented the ability of the player to "time travel" to go to the past I simply took the saved state of the turn in the past and allowed the player to interact in that world from that save point. To go to the future I just simulated events how ever many turns the player wanted to go into the future. Then I wanted to allow for the player to encounter past or future versions of them self. And I tried to consider the probabilities involved in it. As I thought about it I was left with the fact there may be zero to infinite amounts of npc "player" time travelers and from infinite points in time. As I was pondering this I also considered the possibility that any one of those explicit instantiations of time travelers appearing at that point in time, being out of an infinite set approached 0% chance of happening. For me the normal numbers and their stronger restrictions approach a zero percent chance of existing out of all of the possible permutations of numbers that could exist. In response to the know it all number. I am left considering the fate of object oriented code and the implied ability to adapt to future implementations. I can imagine a 1970s set of all possible questions in French and then a later revised 1997 edition. And on considering this I am left to think, despite them being yes or no questions; perhaps depending on who is asked you will get differences on what is yes or no. Maybe due to limitations or lack of understanding. Maybe parts of it were copied over incorrectly. Maybe their is an industry that is funding the universities and black mailing the professors to have certain specific questions answered a certain way. And so you have this thing that is suppose to be universal and absolute but it is far from it and you are actually left with numerous conflicting and non-corresponding and you are left having to implement a specific version to get specific results you are expecting and at times that doesn't even work.

[@robharwood3538]

I've had similar thoughts recently. For example, this rough argument: 1. If an 'infinite choice' Real number system *lacks* the ability to 'exit' the sequence of choices, then it therefore lacks the ability to express any numbers at all, since it can never halt. - Since we want to be able to express at least some numbers, then the system must have the capability to 'exit' from the choices. In other words, there must be a 'choice', let's call it 'exit', available in addition to the choice of digit values. (E.g. in binary, need to be able to not only choose from 0 and 1, but also a third choice 'exit'.) 2. For any 'infinite choice' Real number system to *be able* to express an *arbitrary finite* number, then it must have a non-zero probability to 'exit' at every step in the 'choice' loop. In other words, if any step has 'exit' as 0 probability, then the system is incapable of expressing a *finite* number which exits at that particular step. - Since we want to be able to express not only arbitrary 'infinite decimals' but also arbitrary finite ones, then 'exit' must have a probability > 0 at every step. 3. Since at every step, the probability of choosing 'exit' is > 0, then the probability of continuing (by choosing a digit instead of 'exit') is < 1 at each step. 4. The infinite product of probabilities, all < 1, is 0. Therefore, the probability of any 'infinite choice' Real number system expressing an *actually infinite* number is 0. Likewise, the probability that any number actually expressed by such a system is finite, is 1. This is in contrast to the claim of mainstream math that the 'measure' of the uncomputable Real numbers on the interval [0,1] = 1, and the measure of the computable Real numbers on [0, 1] = 0. It shows that, for any 'infinite choice' Real number system to be able to express any numbers at all, then they will all be finite. [It's not a perfect argument. For instance, I don't think point 4 actually withstands scrutiny, since I could imagine that the infinite product could be converted via logarithms to an infinite sum, and even though the infinite sum will be negative (corresponding to probability of < 1), the probability of 'exit' could be set up in such a way that the corresponding infinite sum of logarithms could still be finite; whereas for probability = 0, the logarithms would have to sum to (in the limit) -infinity. But, it seems like a promising intuition towards a more complete argument. Also, even if point 4 doesn't hold up, the argument still *does* contradict the claim that the 'measure' of uncomputable Real numbers is 1, since measure is (supposedly) connected with probability theory, and if a probability is less than 1, then it cannot have measure = 1.]

[@santerisatama5409]

In contemporary quantum formalism, T in CPT-symmetry is basically a palindrome. So in terms of a suitable formal language, we could write an (open) interval - a duration - that contains both past and future as a palindromic string. Such palindromic strings would be very close to Norman's foundational idea of mark-antimark symmetries. Which could be interpreted also as question-answer symmetries. A suitable formal language of duration palindromes could be written with relational operators and spaces, , < >, < >, etc. The operation 'concatenate mediants' best known from Stern-Brocot generation is very special and allows to generate palindromic strings of increasing resolution. Such duration would be open both internally and externally, a complex mereology of durations/intervals. The "question" about a "point" is not decidable, so it's not a genuine question-answer pair - let's remember Euclid's intuitively valid definition of point is mereological: Point has no part and it is end of a line, hence part of a line. The definition is top down holonomic, it denies that line can consist of points and point-reductionism. Halting problem denies that "end-points" of program runs can be universally defined. Hilbert's geometry of point-reductionism insists on asking the "wrong" kind of impossible questions which can't fit bivalent logic of yes-no answers. Note that "point" is bad terminology also for numerical coordinate systems, much better term for number pairs, triplets etc. of a coordinate vertex is a 'vertex'. For time travel to work, various generator seed strings for concatenating mediants would have to have some global invariant, concretely they would have to have a natural numerical interpretations as Stern-Brocot type rational numbers. Thus time travel between duration structures between various generator seeds with SB-structures could happen by "jumping" between various duration-palindromes.

[@KarmaPeny]

So, all we need to do is to locate an infinitely small point to 'know it all'! Has anyone ever located an infinitely small point? Also, if every possible arrangement of digits occurs an equal amount of times in a 'normal' real number, then we can say that the first n significant digits of pi must occur in that number. Not only that but it must occur infinitely many times. Not only that, but these arguments must hold for all values of n. So all infinitely many digits of pi, and all other real numbers for that matter, must occur infinitely many times in a single 'normal' real number. Wow, who knew we could fit all real numbers into a single real number and all answers to all yes/no questions in an infinitely small point? Perhaps this is what people were thinking about in that film 'Scanners' when their heads exploded.

[@brendawilliams8062]

Lol

[@santerisatama5409]

There are only intervals between rational numbers - as we can see from Stern-Brocot tree etc. No points. Concerning points, we need to stick with Euclid and accept that a "point" is nothing but an end of a line - and the better term for meet of lines is 'vertex'. On a side note, Euclid's definition that line has no width is much less intuitive and much more problematic for coherent analysis and calculus. There can be no Hilbert-style point-reductionism in consistent and coherent approach - which already Zeno proved.

[@brendawilliams8062]

@@santerisatama5409 leave me be or i promise to temper tantrum and quit again. It takes a while to get back once propelled

[@brendawilliams8062]

@@santerisatama5409 it is all in how you want to count. If I want to do it differently. Why not?

[@brendawilliams8062]

@@santerisatama5409 1008 couldn’t jump to 3008 if one was all you added or subtracted. That accordion sings like a choir.

[@dreznik]

Dear Prof. I am a computer scientist. I can see (by dozens of other videos) you are particularly interested in this question. But may I suggest you may be bundling two questions in one: a) the decimal specification of a rational number such as sqrt(2) does require unbounded storage. As you said, in practice there are diminishing returns in engineering settings so one truncates such numbers at a certain precision. b) however, one can uniquely specify an irrational number as an ideal, e.g., as the solution to some equation, where the equation itself is represented finitely. can we treat this finite specification as a suitable representation for the irrational? Why is it unpleasant for us to accept that there are certain problems whose solution "transcends" finite numerical representation? If we were to challenge that, we would have to agree that the circumference of a circle does not exist (we know it can never be described to infinite precision, but is that tantamount to it not existing?) Can we simply continue to think of irrational numbers as "ideals"?

[@Almentoe]

Not a Finitist myself, but I would say that the example of a circle and pi is easily countered with "perfect circles don't exist". If you've watched some of the Profs videos, he doesn't like length as a concept, and does rational trigonometry with "spreads" & "quadrances" rather than angles & length, so the example of a circumference wouldn't bother the guy. Personally, because you must assume some axioms to start doing logic, might as well just accept some form of a ZFC-like system and go forth proving stuff from there. Mathematics is about logical frameworks, not just calculating things.

[@njwildberger]

@Dan Reznik, It is helpful to clearly distinguish between questions which have exact answers and questions which have only approximate answers. In real life, examples of each are " how many gazelles are at the waterhole right now?" and "what is the height of this elephant?" In mathematics, examples of each are: "how many primes numbers are there less than 20?" and " what is the length of the circumference of a unit circle?" Note however that the term "circumference" has two quite distinct meanings: one is as the actual boundary (I walked around the circumference of the cricket field) and the other is as the "length" of that boundary. It turns out that "lengths of boundaries" of circles is only an approximately defined property. There is no need to create a fanciful number system that cannot actually be encoded on a computer, just to be able to convince ourselves that "lengths of boundaries of circles" are exact quantities. We need to face the mathematical world as it REALLY is, not as we might like it to be. After you get used to that, you slowly realize that the way it actually is, is immeasurably more beautiful and elegant than any make-shift dreamings that us pure mathematicians are able to conjure up. I advocate clear-sighted recognition, and acceptance, of what we can actually do and not do, and to clearly separate the two.

[@santerisatama5409]

@@njwildberger The mathematical world as it "really" is, is a philosophical question. Formalist foundations deny validity of empirical arguments, they are not concatenated into coherence conditions of mathematical truth (hence, for an empirist, formalist foundation is post-modern post-truth theory of arbitrary language games) . Intuitionism demands intuitive, empirical coherence for linguistic constructions. Arbitrary if-then speculations of formalism can function as heuristics and game theories, but they can't form empirically coherent foundation of mathematics, and keep failing also in their adherence to bivalent logic. Zeno's proofs by absurdities are empirical proofs, as are many other basic proofs by comparing a more-less relation in a context of geometric structure. More-less relation is foundationally more mereological than numerical. Metaphysical postulation of a "unit" (aka existential quantifier) is less foundational that more-less relation implied in all measuring. However, there are some nuanced but very important differences between coherence theory of truth inclusive of empirism, and pragmatic theory of truth. Ability to "actually do" falls more into category of applied math, on the other hand pure math oozes potential. We can't actually prove undecidability of Halting problem, we deduce that from assuming that LNC is valid for temporal processes of computation.

[@sudippatra1289]

Dear NJ thanks a lot for these deep uncomfortable questions, much needed, I have one perhaps not that sensible question may be: what happens if the list of all questions which has yes / no ans is always indefinite? not infinite...

[@njwildberger]

@Sudip Patra, To be honest, I think that contemplation of "the set of all questions in French which have yes / no answers" is a topic of philosophy, and not serious mathematics, and not even serious CS. That is where a lot of people get confused; they imagine this kind of discussion is part of mathematics, especially when we start to "build real numbers" from "French questions" etc. To be clearer: unless a concept is completely and totally laid out, and supported by explicit examples (in their entirety --not just a few terms and then three dots!), then it really does not deserve to be part of pure mathematics. That also allows us to avoid obvious and not so obvious traps of self - reference, which are easy to fall into. For example, once we are "contemplating abstract questions" we can ask" Does what we are currently doing make any sense?" And one is quickly back to the land of the barber who shaves himself, or doesn't.

[@MrLordCoder]

That's a pretty cool number. But that are even natural numbers that can do almost the same feat. Take a list of all questions that use less than "number of atoms in the universe" words to write them. Order them lexicographically and answer them by yes (1) and no (0). Now put these digits in order and you have a natural number that contains all this information.

[@kyfegte827]

Can you post a link to Borel's 1952 publication? I can read French, so I may assist in interpreting the paper. In fact, rather than a paper it is a small book.

[@njwildberger]

@Ky Fegte I did find this on the internet, but not via a direct link. I am not confident enough about copyright to post it myself publically. But I assure you that you can find it if you look.

[@brendawilliams8062]

You can use 405 and 505 and work it down. I t seems possibilities beside scissor cuts. Thx

[@peng.montreal]

if one construct all french question sequence with russell's paradox flavour about the sequence, can there still be a label ?

[@قصصأمكمال]

تبارك الله عليك

[@christopherellis2663]

《May God Bless You 》he says.

[@njwildberger]

@@christopherellis2663 Thanks for the translation Christopher.

[@mimzim7141]

I get the feeling that in french the "nombres univers" which contain any finite string of decimals are more famous than the normal numbers "nombres normaux" which have any equal length string of decimals equiprobable. In english people tend to mention the normality criteria for a number that contains all possible finite information while the less strong "univers" criteria is sufficient. fr.m.wikipedia.org/wiki/Nombre_univers

[@felixgraphx]

When are tou going to start building a theoretical framework to compare algorithms for such irrational numbers such as sqr(2) pi, etc.?

[@LarryRiedel]

After he solves the halting problem

[@Magis-rt1lw]

"The set of all real numbers is a model of the physical reality". I believe we have been postulating this or a more accurate version of this statement, say for example, speed is just the derivative, the area of a region bounded by some curve defined by a well "behaved" function is the just the integral or measure in some sort. On one hand, would this postulate, if it really is, going to make scientific progress? Hence, I find this approach to the mathematics of reality interesting. On the other hand, we just have to use we currently have and hopefully, soon we may have a better "postulate". Or I could be wrong.

[@katakana1]

The know-it-all number does not exist, because boolean French questions should only be answered with "Oui", "Non", or "Sorry, I don't know French". If the answer is "Que signifie «oui» en anglais?" or "Que signifie «non» en anglais?", then these are the only two French questions that should be answered with "Yes" or "No". QED

[@taliesinbeynon]

So normality in your CS view would be exactly the statement thet the output of some e.g. Turing machine 𝑥ᵢ is maximum entropy with respect to whatever coarse graining : the distribution of individual symbols being the classical form of normality, taking blocks of 𝑛 symbols is then the entropy of the joint distribution 𝑝(xᵢ, 𝑥ᵢ₊₁, ..., 𝑥ᵢ₊ₙ) etc

[@njwildberger]

@Taliesin Yes this to me is the reasonable framework for discussing normality: in CS, not pure mathematics.

[@thomassynths]

@@njwildberger What about Lambda Calculus?

[@santerisatama5409]

@@thomassynths Schönfinkel's original combinators are even more interesting formal language than Lambda Calculus, and Wolfram has very nice articles about those. Computational reducibility being limited to addition and nesting is a foundationally very deep result. And I would say that nesting (cf. mereology) is even more foundational than the additive aspect on which numerical classical arithmetics is based on. The more I been digging, the more interesting and general formal languages seem to be than the standard numerical approaches of "classical" pure math.

[@dansanger5340]

The essence of real numbers seems to be that they are truly continuous and non-discrete in nature. Conceptually, I don't see anything wrong with saying a real number can take on any value between A and B. But then, when an attempt is made to create a rigorous theory of real numbers, it's made by describing them using sequences of discrete symbols that are coerced into being continuous by employing various levels of infinities. Maybe it's the inconsistencies smuggled in with this particular language of mathematics that's the problem. It seems like a graphical language rather than a formal, alphabet-based language could eliminate the problematic mapping, in the same way that Euclidean proofs were able to be done using drawings, and the incommensurable hypotenuse and circumference in the drawings weren't really a problem until they forced the issue by trying to get them to fit into the universe of rational numbers.

[@santerisatama5409]

The main problem with the formalist approach is Hilbert's frankly idiotic point-reductionism, which starts from going where Zeno showed is a no-go and trying to redefine Euclid's definition of 'point' exactly the wrong way. Whitehead became very aware of the problem, and started (again) to develop point-free geometry. Many of the misconceptions and confusion arise from calling vertices of coordinate systems "points". That said, it's very much possible to construct formal language for continuous and non-discrete phenomena, based on the general idea of open interval.

[@elcapitan6126]

"any value" being the slippery slope... what do we consider values to be? algebraic constructions? unbounded sequences defined by certain algorithms / processes? something waay more exotic and basically unreal (real numbers?)

[@KipIngram]

I think we could write a program that would work in any case that was "real," and by that I mean anything other than deliberately designed pathological cases. Just start with N the number of digits you want, and then start looking at the digits to the right in both numbers. As soon as you find a pair of digits, in the same column, that sum to 8 or less, start there. Even if the remainder of the numbers sent a carry into that column, it still would send a carry out. I understand this won't satisfy you, because it won't handle the pathological cases. But it would get our jobs done.

[@eternaldoorman5228]

20:23 On being able to write down the questions in some determinate order, ... that's fraught, isn't it?... You need to be able to decide which questions actually mean the same thing, don't you? As this was in the 1920's he is aware of entscheidungsproblems because Hilbert talked about them in 1901 or sometime, ... I should look this up, and I will come back here if I find anything that seems especially interesting ...

[@KipIngram]

Hmmm. Could we not prove that a number that was strongly normal in any base would be strongly normal in all bases? It just feels like that should hold.

[@JeremyNasmith]

Scaling Borel's know-it-all number up by a factor of 100 brings it remarkably close to 42....

[@vladartemy892]

It seems to me that the "know-it-all" number is a rational number. Otherwise, it means that our knowledge is infinite (because we can ask infinite many correctly formulated questions). If it were true then we could handle any numbers without problems, couldn't we?

[@randyzeitman1354]

Why does capacity to ask mean a solution exists?

[@lukasm5254]

This know it all number needs more explanation with respect to which questions are allowed. Because my gut feeling is there are uncountably many yes no questions but any number would only have countably many digits. So you could construct a question that is not answered.

[@myname1982]

I don't understand why you say you cannot multiply infinite decimals. Suppose we define an infinite decimal as a function f: ℕ→ℚ (say it only outputs truncated decimals). Isn't f also a Cauchy sequence? Further, given a Cauchy sequence we can come up with (i.e. using a terminating program) an infinite decimal and vice versa. So if we can multiply Cauchy sequence (which we most certainly can), we can also multiply infinite decimals. What is wrong with this argument? What we cannot do compare two real numbers (irrespective of which of the three definitions we use), i.e. it is not decidable

[@myname1982]

I am assuming here that f is a computable function, which is also the setting you laid out. And when I say given a Cauchy sequence, we can produce an infinite decimal, I'm assuming that is also specified using a computable function. This assumption is acceptable for the argument I laid out above.

[@adolfninh23]

Analytic Zeno paradox

[@christopherellis2663]

This tells me that it's an even bet that some academicians are more than slightly odd. 😃Are there any aardvarks in Antarctica? Non!

[@whig01]

This makes me think of Kolmogorov complexity and Markov models.

[@theoremus]

Is this what mathematicians discuss at a cocktail party, after a few martinis?

[@tjejojyj]

I thought the current hypothesis was that after enough martinis (between 12 and "infinity") every discussion between mathematicians will contain the phrase "Well the finitists, like Norman Wildberger, would say that, wouldn't they?"

[@thomassynths]

Ah, a good ole Definable but not Computable "real" number. All these real number constructions have definitions on how to construct their elements. But if a number is not Definable (as with most real numbers), how can you construct them? Because if you could construct it, then it would be defined... contradiction! Why doesn't mainstream math introduce the "axiom of constructing undefinables"? This is a huge hurdle for me with any "constructed" uncountable set.

[@vladartemy892]

Looks like the mathematicians did it - see ZPF set theory axioms. There are a couple axioms about the set construction.

[@thomaskember4628]

I suggest you need to be careful with Google Translate. I used it translate a German novella sentence by sentence. Almost every time it came up with rubbish, sometimes hilarious so. I had to make changes to the English text to make it readable.

[@njwildberger]

@Thomas Kember Yes I have already experienced that. But being Canadian, I did learn French in school for many years (on paper at least). So I can use Google Translate as a check on words that I can't remember, which is very helpful.

[@randyzeitman1354]

This is exactly why free will is the ability to distinguish infinity.

[@blakewilliams1478]

I'm not sure what is gained by bringing up the Know-It-All number. It is a pure novelty which is absolutely devoid of content. Of course you can imagine such a thing, but it's not as if it actually encodes any information, all the information is encoded in somehow knowing that this is the correct k for your choice of ordering, which is of course completely impossible to verify. It says nothing about the validity of the real number system and I view invoking it as a bit of a bad faith argument intended to make the listener uncomfortable on a strictly superficial level.

[@njwildberger]

@Blake Williams, Your questions might have been better directed at Borel, as he was the one who came up with this scenario. But a good guess is that he did so exactly to make people uncomfortable with the prevailing blase attitude to the "reality" of the "real numbers". And I happen to think his example is great. It shows just what is possible with the seemingly trivial assumption that we "can do an infinite number of things". Sure -- like consider answers to "all possible questions" etc. Is this what we want mathematics to be about? How about instead restricting ourselves to talking about things which actually make complete sense in this world, not in some imagined parallel universe with vastly different attributes?

[@royaarefiyan6068]

We live in a universe from from minus infinity to plus infinity so this is not true and real number existed. U can change its name but not their existence!!!!!

[@antonioILbig]

You shouldn't care if real numbers have some philosophical issue or not, it's just a model we use for describing things, it doesn't matter wheter they exist or not.

[@robharwood3538]

If they don't exist, then how is it a model for describing things? Sounds like angels dancing on the head of a pin to me.

[@normanwildberger8190]

@Antonio Fusillo That's just how I feel about the leprechauns at the bottom of my garden, who are very adept at modelling all manner of curiosities.

[@santerisatama5409]

Materialistic-reductionisc physicalism based on formalist theory of mathematics fails because the mathematical theory is anti-empirical and counter-intuitive post-modern post-truth gibberish.

[@krisdabrowski5420]

Hi Professor Wildberger, I am working on a project to formulate an Orthodox Christian foundation for philosophy of mathematics, and would like to have a private discussion with you on this topic, since I am a fan of your work and your critical approach. If you are interested, could you let me know how I would be able to reach you to have a discussion? I already sent an email to your university email, but since you're retired, I now realise that email might be out of commission.

[@Velzen5]

Borel's know-it-all number cannot exist. Questions of the form Is position x in the binary number that represents the real number Pi equal to 1 would exhaust it's caoacity and leave no room for any other question,

[@robharwood3538]

In the theory of 'infinite decimals', there is always room for more questions. A better reason that it cannot exist is along the lines of Russell's paradox or Goedel's incompleteness theorems: "Does the know-it-all number have a 0 in the digit whose index represents this question?" If the answer is "Yes" (that the number does have a 0 there) then the number should have a 1 there (since the answer is "Yes"). If the answer is "No" (that the number does *not* have a 0 there) then the number *should* have a 0 there (since the answer is "No"). In neither case can the number have the correct answer to that question. Therefore, the number cannot exist as a fixed number.

[@kvazau8444]

@@robharwood3538 This is not a problem. The know-it-all number presupposes that a question has a definite binary answer. Paradoxical questions are not admitted to the set, by definition (they are not "yes or no" questions, despite their structure suggesting that they would be, if well-formed).