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Crisis in the Foundation of Mathematics | Infinite Series

Crisis in the Foundation of Mathematics | Infinite Series


[MUSIC PLAYING] Mathematics is constructed
from other math, which is constructed from
other math, and so on, but what happens at the bottom? What grounds all of mathematics? Or is it just turtles
all the way down? [MUSIC PLAYING] Mathematics is cumulative. It builds on itself. That’s part of why you take
math courses in a fairly prescribed order. To learn about matrices–
big blocks of numbers– and the procedure for
multiplying matrices, you need to know about numbers. Matrices are defined in
terms of– in other words, constructed from– more fundamental
objects– numbers. For this reason, it’s
not uncommon for people to view mathematics
as a giant pyramid, with objects and
concepts constructed from things below them. Let’s pick as a concrete
example four different types of numbers– real numbers, rational numbers,
integers, and natural numbers. Remember that the natural
numbers are all the counting numbers, usually
taken to include 0, so 0, 1, 2, 3, 4, and so on. Building up the pyramid
from the natural numbers, we can define the integers
as being the result of subtracting natural numbers. They’re all the
possible things you can get by subtracting one
natural number from another. So 3 is an integer because
it’s 3 minus 0 and negative 3 is an integer because
it’s 0 minus 3. More precisely, we can
define each integer as an equivalence class,
or a particular collection of pairs of natural numbers. For example, the integer 3 is
the collection 3, 0, 4, 1, 103, 100, and so on, and
the integer negative 3 is the collection 0, 3,
1, 4, 100, 103, and so on. Under this definition,
the integers also inherit their
structure and properties from the natural numbers. For example, the integer
associated with x, y is less than or equal to the
integer associated with z, w if x plus w is less
than or equal to z plus y. The order on the
integers is defined using the order on the natural
numbers and the integer associated with x, y plus the
integer associated with z, w is equal to the
integer associated with x plus z, y plus w. Integer addition is defined
using natural number addition. You should check
that this structure aligns with your intuition
about the integers. We’ve constructed the integers
using the natural numbers, and then from the integers, we
can build the rational numbers. They’re all the
possible things you can get by dividing one
integer by another, like 1/2, 7/512, or negative
3 divided by 1. And similar to the previous
construction of the integers, we can define the properties
of the rational numbers in terms of the properties
of the integers. One step further, we can also
construct the real numbers from the rationals, but
it’s a bit more tricky, using something
called a Dedekind cut. A Dedekind cut splits the
rational numbers into two sets. One set is made up of all the
rational numbers below the cut and the other set is made up
of all the rational numbers above or at the cut. Let’s make the cut here. All the rational
numbers less than 3 are in one set and all the
rational numbers greater than or equal to 3 are
in another set. This cut, or way of
splitting the rationals, is identified with the
smallest rational number in the upper set, which is 3. The cut defines
the real number 3. Notice that the lower set
doesn’t have a biggest element. There is no rational
number immediately below 3. For any rational number
below 3, for example, 2.9, there will always be another
rational number between that and three, like 2.95. Let’s define a new cut. The lower set is all
the rational numbers x such that x squared
is less than 2 and the upper set is
all the rational numbers y such that y squared is
greater than or equal to 2. In this case, there
is no smallest element in the upper set. It would be the square root
of 2 but that’s not rational. This Dedekind cut defines
the number square root of 2. This particular way to
split the rational numbers can be thought of as the
definition of the real number, square root of 2. The real numbers are defined
as all the possible ways to cut or split the
rational numbers. The cut is identified
with either the smallest element in the upper set,
if one exists, like 3, and otherwise, it’s the
gap between the two sets, like the square root of 2. Starting with the
natural numbers, we subtracted them
in all possible ways to get the integers. Then, we divided the
integers in all possible ways to get the rationals. Then, we split the rationals
in all possible ways to get the real numbers. The structure and properties
of each type of number can be defined in terms
of the previous type. In this way, all
possible statements about the real numbers, like
pi is less than e squared, can be reduced to statements
about natural numbers. Mathematical objects
and their properties– the things we prove
theorems about– are defined in terms of
other mathematical objects and their properties. Math is built out of simpler
math, hence the pyramid idea, but where does this process
of simplification end? In other words, what holds
up the bottom of the pyramid? In the late 1800s
and early 1900s, this was a real crisis
for mathematicians and philosophers. Mathematics has no foundation. Gottlob Frege and
Bertrand Russell were very worried about
mathematics’ seeming lack of a foundation. Along with Richard Dedekind
of the aforementioned Dedekind cuts, they founded and advocated
for a philosophical position known as logicism,
as in logic plus ism. Basically, logicism says that
the bottom of the pyramid of mathematics is logic. Mathematics is founded in logic. Essentially,
mathematics is logic. In Russell’s words,
the goal of logicism is to show that all
pure mathematics follows from purely logical premises
and uses only concepts definable in logical terms. Just as we reduced
all statements about the real
numbers to statements about natural
numbers, the logicist wants to reduce the natural
numbers and everything else to logic. This leads to some
obvious questions, like what exactly is logic? Logicism is a
philosophical position and the intended meaning
of the word “logic” is fundamentally philosophical
and not mathematical. The definition is difficult
to pinpoint and different for each logicist, but
mathematician Ernst Snapper writes that, generally,
the logicist thought that “a logical proposition
is a proposition which has complete generality and
is true in virtue of its form rather than its content.” For example, the law of excluded
middle, for any proposition P, either P or not P. Logic should feel simple,
natural, and never ad hoc. That’s what allows it
to ground mathematics. To sit at the base
of the pyramid. In the late 1800s, Gottlob
Frege became the first person to earnestly attempt to carry
out the logicist project. He spent years developing
an extensive system of logical axioms and notation– a foundational system
from which he derived the basic laws of arithmetic. The project seemed to
be a logicist success until Bertrand Russell
rather famously ruined it. Just as Frege’s book
was going to press, Russell pointed out
that Frege’s system contained a contradiction. Using the Basic Law V, one
could derive Russell’s Paradox– the set of all sets that
do not contain themselves. Even though Russell
witnessed the specific errors in Frege’s work, he was
inspired by its central goal– to give mathematics
a logical foundation. Together with Alfred
North Whitehead, Russell continued to
push the logicist agenda in their three-volume
“Principia Mathematica.” The work succeeded in reducing
large sections of mathematics to an axiomatic
system and famously exhausted the first few hundred
pages proving that 1 plus 1 equals 2 but not all the axioms
they used were pure logic. This is essentially
the same status as mathematics’ most
well-known axiomatic system, Zermelo-Fraenkel Set Theory. From the axioms
of ZF Set Theory, even without the sometimes
controversial axiom of choice, one can derive most of
classical mathematics. That is, you can reduce most
known math to those 8 or 9 axioms– a logicist success. We now know, thanks to
Goedel, that we’ll never have an axiom system which
produces all of math. It’s impossible to prove
the ZF axioms will never produce a contradiction but
none have been found so far– another good sign. Unfortunately, some of the ZF
axioms cannot be considered pure logic. For example, the
axiom of infinity, which asserts that there
exists infinite sets, cannot reasonably be considered
an axiom of pure form. It asserts something
about content. Something more than basic logic. So, as Snapper jokes,
since at least two out of the nine
axioms of ZF are not logical propositions in
the sense of logicism, it is fair to say that this
school failed by about 20% in its efforts to give
mathematics a firm foundation. Logicism had some
successes and huge portions of modern mathematical logic are
the historical or mathematical consequences of the crisis about
the foundations of mathematics. We’ve never really
figured out what’s at the bottom of the
pyramid, but along the way, we’ve discovered a ton of
other fascinating mathematics and philosophy. Hello. You all had a lot
of awesome responses to our episode on
pseudo random numbers and a lot of your comments
had to do with what the nature of randomness is. Where does it come from? And Paradoxically
Excellent said, “I believe I’ve heard
that the digits of pi pass most statistical
randomness tests. And they never repeat. But they are also not random.” That’s a really
interesting point. Maybe they’re pseudo random. They do pass, or we
believe that they pass, a lot of tests for
randomness, and one of those is that we believe pi is
a normal number, which really means it’s just digits
are distributed very evenly. They’re very spread out. All the digits and all
the sequences of digits are equally likely. And we talk about that
a lot in our episode “Combining Pi and E,”
so check that one out. Pan Raphael ask
a great question. “Can’t humans come up
with random numbers?” It also really gets at the
question, what is randomness? And there were also
some awesome responses. So Nathan Rasmussen
says that he teaches a class in cryptography,
and basically, if you ask people to come
up with a random number, there’s only certain numbers
they’re going to come up with. And that’s kind of
a fun experiment. Maybe any of you who are
teachers want to give it a try. A related version that
Ari Reynolds brought up that I’ve actually
tried in class is that if you ask people
to write a sequence of 100 random coin flips–
so say, pretend to flip a coin 100 times and
write heads, tails, heads, tails, over and over and
over again 100 times. And then you have other people
actually flip a coin 100 times and write the sequence out. They look very different. People are afraid to write,
let’s say, five heads in a row. They think that that
doesn’t seem random. But if you actually
flip a coin 100 times, it’s a pretty good
chance that you’ll get something like five heads
in a row or five tails in a row. It’s not that unlikely. And so people write down
very different-looking random sequences than
the actually random ones. And that’s a really
fun experiment to run in the classroom. And in another comment, Co
Phillips linked to a website where you can click the
keys F and D randomly and it will try to guess which
one you’re going to click next. And it’s pretty good. You can see how random you
can make your sequence. Whether it can interpret
any patterns or not. It guessed my next key
about 60% of the time, so I couldn’t really beat
it, but maybe you can. We’ve linked to the
website in the description. It’s really fun. Check it out.

Comments (100)

  1. All mathematics is is a bunch of made up stuff. A thought experiment that makes lots of real-world tasks easier. Since it is all man-made, it is no wonder that issues arise. But until one of these issues results in me only being given $99 in change when I break a hundred-dollar bill because…. "MATH" then nobody should care. Want to be good in math? Learn math. It isn't difficult, it is just not something you will need to use unless you feel like using it. Same as going off and randomly learning how to speak Mongolian. Who the hell cares.

  2. If the universe is infinite then do numbers really exist at all?

  3. There were actually three crises in mathematics. The first was the discovery of irrational numbers. This occurred during the time of the Ancient Greeks. They could not understand how two numbers could be incommensurable (how a ratio of two numbers could not be expressed as a fraction of integers). The Pythagoreans considered this a disaster and kept it a secret. Carl Sagan discussed this in "Cosmos." The second crises came about at the end of "the age calculus." The advance of calculus after Newton, during the time of Euler and Lagrange was rapid, but was not rigorous by current standards. The use of differentials to prove things bore fruit, but sometimes gave wrong results. Richard Courant discusses this explosion in knowledge, acknowledging that this rapid growth would have been severely impeded had extreme rigor been needed for mathematical proof. But this rapid inflation in knowledge did often lead to wrong results. So we payed a price – we got a great amount of knowledge, but with bugs in the system. This caused a reinvention of calculus, started primarily by Cauchy (and deeply advanced by Weierstrass) – the basis of calculus was taken from geometry and put into arithmetic – starting the age of "analysis." The last crisis, the third crisis, was the most serious – the one you mentioned in this video.

    As far as PI, I had read that the distribution of digits is actually not uniform. I think the digit 6 showed some lower frequency than the other numbers.

  4. Isn't it axioms basically ?

  5. What is the host's name?

  6. The hardest question in mathematics? "How many Yo Yo dogs?" And yet… any answer is acceptable under common core.

  7. digits of pi random? ? try google Plouffe spigot algorithm…

  8. I didn't understand what she said because I'm not good at math but she has gorgeous hair.

  9. I would have liked to hear some contributions to math foundations made by the Bourbaki school too. Great video madam!

  10. I was terrified when I first read about this years ago, and in a way I still am.

  11. About randomness, my professor once noted there is a few people that after 10k hours training did get to generate random numbers. But it really requires extensive training.

  12. She does a lot with her hands

  13. Integers seem to be another name for natural numbers.

  14. And then, what is logic founded on? Probably the concept of order and structure. But philosophers use to take logic for granted, and you never read this notion in philosophical literature.

  15. I always come back to this channel. Really love those math videos 😋

  16. Why is everyone talking about turtles?

  17. Happiness = Reality – Expectation

  18. Waddya mean math has no foundation? Bull. You got the pyramid upside down.

  19. This just furthers my delusion that everything is subjective.

  20. TRUE and FALSE – that is the very foundation of logic. From that, we can build circuits that performs mathematical computation.

  21. We are free to make any definitions we like. These defs are jusr word and symbol salad.

  22. _yes yes is crisis in mathematics_sign transsinphinite number theory_archaix lord sith

  23. I think the paradox is solvable.

    So let's name the "set of all sets that are not themselves" "n". What I want to propose now is that n can be simultaneously part of n and exist outside of n. For this we'll need two separate notions:

    1. The paradox's issue is that the number it generates is seemingly variable. This means that the paradox even if still existent is a non-issue as long as we can explain this variation within the realm of numbers and math (without inventing new rules)

    2. N has a double notion within the paradox, one in which it encompasses n and one in which it doesn't, however they provoke one another, thus making it impossible for them to exist separately.

    So, now let's say for a moment that N is not a part of itself, that would make it a part of itself, thus making it so that n+n=n, BUT n+n+n=n and n+n+n+…+n=n. The first instinct is to see this as (X-1)n=0, which would make n=0. Issue is we know some sets that are not part of themselves must exist, in fact most of them are. As a result, the number must be positive and different from 0, this is, n>0. So what number could possibly fit? Well, maybe not a number, but a numerical value called "infinite positive".

    Infinite positive is basically the notion of an ever-growing number, and the thing about it is that it works within our equation. +infinite++infinite=+infinite. Infinite is a variable notion, ever-variable in fact, but so is the "set of all sets that are not themselves", as every separate set not contained within itself generates a new "set of all sets that are not themselves" which in turn does contain it, thus increasing the extension perpetually.

  24. These math people need to get out and get a real job! But its a great gig if you can find people to to pay you to do it…

  25. Randomness comes from entropy

  26. Since subscribing to your Chanel, I find myself picuring you explaining thing to me when I study maths for an examn or had to proove soume engineering theory mathematicly. You restored my love for maths! Thank you for that!

  27. OR IS IT TURTLES ALL THE WAY DOWN LOL I caught that I see you.

  28. i know a few stuff that aren't random: exponents: powers of n and perfect powers and fibonacci sequence.

  29. Warren Ambrose called all this stuff a "can of worms". Lol you start with a consensus point of view and some real mathematics. How about an ultrafilter and hyperreals?

  30. Is it correct to say if I had a number system not in base 10, but in base Pi that Pi would be rational?

  31. I learned that the natural numbers don’t include 0, rather that the whole numbers do

  32. She keeps saying pyramid, but that's an inverted pyramid, no?

  33. I just want to give her a hug.

  34. Let there be 'x'….thus the world begins and all confusion….

  35. Why tf am I watching this I already figured this out. I'm in calculus 4 and differential equations.

  36. Gödel likes this video.

  37. Turtles. Definitely turtles.

  38. The basics of it seems to be axioms, definitions, and undefined terms. Axioms are something that cannot be provide just something we assume to be true.

  39. Math's was all good but one day algebra came everything spoiled

  40. This definition for negative numbers is atrocious.

  41. For the website mentioned in the endnotes, the way I beat (got a 50% guess score for large set) it was by randomly selecting large even numbers and if it was divisible by four, D otherwise F: this worked really well in letting me select a number an outcome randomly by obfuscating the outcome from my brain during the selection process: until of course my brain picked up on things well enough to know whether my number would translate to f or d before the number was fully generated.

    My anecdotal conclusion is that creating random outcomes is easier if you overload your brain's pattern recognition: the catch is that doing so will make the generation process slower.

  42. I quit the video at 1:03

  43. Jokes on you. There is no spoon.

  44. I started adding up an infinite series. I'm not done yet.

  45. What about Peano's Postulates?

  46. Natural number does not include zero! Integer does.

  47. counting is the base.

  48. The foundation of math is not logic, it is the observation that 1+1=2. Even logic is fundamentally built on that observat

  49. i don't like 0 considered as natural number

  50. Women and Math. 😁
    She is a kick in the balls of anyone who dares to say "they cannot".

    Deal with it. 🖕

  51. Is anything possible in an infinite universe? And if so is it possible to have an infinite straight line?

  52. The set of natural numbers doesn't include 0. Natural numbers start from 1. Whole numbers include 0 and natural numbers

  53. So where's the crisis? and whats ur definition of crisis? Cuz it sure ain't mine…

  54. ___yes yes ….sign transsinphinite number theory ….1/3+1/3+1/3=3/3=1 ad 1/3=0,(3) and 0,(3)+0,(3)+0,(3)=0,(9) result 1=0,(9) …it s error numeric callcul…in reality 0,(9)=0,(9)9 with n indice period =infinti and n-1 period …the language nume give transsinphinite number theory is for and because united mathematic numeric operative is situated after inphiniti period…if not accepted transsinphinite number theory is not definit assimptotouse and theorem of darbouxe becam invalited and result imposibille definit derivat ad integrall and analisis mathematics becam insolvable think s innacepted for science mathmatics …the transsinphinite number theory is following for definit space and subspace for for continuumm space time for music for programing corect soft computer and radio and tv hd for callculus financial and etc etc___archaix lord

  55. 00:59 Remember that the natural numbers are all the counting numbers, usually taken to include 0, so 0, 1, 2, 3, 4, and so on.

  56. The title scared me. I was like "Wait, what? Is math itself in danger somehow?"

  57. Sorry. I have and use matrices with variables, it is algebraic matrices, and even other structures.
    They may started with numbers but that is a special case.

  58. #adastra #iamhyperian #projectfallenstar

  59. Real Analysis.

  60. Yikes https://chrome.google.com/webstore/detail/threelly-ai-for-youtube/dfohlnjmjiipcppekkbhbabjbnikkibo

  61. G
    Gr
    Gre
    Grea
    Great
    Great j
    Great jo
    Great job
    https://m.youtube.com/channel/UCLoo2mkhq9oFNlGqwqRW2bA/feed
    https://m.youtube.com/channel/UCLoo2mkhq9oFNlGqwqRW2bA/fee

  62. Well there's obviously not going to be a foundation. Math is fictional.

  63. It's a shame you simply glossed over Godel's incompleteness theorem- profound and incredibly important to the Hilbert program to establish foundations

  64. 8:13 is the wrong principia mathematica https://en.wikipedia.org/wiki/Principia_Mathematica vs https://en.wikipedia.org/wiki/Philosophi%C3%A6_Naturalis_Principia_Mathematica

  65. Is there a bottom?

  66. 0<1
    0.4
    0.41
    0.414
    0.4141
    0.41414

  67. But even logic is built from something else.

  68. I gave this video a like. However, I feel that some of the concepts here deserved better than the somewhat perfunctory treatment they got. I know U guys can do better! 😉 tavi.

  69. Speak with confidence!!!!!

  70. If every real number is a cut between two rationals. How can there be so many more real numbers than rationals?

  71. Its a groundless network, just like any language.

  72. To pee or not to pee.

  73. Just stop waving your hands, gestures are overused only by low IQ people to conpensate their speech communication

  74. Too cute mathematics

  75. 0 isn't natural number

  76. It looks like they pulled up an image of Newton’s Principia, not Russel’s & Whitehead’s lol.

  77. Patterns is random phenomenon too. Base of mathematics maybe is mind, and acsiom what mind exist is true becouse it are, of fact that we know what acsiom exists, we know existace, and it is number I, some 1.

  78. By proper experiment, pi=3.14159 is incorrect. Series is not the correct method to figure out arc lengths. After the perimeter is devided into tiny angles/fractions, the triangle's base line, sinX, tanX and even the "L" AKA [ sinX + (1-cosX) ] will approach the same pi value AKA the trangle's base line length.

  79. Please create a video about the definition of "pretopological space".

  80. Technically, ZFC is an infinite collection of axioms. The assumption that ZFC can be replaced by a finite number of axioms leads to a contradiction.

  81. The real crisis is that Infite Series ended.

  82. Brouwer solved this problem of "what is at the bottom of pyramid" with his intuitionism. Mathematics is creation of human mind and the laws which it is build upon are "intuitions" which are obviously true based on perceived self-evident facts about existence and the way consciousness interacts with reality. These basic laws are just things the could not be otherwise because if it was, we would get nowhere. It's possible to use different sets of axioms and even sometims use higher level concepts (like line in Eclid's synthetic as oposed to analytic geometry) and build further mathematics from this point.

  83. What do you think of this story, 32 digits of pi in an ancient vedas. https://www.indiadivine.org/value-pi-upto-32-decimals-rig-veda/. I saw a way of looking at it in Laws of Form. The "distinction" …. are you "in" or "out" of the set boundary.

  84. At the end of the day.. someone will always hate it after class..

  85. I wonder if this leads back to Godel's incompleteness theorem.

  86. Try mentioning Godel to a particle physicist, the next time they mention the quest for a GUT.

  87. Bertrand Russell was actually on the Germans' lock-up list for after they conquered Britain (I know, I know. They didn't succeed. But they still made the plans). Maybe they were pissed at him for disagreeing with their boy Gottlob.

  88. True whole mathematics is truly too simple (to clever school students level mainly), but what actually does complicate it to turn into many huge businesses and useless branches that always split the human so innocent minds are actually those many alleged most historical and living genius academic professional mathematicians themselves

    Simply because they had never tried to really understand the oldest and most famous historical unsolved problems raised strictly by the ancient Greek (few thousands of years back) and so, unfortunately, up to our current dates so deliberately for the sake of that baseless and huge volume of so unnecessary business mathematics, for never wanting to understand first the following three proven facts about the three unsolved Greek problems (in very and many elementary methods most suitable for school kids)

    1) No circle ever exists, but regular EXISTING polygons with many sides that seems like a circle to the innocent human minds
    due to the very limited ability of visibility of the human eyes and minds as well where a compass and straight unmarked edge can't truly construct a circle, nor any other tool can (by any means), since it doesn't exist physically, conceptionally and theoretically as well

    2) Pi*, therefore, is never any constant number, but truly a varying real constructible number belonging solely for regular existing polygons where the regular existing polygon with a maximum number of sides never exists, hence the well-known REAL number *Pi for circles is, in fact, No existing real number on the real number line as an exact existing and well described distance , where it follows immediately the impossibility of squaring the circle BY ANY MEANS, hence, (problem one Solved)

    3) Pi is actually an exact existing angle and never any real number, where two-thirds of the well-known angles in both old and modern mathematics never exist, like all integer degrees that aren't divisible by 3, and angles like (Pi/7, Pi/9, Pi/11, …) named as non-constructible angles in modern math, where this only explains and proof the impossibility of trisecting the arbitrary angles LIKE (60 Degrees) (BY ANY MEANS)
    , hence (problem 2 Solved)

    4) The real cube root of two denoted by $sqrt[3]{2}$ or [2^{1/3}] IS never any existing real number on the real number line and hence impossible to exactly construct (by any means) since it never exists, but was merely a human-invented like a number, which explains and proof the impossibility of doubling the cube problem, where those old unsolved and most famous historical problems raised strictly by the Greeks had been completely solved

    There are indeed many more THRILLING issues that are too shocking to hear about from an amateur Civil Engineer, and they were all published PUBLICALLY on sci. math immoderate site or SE and Quora (since I'm not a professional mathematician beside those issues are generally forbidden and fought to talk about their facts in any official place for mathematics by the Professionals academic mathematicians since they contradict their own common global education)

    At any case, and for future uprising researchers to verify the facts about any of my many PUBLISHED claims, one of my main public profiles at this link, where one (if interested) can search the suitable topics raised strictly by myself in my profile

    https://groups.google.com/forum/#!profile/sci.math/APn2wQc6Nm-moZ-ytlbuL3jhGbMgub4xj3tyrWvTKOfJGYuut-WbyhFBV06rT-9BkhR3R16XqllA

    I hope to keep visible my comment please for near future historical documentation purposes and many benefits of others as well

    Thanks and Regards

    Bassam Karzeddin

    Oct. 8th, 2019

  89. My math professors never included 0 in the natural numbers. Is including it an American thing or something? You Yankees do everything differently.

  90. Why don't we make an axiom that says "sets cannot contain themselves and statements cannot be self-referential?" That solves all the stupid self-reference paradoxes.

  91. 8:12 That's Philosophiæ Naturalis Principia Mathematica, made in large part my Sir Isaac Newton… way before the 20th century… Russell's books is simply called "Principia Mathematica." It has a white cover without the Philosophiæ Naturalis bit in the title. You can even see in the image you used "Autore S. Newton."

  92. Im sorry I cant listen to a young lady talk about deeper mathematics subjects

  93. A random sequence of letters:
    safghlasfgljsghasrandomnessasfkglasfhfglashfgmakeshasfpifgasosfkgoasgeverythinglsafjgljashfgpksnfpkgjsodjg

  94. I would not expect any old concatenation of terms to produce a valid statement. Surely Russel's paradox boils down to an invalid statement equivalent to A=NOT A?

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