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    "segs": [ {
      "utf8": "Consider the following sentence:\n“This statement is false.”"
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    "segs": [ {
      "utf8": "Is that true?"
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  }, {
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    "dDurationMs": 2375,
    "segs": [ {
      "utf8": "If so, that would make\nthis statement false."
    } ]
  }, {
    "tStartMs": 14538,
    "dDurationMs": 2291,
    "segs": [ {
      "utf8": "But if it’s false, then the statement \nis true."
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  }, {
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    "dDurationMs": 5292,
    "segs": [ {
      "utf8": "By referring to itself directly, this \nstatement creates an unresolvable paradox."
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  }, {
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    "segs": [ {
      "utf8": "So if it’s not true and it’s not false—\nwhat is it?"
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    "segs": [ {
      "utf8": "This question might seem \nlike a silly thought experiment."
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  }, {
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    "segs": [ {
      "utf8": "But in the early 20th century,\nit led Austrian logician Kurt Gödel"
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  }, {
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    "segs": [ {
      "utf8": "to a discovery that would change\nmathematics forever."
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    "segs": [ {
      "utf8": "Gödel’s discovery had to do with \nthe limitations of mathematical proofs."
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    "segs": [ {
      "utf8": "A proof is a logical argument\nthat demonstrates"
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  }, {
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    "segs": [ {
      "utf8": "why a statement about numbers is true."
    } ]
  }, {
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    "segs": [ {
      "utf8": "The building blocks of these arguments\nare called axioms—"
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  }, {
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    "segs": [ {
      "utf8": "undeniable statements \nabout the numbers involved."
    } ]
  }, {
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    "segs": [ {
      "utf8": "Every system built on mathematics,"
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  }, {
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    "segs": [ {
      "utf8": "from the most complex proof \nto basic arithmetic,"
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  }, {
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    "segs": [ {
      "utf8": "is constructed from axioms."
    } ]
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    "segs": [ {
      "utf8": "And if a statement about numbers is true,"
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  }, {
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    "segs": [ {
      "utf8": "mathematicians should be able to confirm\nit with an axiomatic proof."
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    "segs": [ {
      "utf8": "Since ancient Greece, \nmathematicians used this system"
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  }, {
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    "segs": [ {
      "utf8": "to prove or disprove mathematical claims\nwith total certainty."
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    "segs": [ {
      "utf8": "But when Gödel entered the field,"
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    "segs": [ {
      "utf8": "some newly uncovered logical paradoxes\nwere threatening that certainty."
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    "segs": [ {
      "utf8": "Prominent mathematicians were eager\nto prove"
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  }, {
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      "utf8": "that mathematics had no contradictions."
    } ]
  }, {
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    "segs": [ {
      "utf8": "Gödel himself wasn’t so sure."
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    "segs": [ {
      "utf8": "And he was even less confident\nthat mathematics was the right tool"
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  }, {
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    "segs": [ {
      "utf8": "to investigate this problem."
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    "segs": [ {
      "utf8": "While it’s relatively easy to create \na self-referential paradox with words,"
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  }, {
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    "segs": [ {
      "utf8": "numbers don't typically\ntalk about themselves."
    } ]
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    "segs": [ {
      "utf8": "A mathematical statement is simply \ntrue or false."
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    "segs": [ {
      "utf8": "But Gödel had an idea."
    } ]
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    "segs": [ {
      "utf8": "First, he translated mathematical \nstatements and equations into code numbers"
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    "segs": [ {
      "utf8": "so that a complex mathematical idea could\nbe expressed in a single number."
    } ]
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    "segs": [ {
      "utf8": "This meant that mathematical statements\nwritten with those numbers"
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  }, {
    "tStartMs": 127204,
    "dDurationMs": 4459,
    "segs": [ {
      "utf8": "were also expressing something about \nthe encoded statements of mathematics."
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    "segs": [ {
      "utf8": "In this way, the coding allowed\nmathematics to talk about itself."
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    "segs": [ {
      "utf8": "Through this method, he was able to write:"
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  }, {
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    "segs": [ {
      "utf8": "“This statement cannot be proved” \nas an equation,"
    } ]
  }, {
    "tStartMs": 142746,
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    "segs": [ {
      "utf8": "creating the first self-referential\nmathematical statement."
    } ]
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    "segs": [ {
      "utf8": "However, unlike the ambiguous\nsentence that inspired him,"
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  }, {
    "tStartMs": 150913,
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    "segs": [ {
      "utf8": "mathematical statements must be \ntrue or false."
    } ]
  }, {
    "tStartMs": 154579,
    "dDurationMs": 1500,
    "segs": [ {
      "utf8": "So which is it?"
    } ]
  }, {
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    "segs": [ {
      "utf8": "If it’s false, that means the statement\ndoes have a proof."
    } ]
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    "segs": [ {
      "utf8": "But if a mathematical statement has\na proof, then it must be true."
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  }, {
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    "segs": [ {
      "utf8": "This contradiction means that Gödel’s\nstatement can’t be false,"
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    "segs": [ {
      "utf8": "and therefore it must be true that\n“this statement cannot be proved.”"
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    "segs": [ {
      "utf8": "Yet this result is even more surprising,"
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    "segs": [ {
      "utf8": "because it means we now have \na true equation of mathematics"
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    "segs": [ {
      "utf8": "that asserts it cannot be proved."
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    "segs": [ {
      "utf8": "This revelation is at the heart \nof Gödel’s Incompleteness Theorem,"
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  }, {
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    "segs": [ {
      "utf8": "which introduces an entirely new class \nof mathematical statement."
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    "segs": [ {
      "utf8": "In Gödel’s paradigm, statements still \nare either true or false,"
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    "segs": [ {
      "utf8": "but true statements can either be\nprovable or unprovable"
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    "segs": [ {
      "utf8": "within a given set of axioms."
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    "segs": [ {
      "utf8": "Furthermore, Gödel argues these \nunprovable true statements"
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    "segs": [ {
      "utf8": "exist in every axiomatic system."
    } ]
  }, {
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    "segs": [ {
      "utf8": "This makes it impossible to create"
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  }, {
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    "segs": [ {
      "utf8": "because there will always be true\nstatements we cannot prove."
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    "segs": [ {
      "utf8": "Even if you account for these\nunprovable statements"
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  }, {
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    "segs": [ {
      "utf8": "by adding them as new axioms \nto an enlarged mathematical system,"
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  }, {
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    "segs": [ {
      "utf8": "that very process introduces new\nunprovably true statements."
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    "segs": [ {
      "utf8": "No matter how many axioms you add,"
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    "segs": [ {
      "utf8": "there will always be unprovably true \nstatements in your system."
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    "segs": [ {
      "utf8": "It’s Gödels all the way down!"
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    "segs": [ {
      "utf8": "This revelation rocked the foundations \nof the field,"
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    "segs": [ {
      "utf8": "crushing those who dreamed that every\nmathematical claim would one day"
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    "segs": [ {
      "utf8": "be proven or disproven."
    } ]
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    "segs": [ {
      "utf8": "While most mathematicians accepted this \nnew reality, some fervently debated it."
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    "segs": [ {
      "utf8": "Others still tried to ignore \nthe newly uncovered a hole"
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  }, {
    "tStartMs": 262496,
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    "segs": [ {
      "utf8": "in the heart of their field."
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    "segs": [ {
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    "segs": [ {
      "utf8": "some began to worry their life's work\nwould be impossible to complete."
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    "segs": [ {
      "utf8": "Still, Gödel’s theorem opened \nas many doors as a closed."
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    "segs": [ {
      "utf8": "inspired key innovations \nin early computers."
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    "segs": [ {
      "utf8": "And today, some mathematicians dedicate\ntheir careers"
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      "utf8": "So while mathematicians may have \nlost some certainty,"
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      "utf8": "thanks to Gödel they can embrace \nthe unknown"
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