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  • Consider the following sentence: “This statement is false.”

  • Is that true?

  • If so, that would make this statement false.

  • But if it's false, then the statement is true.

  • By referring to itself directly, this statement creates an unresolvable paradox.

  • So if it's not true and it's not falsewhat is it?

  • This question might seem like a silly thought experiment.

  • But in the early 20th century, it led Austrian logician Kurtdel

  • to a discovery that would change mathematics forever.

  • del's discovery had to do with the limitations of mathematical proofs.

  • A proof is a logical argument that demonstrates

  • why a statement about numbers is true.

  • The building blocks of these arguments are called axioms

  • undeniable statements about the numbers involved.

  • Every system built on mathematics,

  • from the most complex proof to basic arithmetic,

  • is constructed from axioms.

  • And if a statement about numbers is true,

  • mathematicians should be able to confirm it with an axiomatic proof.

  • Since ancient Greece, mathematicians used this system

  • to prove or disprove mathematical claims with total certainty.

  • But whendel entered the field,

  • some newly uncovered logical paradoxes were threatening that certainty.

  • Prominent mathematicians were eager to prove

  • that mathematics had no contradictions.

  • del himself wasn't so sure.

  • And he was even less confident that mathematics was the right tool

  • to investigate this problem.

  • While it's relatively easy to create a self-referential paradox with words,

  • numbers don't typically talk about themselves.

  • A mathematical statement is simply true or false.

  • Butdel had an idea.

  • First, he translated mathematical statements and equations into code numbers

  • so that a complex mathematical idea could be expressed in a single number.

  • This meant that mathematical statements written with those numbers

  • were also expressing something about the encoded statements of mathematics.

  • In this way, the coding allowed mathematics to talk about itself.

  • Through this method, he was able to write:

  • This statement cannot be provedas an equation,

  • creating the first self-referential mathematical statement.

  • However, unlike the ambiguous sentence that inspired him,

  • mathematical statements must be true or false.

  • So which is it?

  • If it's false, that means the statement does have a proof.

  • But if a mathematical statement has a proof, then it must be true.

  • This contradiction means thatdel's statement can't be false,

  • and therefore it must be true thatthis statement cannot be proved.”

  • Yet this result is even more surprising,

  • because it means we now have a true equation of mathematics

  • that asserts it cannot be proved.

  • This revelation is at the heart ofdel's Incompleteness Theorem,

  • which introduces an entirely new class of mathematical statement.

  • Indel's paradigm, statements still are either true or false,

  • but true statements can either be provable or unprovable

  • within a given set of axioms.

  • Furthermore, Gödel argues these unprovable true statements

  • exist in every axiomatic system.

  • This makes it impossible to create

  • a perfectly complete system using mathematics,

  • because there will always be true statements we cannot prove.

  • Even if you account for these unprovable statements

  • by adding them as new axioms to an enlarged mathematical system,

  • that very process introduces new unprovably true statements.

  • No matter how many axioms you add,

  • there will always be unprovably true statements in your system.

  • It's Gödels all the way down!

  • This revelation rocked the foundations of the field,

  • crushing those who dreamed that every mathematical claim would one day

  • be proven or disproven.

  • While most mathematicians accepted this new reality, some fervently debated it.

  • Others still tried to ignore the newly uncovered a hole

  • in the heart of their field.

  • But as more classical problems were proven to be unprovably true,

  • some began to worry their life's work would be impossible to complete.

  • Still, Gödel's theorem opened as many doors as a closed.

  • Knowledge of unprovably true statements

  • inspired key innovations in early computers.

  • And today, some mathematicians dedicate their careers

  • to identifying provably unprovable statements.

  • So while mathematicians may have lost some certainty,

  • thanks todel they can embrace the unknown

  • at the heart of any quest for truth.

Consider the following sentence: “This statement is false.”

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B1 中級 美國腔

數學核心的悖論:哥德爾不完備性定理 - Marcus du Sautoy(The paradox at the heart of mathematics: Gödel's Incompleteness Theorem - Marcus du Sautoy)

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    ocean 發佈於 2021 年 07 月 20 日
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