 字幕表 動画を再生する

• 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

• 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.

• 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.”

B1 中級

The paradox at the heart of mathematics: Gödel's Incompleteness Theorem - Marcus du Sautoy

• 4 0
ocean に公開 2021 年 07 月 20 日