How big is infinity? - Dennis Wildfogel
TED-Ed・7 minutes read
In the late 1800s, Georg Cantor created a list of all fractions, showing a one-to-one match with whole numbers, while also demonstrating that irrational numbers like the square root of two and pi represent a bigger infinity. Cantor's work on infinite sets led to the development of the theory of multiple infinities, including his continuum hypothesis, which remained unsolved until the 20th century when Gödel and Cohen showed its unprovability, emphasizing the presence of unanswerable questions in mathematics.
Insights
- Cantor demonstrated that subsets of infinite sets can lead to larger infinities, introducing the concept of multiple infinities of varying sizes beyond whole numbers.
- Gödel and Cohen's research showed that the continuum hypothesis, which questions infinities between whole and real numbers, is unprovable, emphasizing the presence of unsolvable mathematical questions.
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Recent questions
What is the concept of sets having the same size?
Matching elements in sets to determine equality.
Who created a list of all fractions in the late 1800s?
Georg Cantor
What are irrational numbers like the square root of two and pi?
Numbers that cannot be matched one-to-one with whole numbers.
What did Cantor prove about forming subsets of an infinite set?
Results in a larger infinity.
What did Gödel and Cohen's work reveal about the continuum hypothesis?
It is unprovable, highlighting unanswerable questions in mathematics.
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Summary
00:00
Exploring Infinities: Cantor's Unsolvable Continuum Hypothesis
- In fourth grade, the concept of sets having the same size was introduced, illustrated by matching fingers on hands and sheep counting with stones.
- A list of all fractions was created by Georg Cantor in the late 1800s, demonstrating a one-to-one match with whole numbers.
- Irrational numbers, like the square root of two and pi, cannot be matched one-to-one with whole numbers, showcasing a bigger infinity.
- Cantor proved that forming subsets of an infinite set results in a larger infinity, leading to multiple infinities of varying sizes.
- Cantor's continuum hypothesis, questioning infinities between whole numbers and real numbers, was unsolved until the 20th century.
- Gödel and Cohen's work in the 1920s and 1960s revealed that the continuum hypothesis is unprovable, highlighting the existence of unanswerable questions in mathematics.
