How the Axiom of Choice Gives Sizeless Sets | Infinite Series
PBS Infinite Series・2 minutes read
The Lebesgue measure in mathematics formalizes size, with properties like translation invariance and additivity. Non-measurable sets, created using the axiom of choice, challenge physical intuitions and involve understanding higher dimensions through improved intuition.
Insights
- The Lebesgue measure in mathematics defines size based on dimensions, with properties like translation invariance and countable additivity.
- Non-measurable sets, like S created by the Banach-Tarski paradox, challenge traditional notions of size and rely on the axiom of choice, illustrating complex mathematical concepts beyond simple visualization.
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Recent questions
What is the Lebesgue measure in mathematics?
The Lebesgue measure formalizes size, representing length, area, and volume in different dimensions.
What are some properties of the Lebesgue measure?
The Lebesgue measure exhibits translation invariance, additivity, and countable additivity.
Why do countable collections of points have a size of 0?
Countable collections, such as all integers, have a size of 0 due to countable additivity.
Do intervals with uncountably many points have a simple size?
Intervals with uncountably many points, like 0 to 1, do not have a size that is the sum of the sizes of the points.
How is a non-measurable set created in mathematics?
Non-measurable sets, like set S, are formed by sorting numbers into bins based on rational differences and selecting one representative from each bin.
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Summary
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"Lebesgue Measure and Non-Measurable Sets"
- Size in mathematics is formalized as the Lebesgue measure, representing length in one dimension, area in two dimensions, and volume in three dimensions.
- Properties of the Lebesgue measure include translation invariance, additivity, and countable additivity.
- Countable collections of points, like all integers, have a size of 0 due to countable additivity.
- Intervals with uncountably many points, like 0 to 1, do not have a size that is simply the sum of the sizes of the points.
- A set without a size, or non-measurable set, can be created by sorting numbers in an interval into bins based on rational differences.
- The set S, formed by selecting one representative from each bin, is non-measurable and has no size.
- The axiom of choice is used to create non-measurable sets like S, allowing for the selection of one element from each set.
- The Banach-Tarski paradox demonstrates the doubling of a solid sphere by breaking it into non-measurable pieces and reassembling them using the axiom of choice.
- The Banach-Tarski paradox relies on non-measurable sets and the axiom of choice, challenging physical intuitions.
- Understanding higher dimensions, like the fourth dimension, involves improving intuition rather than literal visualization, aiding in mathematical analysis.
