- #1
Dragonfall
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How do you show that the surreal numbers form a proper class?
Dragonfall said:If a metric needs to be real-valued,
they do - apparently.and if the surreals form a proper class
Or by trichotomy...Dragonfall said:If X is the set of all surreal numbers, then so is [tex]\{ X|\emptyset\}[/tex]. But this only leads to a contradiction assuming well-foundedness.
Surreal numbers are a mathematical concept introduced by John Horton Conway in 1976. They can be thought of as a generalization of real numbers, including both finite and infinite numbers.
Surreal numbers differ from real numbers in that they include both finite and infinite numbers, allowing for numbers that are infinitely larger or smaller than any real number. They also have the property of being a proper class, meaning that they cannot be contained in a set.
A proper class is a collection of objects that is too large to be considered a set. In other words, it is a collection that cannot be contained in a set. In terms of surreal numbers, this means that they cannot be listed or counted in a traditional way, as they are infinite in number.
Surreal numbers are typically represented using a decimal-like notation, with the left and right halves of the number separated by a vertical bar. The left and right halves can contain any number of other surreal numbers, including infinite numbers.
Surreal numbers have applications in fields such as game theory, combinatorics, and topology. They can also be used to solve problems in other areas of mathematics, such as calculus and algebra.