Description: Truss [1995] studies the various structures an amorphous set can carry.
Content:
Truss [1995] studies the various structures an amorphous set can carry. (A set is amorphous if it is infinite but is not the disjoint union of two infinite sets.) If \(\pi\) is a partition of an amorphous set \(U\) into infinitely many parts, then each part must be finite. (Otherwise, if \(X\) were an infinite part, then \(U=X\cup (U-X)\), contradicting the fact that \(U\) is amorphous.) Define a map \(f\) from \(U\) into \(\omega\) such that for each \(x\in U\), \(f(x)=n\) where \(x\in\xi\in\pi\) and \(|\xi|=n\). Since \(U\) is amorphous, the range of \(f\) must be finite. Since \(U=\bigcup\{f^{-1}(n): n\in\hbox{ range of } f\}\), there is a unique \(n\) such that \(f^{-1}(n)\) is infinite. Let \(n(\pi)\) be this unique \(n\), then \(\{\xi\in\pi: |\xi|=n(\pi)\}\) is infinite.
Let \(\Pi(U)\) be the set of finitary partitions of \(U\) and let \(n(U)=\sup\{n(\pi): \pi\in\Pi(U)\}\). If \(n(U)\) is finite, \(U\) is called bounded, and if \(n(U)\) infinite, \(U\) is called unbounded.
Two interesting results that Truss obtains are:
Theorem: The following are consistent with \(ZF\):
See Truss [1995] for additional results.
Howard-Rubin number: 57
Type: Definitions and summaries
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