Description:
Form 30 implies Form 83
Content:
We argue that
Form 30 (the ordering principle) implies Form 83 (If every subset of \({\cal P}(X)\) which is linearly ordered by \(\subseteq \) has a maximum element then \(X\) is finite.)
Assume \(X\) is infinite and let \(\le \) be a linear ordering of \(X\).There must be either a non-empty subset \(Y\) of \(X\) with no least element (under \(\le\)) or a non-empty subset with no greatest element. (If every non-empty subset of \(X\) has a least element then let \(x_{1}\) be the least element of \(X\), \(x_{2}\) be the least element of \(X - \{x_{1}\}\), etc. This produces a subset \(\{x_{1}, x_{2}, \ldots \}\) with no greatest element.) If \(Y\) is an infinite subset of \(X\) with no greatest element, we get a linearly ordered subset \(A\) of \({\cal P}(X)\) with no maximum by letting \(A = \left\{\{x: x\in Y \wedge x\le y\}: y\in Y \right\}\). We can proceed similarly if \(Y\)is an infinite subset of \(X\) with no smallest element.
Howard-Rubin number:
81
Type:
Theorem
Back