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Another paradox in naive set-theory. (English) Zbl 1121.03066

Let \((X, <)\) be a well-founded set. Then \((\text{Succ}(X),\prec)\) is the well-founded set which arises from \((X, <)\) by adding an element \(\infty\) to \(X\) which is greater than all elements of \(X\). The author considers a paradox which can be seen as a variant of the Burali-Forti paradox. To this purpose, let \(I_0\) be the “set” of all well-founded sets and \(\Sigma:=\sum\{i\mid i\in I_0\}\) their disjoint ordered sum. Then \(\text{Succ}(\Sigma)\) can be mapped \(\prec\)-preserving into \(\Sigma\), which leads to a contradiction.

MSC:

03E30 Axiomatics of classical set theory and its fragments
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