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Finite-to-one maps. (English) Zbl 1057.03035

From the text: In ZFC we can argue as follows: if there is a finite-to-one map from \(Y\) onto \(X\), and \(X\) is infinite, then \(|Y|\leq|X|\cdot \aleph_0=|X|\), and \(|Y|\geq|X|\) by AC, so \(|Y|=|X|\). So there can be a finite-to-one map from \({\mathcal P}(X)\twoheadrightarrow X\) only if \(X\) is finite. (Here finite means “has cardinal in \(\omega\)”, and “finite-to-one” means that the preimage of every singleton is finite.) It is the purpose of this (self-contained) note to show that the result can be proved in ZF even without AC. The proof provided here uses replacement and cannot be conducted in Zermelo set theory. Nor does the proof generalise to show, for an arbitrary strongly inaccessible aleph \(\kappa\), that if there is a surjection \(f:{\mathcal P}(X)\twoheadrightarrow X\) where \(|f^{-1''} \{x\}|<\kappa\) for all \(x\in X\) then \(|X|<\kappa\).
Theorem 1. If there is a finite-to-one map \({\mathcal P}(X)\twoheadrightarrow X\), then \(X\) is finite.

MSC:

03E25 Axiom of choice and related propositions
03E30 Axiomatics of classical set theory and its fragments
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References:

[1] E. Specker Verallgemeinerte Kontinuumshypothese und Auswahlaxiom , Archiv der Mathematik , vol. 5 (1954), pp. 332–337. · Zbl 0056.05001
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