Published online by Cambridge University Press: 12 March 2014
In this paper, we consider certain cardinals in ZF (set theory without AC, the axiom of choice). In ZFC (set theory with AC), given any cardinals and
, either
≤
or
≤
. However, in ZF this is no longer so. For a given infinite set A consider seq1-1(A), the set of all sequences of A without repetition. We compare |seq1-1(A)|, the cardinality of this set, to |
|, the cardinality of the power set of A. What is provable about these two cardinals in ZF? The main result of this paper is that ZF ⊢ ∀A(| seq1-1(A)| ≠ |
|), and we show that this is the best possible result. Furthermore, it is provable in ZF that if B is an infinite set, then | fin(B)| < |
(B*)| even though the existence for some infinite set B* of a function ƒ from fin(B*) onto
(B*) is consistent with ZF.
Full text views reflects PDF downloads, PDFs sent to Google Drive, Dropbox and Kindle and HTML full text views.
* Views captured on Cambridge Core between September 2016 - 1st March 2021. This data will be updated every 24 hours.