Characterization of uncountable set I was looking at this question here and was wondering, if also the opposite direction holds. Specifically, I want to prove/disprove the statement: a set $A$ is uncountable, if and only if there exists an injective function $f:(0,1) \to A$.
 A: The statement "Every uncountable set admits an injection from $(0, 1)$" is the continuum hypothesis (C); it is known that CH is independent from the ZFC axioms, meaning that ZFC can neither prove nor disprove CH.
However, it is not true that "assuming that the hypothesis is true or that its negation is true does not affect any other statement:" CH does interact with other independent sentences. For instance, Freiling's axiom of symmetry is an arguably-reasonable principle which is equivalent to the negation of CH: if one is true, the other is false. On the other hand, certain natural combinatorial principles or statements saying that all sets are "reasonably definable" in a certain sense imply CH
There is a huge variety of statements independent of ZFC which interact with each other in really interesting ways; indeed, one of the main consequences of the discover of forcing was the realization that beyond a certain point, set-theoretic questions are arguably more frequently undecidable-from-ZFC than decidable-from-ZFC; this led to a push towards discovery strong hypotheses which, when added as axioms, would resolve large classes of questions. These include:


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*The axiom of constructibility (linked above).

*Large cardinal hypotheses, which imply that reasonably definable sets of real numbers have nice properties; also their generic versions, which resolve combinatorial principles about the continuum (unlike large cardinals themselves).

*Forcing axioms, which resolve combinatorial questions about the continuum (and imply the negation of CH!).
If you're interested in the search for new axioms of set theory, this is a great paper to read.
