The following assertion is trivial in ZFC, or even in much weaker theories. Is it also true in ZF?

(I couldn't find it in the Consequences site so far.)


  If $A$ is an infinite set such that $A$ can be mapped onto $A\times 2$ then $|A\times 2|=|A|$


The problem is that we cannot necessarily choose from every fiber of $f$, so we cannot construct an injection from $A$ to $f^{-1}(A\times\lbrace 0\rbrace)$, which will prove the assertion.

While I'm on the topic, is it possible for a D-finite set to have such property? It is possible for a D-finite set to be surjected onto a larger set than itself, but what about that large?


Question: Can tab just mean "insert tab character"?
Answer: You can try (for a single notebook)

CreateDocument[{},
 NotebookEventActions -&gt; {{"KeyDown", "\t"} :&gt; 
  NotebookWrite[SelectedNotebook[], "\t"]}]


or (for global application)

SetOptions[$FrontEnd,
 FrontEndEventActions -&gt; {{"KeyDown", "\t"} :&gt; 
  NotebookWrite[SelectedNotebook[], "\t"]}]


Does this meet your needs? For me it had an effect in Input cells but didn't block autocomplete behavior, though if that's what's getting in your way it can be disabled in Preferences.