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Asaf Karagila ♦
Asaf Karagila ♦
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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?

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?

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?

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Asaf Karagila ♦
Asaf Karagila ♦
  • 39.7k
  • 8
  • 134
  • 282

On surjections, idempotence and axiom of choice

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?