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This might be a terribly simple question, but I cannot convince myself whether the answer is yes or no. Maybe I am missing something simple. I am not well-versed in the area of elementary set theory so excuse the simplicity of the question. Note within a world where $\mathsf{GCH}$ is true we clearly have a yes; but with $\mathsf{GCH}$ false the answer is no longer obvious to me.

Exclude the triviality of finite sets. If $|X|<|Y|$ then $|X|<|Y\setminus X|$ so say $|X|=\kappa$, then $|Y|=\kappa+\lambda$ with $\kappa<\lambda$. But then proving $2^{\kappa}<2^{\lambda}$ is equivalent to the initial problem.

Am I being silly or is it consistent with $\mathsf{ZFC}$ that this implication is false?

user111064
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In ZFC the Generalized Continuum Hypothesis (GCH) is that $2^k=k^+$ for every infinite cardinal. GCH implies that if $k<l$ then $2^k=k^+\leq l<2^l$ for infinite cardinals $k,l.$ It has been shown by the method of Forcing (invented by Paul Cohen, in, I think, the late 1960's) that if ZFC is consistent, then $2^{\omega}=2^{\omega_1}$ is not disprovable by ZFC. There are many more examples violating your inequality that are also equiconsistent with ZFC. Some of these satisfy CH. For example $2^{\omega}=\omega_1 \land 2^{\omega_1}=2^{\omega_2}.$ Note: $k^+$ is the least cardinal greater than $k$.

DanielWainfleet
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  • Certainly not a "silly question". Likely dates back to the founding of set theory by Georg Cantor and others in late 19th century. Consistency of $ GCH\land AC $(Axiom of Choice) with the axiom system ZF was shown by Kurt Godel in the 1930's. Not until Paul Cohen did anyone make any progress on the consistency of $\neg CH$ or on your Q. – DanielWainfleet Feb 27 '16 at 05:20
  • Thanks, sorry about the CH/"no" rubbish in my post, I did mean GHC and "yes", as you pointed out in your first line. Anyhow this is very interesting, many thanks for your answer. – user111064 Feb 27 '16 at 15:16