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Is there any example that for cardinal numbers $\kappa < \lambda$, we have $2^\kappa = 2^\lambda$?

My guess is that it only depends on whether GCH holds. Is it true?

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I just noticed the exact wording of the question was not answered by my answer previously. I added more. –  Asaf Karagila Nov 28 '12 at 2:02
Thank you very much. I'm so amazed. It seems to me that you ponder upon these problems like crazy :) –  Metta World Peace Nov 28 '12 at 2:10
I do (and I am). I mean, seriously. 4am. I am going to sleep! (I have to give a talk tomorrow afternoon, and I won't have time to doze off around noon.) –  Asaf Karagila Nov 28 '12 at 2:11
What should I say?! I don't want to sound like Cee lo in a talent show. But what you have been doing do amaze and impress me。 –  Metta World Peace Nov 28 '12 at 5:59
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1 Answer

up vote 4 down vote accepted

This is independent of ZFC. It is consist that there are no cardinals, for example if GCH holds. Note that $\lambda\leq\kappa\implies2^\lambda\leq2^\kappa$, so it is enough to show that the continuum function is injective.

However it is consistent that $2^{\aleph_0}=2^{\aleph_1}=\aleph_3$.

There is not much we can say about the continuum function in ZFC. This is a dire consequence from Easton's theorem.

Easton theorem tells us that if $F$ is a function whose domain is the regular cardinals and:

  1. $\kappa<\lambda\implies F(\kappa)\leq F(\lambda)$,
  2. $\operatorname{cf}(\kappa)<\operatorname{cf}(F(\kappa))$

Then there is a forcing extension which does not collapse cardinals and for every regular $\kappa$, $2^\kappa=F(\kappa)$ in the extension.

Assume GCH holds and take the function $F(\kappa)=\kappa^{++}$. We can show that in the extension where $F$ describes the continuum function we have $2^{\kappa}=\kappa^{++}$ for regular cardinals, and $F(\mu)=\mu^+$ for singular $\mu$. This means that GCH fails for all regular cardinals, but $2^\lambda=2^\kappa\iff\lambda=\kappa$. So the injectivity of the continuum function holds, while GCH fails.

(If one is not in the mood for a class-forcing, which can be a bit complicated, one can simply start with GCH and set $2^{\aleph_n}=\aleph_{n+2}$ for $n<\omega$, and GCH to hold otherwise instead.)

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Thank you. It sounds fairly interesting. –  Metta World Peace Nov 26 '12 at 10:27
You're welcome. You should search for the term Easton on this site, it has been mentioned and explained before (I am writing from a cellphone, so I am being brief). –  Asaf Karagila Nov 26 '12 at 10:33
Good advise. Although it is probably too advanced for me, I'll have a look. –  Metta World Peace Nov 26 '12 at 10:39
Now that I am by a computer again, I added a bit. –  Asaf Karagila Nov 26 '12 at 11:39
+1 Nice approach Asaf. –  B. S. Nov 26 '12 at 12:09
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