Is $2^{\alpha} < 2^{\beta}$, where $\alpha$ and $\beta$ are cardinal numbers, such that $\alpha < \beta$?

Asked Jul 03 '14 at 15:48

Active Jul 03 '14 at 16:03

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Let $\alpha$ and $\beta$ be cardinal numbers such that $\alpha < \beta$. Isn't it always true that $2^{\alpha} < 2^{\beta}$ ? Because if I am not wrong, $2^{\alpha}$ denotes the immediate successor of $\alpha$ by GCH. Can anyone please suggest me some good book which can clarify my concepts on cardinals, ordinals and cofinality. Thank you in advance.

edited Jul 03 '14 at 16:03

amWhy

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asked Jul 03 '14 at 15:48

akansha

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Already answered: http://math.stackexchange.com/questions/244870/why-continuum-function-isnt-strictly-increasing. – Martín-Blas Pérez Pinilla Jul 03 '14 at 15:58
You could try 'Classic set Theory' by Derek Goldrei, although I don't know if it has exactly what you want. – Jessica B Jul 03 '14 at 16:07

Is $2^{\alpha} < 2^{\beta}$, where $\alpha$ and $\beta$ are cardinal numbers, such that $\alpha < \beta$?

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