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edited Jan 4, 2016 at 6:04

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The first question asks about the global behavior of the power function in the case of finite gaps.

Question 1. Fix a natural number $m>1.$ For each limit ordinal $\alpha,$ including $0$, let $\phi_\alpha: \omega \to \omega$ be a function such that $\phi_\alpha(0)=m, \phi_\alpha(n)>n$ and $n_1 < n_2 \implies \phi_\alpha(n_1) \leq \phi_\alpha(n_2).$ Is there a model of set theory (ZFC) in which for each limit ordinal $\alpha$ and each natural number $n, 2^{\aleph_{\alpha+n}}=\aleph_{\alpha+\phi_\alpha(n)}$?

See Possible values for $2^{\aleph_n}$ and $2^{\aleph_\omega}$, where the case $\alpha=0$ is considered.

The second question simply asks if an Easton like theorem in the case of finite gaps exists below $\aleph_{\omega_1}$.

Question 2. For each limit ordinal $\alpha < \omega_1,$ including $0$, let $\phi_\alpha: \omega \to \omega$ be a function such that $ \phi_\alpha(n)>n$ and $n_1 < n_2 \implies \phi_\alpha(n_1) \leq \phi_\alpha(n_2).$ Is there a model of set theory (ZFC) in which for each limit ordinal $\alpha< \omega_1$ and each natural number $n, 2^{\aleph_{\alpha+n}}=\aleph_{\alpha+\phi_\alpha(n)}$?

See Two Stationary Sets with Different Gaps of the Power Function and Power function on stationry classes for partial results.

Also note that in the case of non-AC, the answer to the above questions is yes, in fact more is true. See An Easton-like Theorem for Zermelo-Fraenkel Set Theory Without Choice (Preliminary Report)

The first question asks about the global behavior of the power function in the case of finite gaps.

Question 1. Fix a natural number $m>1.$ For each limit ordinal $\alpha,$ including $0$, let $\phi_\alpha: \omega \to \omega$ be a function such that $\phi_\alpha(0)=m, \phi_\alpha(n)>n$ and $n_1 < n_2 \implies \phi_\alpha(n_1) \leq \phi_\alpha(n_2).$ Is there a model of set theory in which for each limit ordinal $\alpha$ and each natural number $n, 2^{\aleph_{\alpha+n}}=\aleph_{\alpha+\phi_\alpha(n)}$?

See Possible values for $2^{\aleph_n}$ and $2^{\aleph_\omega}$, where the case $\alpha=0$ is considered.

The second question simply asks if an Easton like theorem in the case of finite gaps exists below $\aleph_{\omega_1}$.

Question 2. For each limit ordinal $\alpha < \omega_1,$ including $0$, let $\phi_\alpha: \omega \to \omega$ be a function such that $ \phi_\alpha(n)>n$ and $n_1 < n_2 \implies \phi_\alpha(n_1) \leq \phi_\alpha(n_2).$ Is there a model of set theory in which for each limit ordinal $\alpha< \omega_1$ and each natural number $n, 2^{\aleph_{\alpha+n}}=\aleph_{\alpha+\phi_\alpha(n)}$?

See Two Stationary Sets with Different Gaps of the Power Function and Power function on stationry classes for partial results.

Also note that in the case of non-AC, the answer to the above questions is yes, in fact more is true. See An Easton-like Theorem for Zermelo-Fraenkel Set Theory Without Choice (Preliminary Report)

The first question asks about the global behavior of the power function in the case of finite gaps.

Question 1. Fix a natural number $m>1.$ For each limit ordinal $\alpha,$ including $0$, let $\phi_\alpha: \omega \to \omega$ be a function such that $\phi_\alpha(0)=m, \phi_\alpha(n)>n$ and $n_1 < n_2 \implies \phi_\alpha(n_1) \leq \phi_\alpha(n_2).$ Is there a model of set theory (ZFC) in which for each limit ordinal $\alpha$ and each natural number $n, 2^{\aleph_{\alpha+n}}=\aleph_{\alpha+\phi_\alpha(n)}$?

See Possible values for $2^{\aleph_n}$ and $2^{\aleph_\omega}$, where the case $\alpha=0$ is considered.

The second question simply asks if an Easton like theorem in the case of finite gaps exists below $\aleph_{\omega_1}$.

Question 2. For each limit ordinal $\alpha < \omega_1,$ including $0$, let $\phi_\alpha: \omega \to \omega$ be a function such that $ \phi_\alpha(n)>n$ and $n_1 < n_2 \implies \phi_\alpha(n_1) \leq \phi_\alpha(n_2).$ Is there a model of set theory (ZFC) in which for each limit ordinal $\alpha< \omega_1$ and each natural number $n, 2^{\aleph_{\alpha+n}}=\aleph_{\alpha+\phi_\alpha(n)}$?

See Two Stationary Sets with Different Gaps of the Power Function and Power function on stationry classes for partial results.

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asked Jan 4, 2016 at 5:48

Mohammad Golshani

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Two questions about the behavior of the continuum function

The first question asks about the global behavior of the power function in the case of finite gaps.

Question 1. Fix a natural number $m>1.$ For each limit ordinal $\alpha,$ including $0$, let $\phi_\alpha: \omega \to \omega$ be a function such that $\phi_\alpha(0)=m, \phi_\alpha(n)>n$ and $n_1 < n_2 \implies \phi_\alpha(n_1) \leq \phi_\alpha(n_2).$ Is there a model of set theory in which for each limit ordinal $\alpha$ and each natural number $n, 2^{\aleph_{\alpha+n}}=\aleph_{\alpha+\phi_\alpha(n)}$?

See Possible values for $2^{\aleph_n}$ and $2^{\aleph_\omega}$, where the case $\alpha=0$ is considered.

The second question simply asks if an Easton like theorem in the case of finite gaps exists below $\aleph_{\omega_1}$.

Question 2. For each limit ordinal $\alpha < \omega_1,$ including $0$, let $\phi_\alpha: \omega \to \omega$ be a function such that $ \phi_\alpha(n)>n$ and $n_1 < n_2 \implies \phi_\alpha(n_1) \leq \phi_\alpha(n_2).$ Is there a model of set theory in which for each limit ordinal $\alpha< \omega_1$ and each natural number $n, 2^{\aleph_{\alpha+n}}=\aleph_{\alpha+\phi_\alpha(n)}$?

See Two Stationary Sets with Different Gaps of the Power Function and Power function on stationry classes for partial results.

Also note that in the case of non-AC, the answer to the above questions is yes, in fact more is true. See An Easton-like Theorem for Zermelo-Fraenkel Set Theory Without Choice (Preliminary Report)