Return to Question

Gandy's basis theorem says that any nonempty $\Sigma^1_1$ set $A$ contains a real $x$ with $\omega_1^x=\omega_1^{CK}$, the least nonrecursive ordinal.

Now the following question seems quite interesting to me:

Question: Is it true that for any real $z$ and nonempty $\Sigma^1_1(z)$ set $A$ containing a real $y>_hz$, $A$ must contain a real $y_0>_h z$ so that $\omega_1^{y_0}=\omega_1^z$?

Here $\geq_h$ is the hyperarithmetic reduction. Note that to give a positive answer to the question, it is sufficient to show that every such $A$ must contain a real $y_0\geq_h z$ so that $\omega_1^{y_0}=\omega_1^z$ since the set $\{y\mid y\not\leq_h z\}$ is $\Sigma^1_1(z)$.

The question has a negative answer (see below). Then how about for $\Delta^1_1(z)$ sets?

Gandy's basis theorem says that any nonempty $\Sigma^1_1$ set $A$ contains a real $x$ with $\omega_1^x=\omega_1^{CK}$, the least nonrecursive ordinal.

Now the following question seems quite interesting to me:

Question: Is it true that for any real $z$ and nonempty $\Sigma^1_1(z)$ set $A$ containing a real $y>_hz$, $A$ must contain a real $y_0>_h z$ so that $\omega_1^{y_0}=\omega_1^z$?

Here $\geq_h$ is the hyperarithmetic reduction. Note that to give a positive answer to the question, it is sufficient to show that every such $A$ must contain a real $y_0\geq_h z$ so that $\omega_1^{y_0}=\omega_1^z$ since the set $\{y\mid y\not\leq_h z\}$ is $\Sigma^1_1(z)$.

The question has a negative answer (see below). Then the question can be modified to be

Question' Is it true that for any real $z$ and nonempty $\Delta^1_1(z)$ set $A$ containing a real $y>_hz$, $A$ must contain a real $y_0>_h z$ so that $\omega_1^{y_0}=\omega_1^z$?

Gandy's basis theorem says that any nonempty $\Sigma^1_1$ set $A$ contains a real $x$ with $\omega_1^x=\omega_1^{CK}$, the least nonrecursive ordinal.

Now the following question seems quite interesting to me:

Question: Is it true that for any real $z$ and nonempty $\Sigma^1_1(z)$ set $A$ containing a real $y>_hz$, $A$ must contain a real $y_0>_h z$ so that $\omega_1^{y_0}=\omega_1^z$?

Here $\geq_h$ is the hyperarithmetic reduction. Note that to give a positive answer to the question, it is sufficient to show that every such $A$ must contain a real $y_0\geq_h z$ so that $\omega_1^{y_0}=\omega_1^z$ since the set $\{y\mid y\not\leq_h z\}$ is $\Sigma^1_1(z)$.

Gandy's basis theorem says that any nonempty $\Sigma^1_1$ set $A$ contains a real $x$ with $\omega_1^x=\omega_1^{CK}$, the least nonrecursive ordinal.

Now the following question seems quite interesting to me:

Question: Is it true that for any real $z$ and nonempty $\Sigma^1_1(z)$ set $A$ containing a real $y>_hz$, $A$ must contain a real $y_0>_h z$ so that $\omega_1^{y_0}=\omega_1^z$?

Here $\geq_h$ is the hyperarithmetic reduction. Note that to give a positive answer to the question, it is sufficient to show that every such $A$ must contain a real $y_0\geq_h z$ so that $\omega_1^{y_0}=\omega_1^z$ since the set $\{y\mid y\not\leq_h z\}$ is $\Sigma^1_1(z)$.

The question has a negative answer (see below). Then how about for $\Delta^1_1(z)$ sets?

Gandy's basis theorem says that any nonempty $\Sigma^1_1$ set $A$ contains a real $x$ with $\omega_1^x=\omega_1^{CK}$, the least nonrecursive ordinal.

Now the following question seems quite interesting to me:

Question: Is it true that for any real $z$ and nonempty $\Sigma^1_1(z)$ set $A$ containing a real $y>_hz$, $A$ must contain a real $y_0>_h z$ so that $\omega_1^{y_0}=\omega_1^z$?

Here $\geq_h$ is the hyperarithmetic reduction.

Gandy's basis theorem says that any nonempty $\Sigma^1_1$ set $A$ contains a real $x$ with $\omega_1^x=\omega_1^{CK}$, the least nonrecursive ordinal.

Now the following question seems quite interesting to me:

Question: Is it true that for any real $z$ and nonempty $\Sigma^1_1(z)$ set $A$ containing a real $y>_hz$, $A$ must contain a real $y_0>_h z$ so that $\omega_1^{y_0}=\omega_1^z$?

Here $\geq_h$ is the hyperarithmetic reduction. Note that to give a positive answer to the question, it is sufficient to show that every such $A$ must contain a real $y_0\geq_h z$ so that $\omega_1^{y_0}=\omega_1^z$ since the set $\{y\mid y\not\leq_h z\}$ is $\Sigma^1_1(z)$.