Return to Question

This question is related to my question "http://mathoverflow.net/questions/146800/forcing-with-c-c-c-forcing-notions-cohen-reals-and-random-reals".

A natural way to answer Prikry's conjecture is to build a c.c.c. forcing notion which adds a minimal generic real.

I arrived to the idea of constructing such a forcing notion by considering the following:

1. Mathis forcing at $\omega$ is essentially the same as Prikry's forcing at some measurable cardinal,

2. Recently, a minimal Prikry type forcing at a measurable cardinal is constructed by Koepke-Räsch-Schlicht (see "A minimal Prikry-type forcing for singularizing a measurable cardinal, J. Symbolic Logic Volume 78, Issue 1 (2013), 85-100." and also Is Prikry forcing minimal? ).

So a natural idea is to define a version of the above forcing for $\omega$. It turns out to me that such a forcing is constructed many years ago by Judah-Shelah in "Forcing minimal degree of constructibility" and there it is shown that under $CH$ we can choose nice ultrafilters so that the resulting forcing adds a minimal generic real which is minimal.

Let me describe the forcing I have in mind: Let $(A_n: n<\omega)$ be a partition of $\omega$ into infinite sets and for each $n$ let $U_n$ be a non-principal ultrafilter over $A_n$. Let $\mathbb{P}$ consists of all pairs $(t, T)$ where $T \subseteq [\omega]^{<\omega}$ is a tree with trunk $t$ and for all $t \unlhd u\in T$ (where $\unlhd$ denotes end-extension relation) $Suc_T(u) \in U_{max(u)}.$ We define $(t, T) \leq (s, S)$ iff $T \subseteq S$ and $(t, T) \leq^* (s, S)$ iff $T \subseteq S$ and $t=s.$ The following can be proved easily:

1. $(\mathbb{P}, \leq)$ satisfies the c.c.c.,

2. $(\mathbb{P}, \leq, \leq^*)$ satisfies the Prikry property,

3. Let $G$ be $\mathbb{P}-$generic over $V$ and let $f_G=\bigcup \{t: \exists T, (t, T)\in G \}.$ Then $f_G$ is a real,

4. Suppose $d \subseteq f_G$ is infinite, then $V[d]=V[f_G]$ (the proof is the same as in the proof of Proposition 4.4 in Koepke-Räsch-Schlicht paper).

Question. Is it consistent with $ZFC$ that there is no sequence of ultrafilters as above such that the corresponding forcing defined above adds a minimal generic real?

Note that in such a model $CH$ or $MA+\sim CH$ should fail.

Remark 1. If we can show that the forcing does not add a minimal generic real when all of our ultrafilters are on the same class of near-coherence ultrafilters, then the answer to question 1 will become positive, as it is consistent that there is just one class of near-coherence non-principal ultrafilters.

Remark 2. As pointed out by Dorais, if all of the ultrafilters are RK equivalent, then the resulting forcing does not add a minimal generic real.

This question is related to my question "https://mathoverflow.net/questions/146800/forcing-with-c-c-c-forcing-notions-cohen-reals-and-random-reals".

A natural way to answer Prikry's conjecture is to build a c.c.c. forcing notion which adds a minimal generic real.

I arrived to the idea of constructing such a forcing notion by considering the following:

1. Mathis forcing at $\omega$ is essentially the same as Prikry's forcing at some measurable cardinal,

2. Recently, a minimal Prikry type forcing at a measurable cardinal is constructed by Koepke-Räsch-Schlicht (see "A minimal Prikry-type forcing for singularizing a measurable cardinal, J. Symbolic Logic Volume 78, Issue 1 (2013), 85-100." and also Is Prikry forcing minimal? ).

So a natural idea is to define a version of the above forcing for $\omega$. It turns out to me that such a forcing is constructed many years ago by Judah-Shelah in "Forcing minimal degree of constructibility" and there it is shown that under $CH$ we can choose nice ultrafilters so that the resulting forcing adds a minimal generic real which is minimal.

Let me describe the forcing I have in mind: Let $(A_n: n<\omega)$ be a partition of $\omega$ into infinite sets and for each $n$ let $U_n$ be a non-principal ultrafilter over $A_n$. Let $\mathbb{P}$ consists of all pairs $(t, T)$ where $T \subseteq [\omega]^{<\omega}$ is a tree with trunk $t$ and for all $t \unlhd u\in T$ (where $\unlhd$ denotes end-extension relation) $Suc_T(u) \in U_{max(u)}.$ We define $(t, T) \leq (s, S)$ iff $T \subseteq S$ and $(t, T) \leq^* (s, S)$ iff $T \subseteq S$ and $t=s.$ The following can be proved easily:

1. $(\mathbb{P}, \leq)$ satisfies the c.c.c.,

2. $(\mathbb{P}, \leq, \leq^*)$ satisfies the Prikry property,

3. Let $G$ be $\mathbb{P}-$generic over $V$ and let $f_G=\bigcup \{t: \exists T, (t, T)\in G \}.$ Then $f_G$ is a real,

4. Suppose $d \subseteq f_G$ is infinite, then $V[d]=V[f_G]$ (the proof is the same as in the proof of Proposition 4.4 in Koepke-Räsch-Schlicht paper).

Question. Is it consistent with $ZFC$ that there is no sequence of ultrafilters as above such that the corresponding forcing defined above adds a minimal generic real?

Note that in such a model $CH$ or $MA+\sim CH$ should fail.

Remark 1. If we can show that the forcing does not add a minimal generic real when all of our ultrafilters are on the same class of near-coherence ultrafilters, then the answer to question 1 will become positive, as it is consistent that there is just one class of near-coherence non-principal ultrafilters.

Remark 2. As pointed out by Dorais, if all of the ultrafilters are RK equivalent, then the resulting forcing does not add a minimal generic real.

This question is related to my question "http://mathoverflow.net/questions/146800/forcing-with-c-c-c-forcing-notions-cohen-reals-and-random-reals".

A natural way to answer Prikry's conjecture is to build a c.c.c. forcing notion which adds a minimal generic real.

I arrived to the idea of constructing such a forcing notion by considering the following:

1. Mathis forcing at $\omega$ is essentially the same as Prikry's forcing at some measurable cardinal,

2. Recently, a minimal Prikry type forcing at a measurable cardinal is constructed by Koepke-Räsch-Schlicht (see "A minimal Prikry-type forcing for singularizing a measurable cardinal, J. Symbolic Logic Volume 78, Issue 1 (2013), 85-100." and also Is Prikry forcing minimal? ).

So a natural idea is to define a version of the above forcing for $\omega$. It turns out to me that such a forcing is constructed many years ago by Judah-Shelah in "Forcing minimal degree of constructibility" and there it is shown that under $CH$ we can choose nice ultrafilters so that the resulting forcing adds a minimal generic real which is minimal.

Let me describe the forcing I have in mind: Let $(A_n: n<\omega)$ be a partition of $\omega$ into infinite sets and for each $n$ let $U_n$ be a non-principal ultrafilter over $A_n$. Let $\mathbb{P}$ consists of all pairs $(t, T)$ where $T \subseteq [\omega]^{<\omega}$ is a tree with trunk $t$ and for all $t \unlhd u\in T$ (where $\unlhd$ denotes end-extension relation) $Suc_T(u) \in U_{max(u)}.$ We define $(t, T) \leq (s, S)$ iff $T \subseteq S$ and $(t, T) \leq^* (s, S)$ iff $T \subseteq S$ and $t=s.$ The following can be proved easily:

1. $(\mathbb{P}, \leq)$ satisfies the c.c.c.,

2. $(\mathbb{P}, \leq, \leq^*)$ satisfies the Prikry property,

3. Let $G$ be $\mathbb{P}-$generic over $V$ and let $f_G=\bigcup \{t: \exists T, (t, T)\in G \}.$ Then $f_G$ is a real,

4. Suppose $d \subseteq f_G$ is infinite, then $V[d]=V[f_G]$ (the proof is the same as in the proof of Proposition 4.4 in Koepke-Räsch-Schlicht paper).

question 1. Is it consistent with $ZFC$ that there is no sequence of ultrafilters as above such that the corresponding forcing defined above adds a minimal generic real?

Note that in such a model $CH$ or $MA+\sim CH$ should fail.

More generally is the following situation consistent:

Question 2. Is there a generic real $f$ (possibly added by a c.c.c forcing notion) such that:

1. for all infinite $d \subseteq f, V[d]=V[f],$

2. there exists $d \subset \omega, d \notin V$ such that $V[d]\neq V[f].$

Remark. As pointed out by Dorais, this situation can happen if in the above forcing construction all the ultrafilters are RK equivalent.

This question is related to my question "http://mathoverflow.net/questions/146800/forcing-with-c-c-c-forcing-notions-cohen-reals-and-random-reals".

A natural way to answer Prikry's conjecture is to build a c.c.c. forcing notion which adds a minimal generic real.

I arrived to the idea of constructing such a forcing notion by considering the following:

1. Mathis forcing at $\omega$ is essentially the same as Prikry's forcing at some measurable cardinal,

2. Recently, a minimal Prikry type forcing at a measurable cardinal is constructed by Koepke-Räsch-Schlicht (see "A minimal Prikry-type forcing for singularizing a measurable cardinal, J. Symbolic Logic Volume 78, Issue 1 (2013), 85-100." and also Is Prikry forcing minimal? ).

So a natural idea is to define a version of the above forcing for $\omega$. It turns out to me that such a forcing is constructed many years ago by Judah-Shelah in "Forcing minimal degree of constructibility" and there it is shown that under $CH$ we can choose nice ultrafilters so that the resulting forcing adds a minimal generic real which is minimal.

Let me describe the forcing I have in mind: Let $(A_n: n<\omega)$ be a partition of $\omega$ into infinite sets and for each $n$ let $U_n$ be a non-principal ultrafilter over $A_n$. Let $\mathbb{P}$ consists of all pairs $(t, T)$ where $T \subseteq [\omega]^{<\omega}$ is a tree with trunk $t$ and for all $t \unlhd u\in T$ (where $\unlhd$ denotes end-extension relation) $Suc_T(u) \in U_{max(u)}.$ We define $(t, T) \leq (s, S)$ iff $T \subseteq S$ and $(t, T) \leq^* (s, S)$ iff $T \subseteq S$ and $t=s.$ The following can be proved easily:

1. $(\mathbb{P}, \leq)$ satisfies the c.c.c.,

2. $(\mathbb{P}, \leq, \leq^*)$ satisfies the Prikry property,

3. Let $G$ be $\mathbb{P}-$generic over $V$ and let $f_G=\bigcup \{t: \exists T, (t, T)\in G \}.$ Then $f_G$ is a real,

4. Suppose $d \subseteq f_G$ is infinite, then $V[d]=V[f_G]$ (the proof is the same as in the proof of Proposition 4.4 in Koepke-Räsch-Schlicht paper).

Question. Is it consistent with $ZFC$ that there is no sequence of ultrafilters as above such that the corresponding forcing defined above adds a minimal generic real?

Note that in such a model $CH$ or $MA+\sim CH$ should fail.

Remark 1. If we can show that the forcing does not add a minimal generic real when all of our ultrafilters are on the same class of near-coherence ultrafilters, then the answer to question 1 will become positive, as it is consistent that there is just one class of near-coherence non-principal ultrafilters.

Remark 2. As pointed out by Dorais, if all of the ultrafilters are RK equivalent, then the resulting forcing does not add a minimal generic real.

This question is related to my question "http://mathoverflow.net/questions/146800/forcing-with-c-c-c-forcing-notions-cohen-reals-and-random-reals".

A natural way to answer Prikry's conjecture is to build a c.c.c. forcing notion which adds a minimal generic real.

I arrived to the idea of constructing such a forcing notion by considering the following:

1. Mathis forcing at $\omega$ is essentially the same as Prikry's forcing at some measurable cardinal,

2. Recently, a minimal Prikry type forcing at a measurable cardinal is constructed by Koepke-Räsch-Schlicht (see "A minimal Prikry-type forcing for singularizing a measurable cardinal, J. Symbolic Logic Volume 78, Issue 1 (2013), 85-100." and also Is Prikry forcing minimal? ).

So a natural idea is to define a version of the above forcing for $\omega$. It turns out to me that such a forcing is constructed many years ago by Judah-Shelah in "Forcing minimal degree of constructibility" and there it is shown that under $CH$ we can choose nice ultrafilters so that the resulting forcing adds a minimal generic real which is minimal.

Let me describe the forcing I have in mind: Let $(A_n: n<\omega)$ be a partition of $\omega$ into infinite sets and for each $n$ let $U_n$ be a non-principal ultrafilter over $A_n$. Let $\mathbb{P}$ consists of all pairs $(t, T)$ where $T \subseteq [\omega]^{<\omega}$ is a tree with trunk $t$ and for all $t \unlhd u\in T$ (where $\unlhd$ denotes end-extension relation) $Suc_T(u) \in U_{max(u)}.$ We define $(t, T) \leq (s, S)$ iff $T \subseteq S$ and $(t, T) \leq^* (s, S)$ iff $T \subseteq S$ and $t=s.$ The following can be proved easily:

1. $(\mathbb{P}, \leq)$ satisfies the c.c.c.,

2. $(\mathbb{P}, \leq, \leq^*)$ satisfies the Prikry property,

3. Let $G$ be $\mathbb{P}-$generic over $V$ and let $f_G=\bigcup \{t: \exists T, (t, T)\in G \}.$ Then $f_G$ is a real,

4. Suppose $d \subseteq f_G$ is infinite, then $V[d]=V[f_G]$ (the proof is the same as in the proof of Proposition 4.4 in Koepke-Räsch-Schlicht paper).

question 1. If the forcing is not minimal with respect to reals, then there should be some $d \subset \omega, d \notin V$ such that $V[d] \neq V[f_G].$ Can we have such a situation in the above forcing?

More generally is the following situation consistent:

Question 2. Is there a generic real $f$ (possibly added by a c.c.c forcing notion) such that:

1. for all infinite $d \subseteq f, V[d]=V[f],$

2. there exists $d \subset \omega, d \notin V$ such that $V[d]\neq V[f].$

This question is related to my question "http://mathoverflow.net/questions/146800/forcing-with-c-c-c-forcing-notions-cohen-reals-and-random-reals".

A natural way to answer Prikry's conjecture is to build a c.c.c. forcing notion which adds a minimal generic real.

I arrived to the idea of constructing such a forcing notion by considering the following:

1. Mathis forcing at $\omega$ is essentially the same as Prikry's forcing at some measurable cardinal,

2. Recently, a minimal Prikry type forcing at a measurable cardinal is constructed by Koepke-Räsch-Schlicht (see "A minimal Prikry-type forcing for singularizing a measurable cardinal, J. Symbolic Logic Volume 78, Issue 1 (2013), 85-100." and also Is Prikry forcing minimal? ).

So a natural idea is to define a version of the above forcing for $\omega$. It turns out to me that such a forcing is constructed many years ago by Judah-Shelah in "Forcing minimal degree of constructibility" and there it is shown that under $CH$ we can choose nice ultrafilters so that the resulting forcing adds a minimal generic real which is minimal.

Let me describe the forcing I have in mind: Let $(A_n: n<\omega)$ be a partition of $\omega$ into infinite sets and for each $n$ let $U_n$ be a non-principal ultrafilter over $A_n$. Let $\mathbb{P}$ consists of all pairs $(t, T)$ where $T \subseteq [\omega]^{<\omega}$ is a tree with trunk $t$ and for all $t \unlhd u\in T$ (where $\unlhd$ denotes end-extension relation) $Suc_T(u) \in U_{max(u)}.$ We define $(t, T) \leq (s, S)$ iff $T \subseteq S$ and $(t, T) \leq^* (s, S)$ iff $T \subseteq S$ and $t=s.$ The following can be proved easily:

1. $(\mathbb{P}, \leq)$ satisfies the c.c.c.,

2. $(\mathbb{P}, \leq, \leq^*)$ satisfies the Prikry property,

3. Let $G$ be $\mathbb{P}-$generic over $V$ and let $f_G=\bigcup \{t: \exists T, (t, T)\in G \}.$ Then $f_G$ is a real,

4. Suppose $d \subseteq f_G$ is infinite, then $V[d]=V[f_G]$ (the proof is the same as in the proof of Proposition 4.4 in Koepke-Räsch-Schlicht paper).

question 1. Is it consistent with $ZFC$ that there is no sequence of ultrafilters as above such that the corresponding forcing defined above adds a minimal generic real?

Note that in such a model $CH$ or $MA+\sim CH$ should fail.

More generally is the following situation consistent:

Question 2. Is there a generic real $f$ (possibly added by a c.c.c forcing notion) such that:

1. for all infinite $d \subseteq f, V[d]=V[f],$

2. there exists $d \subset \omega, d \notin V$ such that $V[d]\neq V[f].$

Remark. As pointed out by Dorais, this situation can happen if in the above forcing construction all the ultrafilters are RK equivalent.