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First, I would like to say that I asked this question (a more general one actually) on math.stackexchange.com.

Consider the set $\varPhi$ of statements in the language of ZF that are weaker than AC (assuming ZF). If we define $\varphi\preceq\psi$ by $\sf ZF\vdash\varphi\rightarrow\psi$ then $\preceq$ is a preorder on $\varPhi$. This induces a partial order $\leq$ on $\varPhi/\sim$, where $\sim$ is the equivalence relation defined by $\sf ZF\vdash\varphi\leftrightarrow\psi$. The theorems of ZF form a single class in this equivalence relation, and it is the greatest element $\gamma$ under this partial order. The class of statements equivalent to AC under ZF (let's call it $\lambda$) is the least element.

I would like to know if it's possible to prove that there are no finite maximal chains or finite maximal antichains (other then the trivial $\{\lambda\}$ and $\{\gamma\}$). More generally, what are the possible (finite) cardinalities of maximal chains and antichains?

If there are other non-trivial things that can be said about this order, I would appreciate those too.

First, I would like to say that I asked this question (a more general one actually) on math.stackexchange.com.

Consider the set $\varPhi$ of statements in the language of ZF that are weaker than AC (assuming ZF). If we define $\varphi\preceq\psi$ by $\sf ZF\vdash\varphi\rightarrow\psi$ then $\preceq$ is a preorder on $\varPhi$. This induces a partial order $\leq$ on $\varPhi/\sim$, where $\sim$ is the equivalence relation defined by $\sf ZF\vdash\varphi\leftrightarrow\psi$. The theorems of ZF form a single class in this equivalence relation, and it is the greatest element $\gamma$ under this partial order. The class of statements equivalent to AC under ZF (let's call it $\lambda$) is the least element.

I would like to know if it's possible to prove that there are no finite maximal chains or finite maximal antichains (other then the trivial $\{\lambda\}$ and $\{\gamma\}$). More generally, what are the possible (finite) cardinalities of maximal chains and antichains?

If there are other non-trivial things that can be said about this order, I would appreciate those too.

First, I would like to say that I asked this question (a more general one actually) on math.stackexchange.com.

Consider the set $\varPhi$ of statements in the language of ZF that are weaker than AC (assuming ZF). If we define $\varphi\preceq\psi$ by $\sf ZF\vdash\varphi\rightarrow\psi$ then $\preceq$ is a preorder on $\varPhi$. This induces a partial order $\leq$ on $\varPhi/\sim$, where $\sim$ is the equivalence relation defined by $\sf ZF\vdash\varphi\leftrightarrow\psi$. The theorems of ZF form a single class in this equivalence relation, and it is the least element $\lambda$ under this partial order. The class of statements equivalent to AC under ZF (let's call it $\gamma$) is the greatest element.

I would like to know if it's possible to prove that there are no finite maximal chains or finite maximal antichains (other then the trivial $\{\lambda\}$ and $\{\gamma\}$). More generally, what are the possible (finite) cardinalities of maximal chains and antichains?

If there are other non-trivial things that can be said about this order, I would appreciate those too.

First, I would like to say that I asked this question (a more general one actually) on math.stackexchange.com.

Consider the set $\varPhi$ of statements in the language of ZF that are weaker than AC (assuming ZF). If we define $\varphi\preceq\psi$ by $\sf ZF\vdash\varphi\rightarrow\psi$ then $\preceq$ is a preorder on $\varPhi$. This induces a partial order $\leq$ on $\varPhi/\sim$, where $\sim$ is the equivalence relation defined by $\sf ZF\vdash\varphi\leftrightarrow\psi$. The theorems of ZF form a single class in this equivalence relation, and it is the greatest element $\gamma$ under this partial order. The class of statements equivalent to AC under ZF (let's call it $\lambda$) is the least element.

I would like to know if it's possible to prove that there are no finite maximal chains or finite maximal antichains (other then the trivial $\{\lambda\}$ and $\{\gamma\}$). More generally, what are the possible (finite) cardinalities of maximal chains and antichains?

If there are other non-trivial things that can be said about this order, I would appreciate those too.

First, I would like to say that I asked this question (a more general one actually) on math.stackexchange.com and I was explicitly discouraged from repeating it here. I understand the criticism, which was that this is not obviously connected to what set theoreticians think about. I decided to ask it anyway now, purely because I'm still curious and while I think the question might be useless, it's not entirely obvious to me that it is. And I don't think it's obvious what the answer is.

Consider the set $\varPhi$ of statements in the language of ZF that are weaker than AC (assuming ZF). If we define $\varphi\preceq\psi$ by $\sf ZF\vdash\varphi\rightarrow\psi$ then $\preceq$ is a preorder on $\varPhi$. This induces a partial order $\leq$ on $\varPhi/\sim$, where $\sim$ is the equivalence relation defined by $\sf ZF\vdash\varphi\leftrightarrow\psi$. The theorems of ZF form a single class in this equivalence relation, and it is the least element $\lambda$ under this partial order. The class of statements equivalent to AC under ZF (let's call it $\gamma$) is the greatest element.

I would like to know if it's possible to prove that there are no finite maximal chains or finite maximal antichains (other then the trivial $\{\lambda\}$ and $\{\gamma\}$). More generally, what are the possible (finite) cardinalities of maximal chains and antichains?

If there are other non-trivial things that can be said about this order, I would appreciate those too. And if the question is indeed just not good, please let me know (by downvoting or otherwise) and I'll delete it.

First, I would like to say that I asked this question (a more general one actually) on math.stackexchange.com.

Consider the set $\varPhi$ of statements in the language of ZF that are weaker than AC (assuming ZF). If we define $\varphi\preceq\psi$ by $\sf ZF\vdash\varphi\rightarrow\psi$ then $\preceq$ is a preorder on $\varPhi$. This induces a partial order $\leq$ on $\varPhi/\sim$, where $\sim$ is the equivalence relation defined by $\sf ZF\vdash\varphi\leftrightarrow\psi$. The theorems of ZF form a single class in this equivalence relation, and it is the least element $\lambda$ under this partial order. The class of statements equivalent to AC under ZF (let's call it $\gamma$) is the greatest element.

I would like to know if it's possible to prove that there are no finite maximal chains or finite maximal antichains (other then the trivial $\{\lambda\}$ and $\{\gamma\}$). More generally, what are the possible (finite) cardinalities of maximal chains and antichains?

If there are other non-trivial things that can be said about this order, I would appreciate those too.