Let $\Gamma = \{\exists x \forall y (x\mathrel R y),\exists x \forall y(y\mathrel R x),\forall x\exists y(x\mathrel R y)\}$. Is $\Gamma$ consistent?

Question

Consider the language consisting of one symbol $R$ for a binary relation. Let $\Gamma = \{\exists x \forall y (x\mathrel R y),\exists x \forall y(y\mathrel R x),\forall x\exists y(x\mathrel R y)\}$. Is $\Gamma$ consistent?

I know that $\Gamma$ is consistent if and only if it has a model. So I believe my task is to find a model that these sentences are satisfied by. Is the natural numbers together with the less than or equal to relation a valid model?

Answer 1

If you think about $R$ being a partial order, then these mean:

1. $\exists x\forall y(x\mathrel R y)$, there is a minimum.
2. $\exists x\forall y(y\mathrel R x)$, there is a maximum.
3. $\forall x\exists y(x\mathrel R y)$, every element is related to someone.

Of course, if you want $R$ to be a partial order, then for $x$ being the maximum element the only $y$ such that $x\mathrel R y$ can be true is $x=y$. But partial orders are reflexive. So that's not a problem.

Does the natural numbers have a maximum?

What is the simplest partial order which has a minimum and maximum? (recall, these need not be distinct!)

Answer 2

Hint: There is not a single negation symbol to be seen in $\Gamma$.