The Problem
My professor gave us some homework exercises and there is one my colleague and I could not solve. First the definitions:
Let $A$ be a non-empty set and $R \subseteq A \times A$ a relation. Elements of the relation $R$ will be denoted as $xRy$, the shorthand notation for $\langle x,y\rangle \in R$. Let $\mathfrak{A}$ denote the structure of $(A,R)$. Let $V$ be a subset of $A$. We call $V$ $\mathfrak{A}-$small if and only if $\exists_{a \in A} [V = \{b \in A \mid bRa\}]$. $V$ is $\mathfrak{A}-$big iff $V$ is not $\mathfrak{A}-$small.
The exercise showed us some sets $V$ and most of them were determined by us to be $\mathfrak{A}-$big. However, this last problem is giving the headaches
Determine if $$V = \{a\in A \mid \neg\exists_{f:\mathbb{N} \to A}[\forall_{n \in \mathbb{N}}[f(n+1)Rf(n)] \land f(0) = a]\}$$ is $\mathfrak{A}-$big or not. Prove the result.
Attempts
First of all, we believe that $V$ is $\mathfrak{A}-$big. We tried to get to a contradiction when assuming that $V$ is $\mathfrak{A}-$small. There must therefore be an $a \in A$ such that $V = \{b\in A \mid bRa\}$.
We then made a case distinction: $a \in V$ and $a\notin V$. We managed to determine that if $a\notin V$ then a contradiction occurs and therefore $a \in V$. If you assume that $a \notin V$ then there is a function $f$ for which $f(0) = a$ and $\forall_n[ f(n+1)Rf(n)]$. It must hold then that $f(1)Ra$ and thus $f(1) \in V$. But we can construct a function $g:\mathbb{N} \to A$ as $n \mapsto f(n+1)$. It can now be seen that $g(0) = f(1)$ and $g(n+1)Rg(n)$ (This last one because $f(n+2)Rf(n+1)$. Thus $f(1)$ cannot be in $V$ so $a$ must be in $V$.
We tried something similar for the assumption that $a \in V$ but we soon realized that this method wouldn't work because we now cannot determine an element which is not $a$ to be in $V$. Even if we assume more statements (such as $\forall_n [f(n+1)Rf(n)]$), we ended up in a dead end because we need more assumptions we cannot rectify.
Can someone give us a hint into the right direction?
Thank you in advance!
If $a\in V$, then $aRa$ by your choice of $a$. Now consider the function $:\mathbb N\to A$ that is constant with value $a$.