I have a question which formulas are in the set $\kappa$ (which is defined below). Sadly, for this I have to introduce some definitions and I apologize in advance for making the reader go through this:
We have defined the set $\Sigma_{Form}$of $\Sigma$-formulas as the smallest set of strings over the alphabet $\Sigma\cup\mathbb{V}\cup \left\{ \boldsymbol{\exists},\boldsymbol{\&},\boldsymbol{\neg},\boldsymbol{=} \right\}$ (where $\Sigma$ is the signature and $\mathbb{V}=\left\{v_0,v_1,v_2,\ldots \right\}$ is a countable set of variables) such that:
$ \blacktriangleright \ \boldsymbol{\exists}v \psi\in \Sigma_{Form} \ $ if $\psi$ were in $\Sigma_{Form}$ and $v\in \mathbb{V}$
$ \blacktriangleright \ \Omega \boldsymbol{\&} \psi\in \Sigma_{Form}$ if $\Omega,\psi$ were in $\Sigma_{Form}$
$ \blacktriangleright \ \boldsymbol{\neg} \psi\in \Sigma_{Form}$ if $\psi$ were in $\Sigma_{Form}$
$ \blacktriangleright \ \mathtt{t_1\boldsymbol{=}t_2}\in \Sigma_{Form}$ if the $\Sigma$-terms $\mathtt{t_1,t_2}$ were in $\Sigma$-terms (see my previous post , to see what a $\Sigma$-term is)
$ \blacktriangleright \ \ \ \mathtt{Rt_1\ldots t_n}\in \Sigma_{Form}$ if the $\Sigma$-terms $\mathtt{t_1,\ldots ,t_n}$ were $\Sigma$-terms and $\mathtt{R}$ was a symbol for a relation of arity $n$ in the signature $\Sigma$
and building on this we have introduced the following "deductive calculus": The set $\kappa$ is the smallest set of $\sigma$ - formulas such that
$ \blacktriangleright \ v_0 \boldsymbol{=}v_0$, $\ \boldsymbol{\neg}(v_0\boldsymbol{=} v_1 \boldsymbol{\&} \boldsymbol{\neg}(v_1 \boldsymbol{=}v_0)),$
$\ \boldsymbol{\neg}(v_0\boldsymbol{=} v_1 \boldsymbol{\&} v_1 \boldsymbol{=}v_2 \boldsymbol{\&} \boldsymbol{\neg}(v_0 \boldsymbol{=}v_2)),$
$ \boldsymbol{\neg}(v_1\boldsymbol{=} v_{n+1} \boldsymbol{\&} v_2\boldsymbol{=} v_{n+2}\boldsymbol{\&} \ldots \boldsymbol{\&}v_n\boldsymbol{=} v_{2n} \boldsymbol{\&} \boldsymbol{\neg}(\mathtt{f}v_1\ldots v_{2n} ))$ (where $\mathtt{f}$ is an $n$-ary function symbol from $\Sigma$),
$ \boldsymbol{\neg}(v_1\boldsymbol{=} v_{n+1} \boldsymbol{\&} v_2\boldsymbol{=} v_{n+2}\boldsymbol{\&} \ldots \boldsymbol{\&}v_n\boldsymbol{=} v_{2n} \boldsymbol{\&} \boldsymbol{\neg}(\mathtt{R}v_1\ldots v_{2n} ))$ (where $\mathtt{f}$ is an $n$-ary relation symbol from $\Sigma$)
are all in $\kappa$
$ \blacktriangleright $ all tautologies are in $\kappa$ (meaning if we replace the predicates in a tautology from propositional calculus with $\Sigma$-formulas, then we have a tautology in $\kappa$).
$ \blacktriangleright $ $\Omega \in \kappa$ if the $\Sigma$-formulas $\psi,\ \boldsymbol{\neg}(\psi \boldsymbol{\&} \boldsymbol{\neg} \Omega)$ were in $\kappa$ (this is the modus ponens)
$ \blacktriangleright $ $ \boldsymbol{\neg}(\psi_{v\rightarrow \mathtt{t}} \boldsymbol{\&} \boldsymbol{\neg} \boldsymbol{\exists}v\psi) \in \kappa$ if the $\Sigma$-formula $\psi$ were in $\kappa$, $v\in \mathbb{V}$ and $\mathtt{t}$ were a $\Sigma$-term, whose variables do not appear in $\psi$ ("$\psi_{v \rightarrow \mathtt{t}}$" means, the variable $v$ in $\psi$ is replaced with the term $\mathtt{t}$)
$ \blacktriangleright $ $ \boldsymbol{\neg} ( \boldsymbol{\exists} v \psi \boldsymbol{\&} \boldsymbol{\neg} \Omega ) \in \kappa$ if $ \boldsymbol{\neg}(\psi \boldsymbol{\&} \boldsymbol{\neg} \Omega)$ were in $\kappa$ and $v\in \mathbb{V}$ is not a free variable in $\Omega$
Can I prove for example, that $v_1 \boldsymbol{=} v_1 \in \kappa$. I don't see any way how I could do this...Could it maybe be proved that $v_1 \boldsymbol{=} v_1 \not \in \kappa$?
If $\mathtt{0,1,2} $ are function symbols of arity $0$ (they are constants) can I then prove, that $\boldsymbol{\neg}(0\boldsymbol{=}2 \ \boldsymbol{\&} \ 1\boldsymbol{=}2)$ is not in $\kappa$ ?