Ytukyg

This resulted from trying to write a proof for the definition of equality. In the end I gave up, but I did realise that under the specifications I had provided, any definition for anything would ultimately be either circular or tautological.

I thought this was worth working on, because it could serve to explain and justify some of the seemingly arbitrary conventions in different areas of mathematics.

But, I don't usually write proofs, I'm not a mathematician, and my vocabulary is somewhat limited - so I would like

Definition: A system is a set $S$, along with all operations and relations defined on $S$.

Definition: The definition of an element $sin S$ is a unique, true statement $s=x$, for which $s,xin S$. Each element of $S$ may have one and only one definition.

Definition: A definition is a tautology, when it is written as $x=x$.

Definition: A definition is circular when it is written as $x=yast_0z,y=xast_1z,z=xast_2y$, thus $x=(xast_1z)ast_0z=(xast_1xast_2y)ast_0(xast_2 y)=(xast_1xast_2(xast_1z))ast_0(xast_2(xast_1z))=cdots$ - i.e. the definition contains terms which are defined by, or which contain terms which are defined by, etc., $x$.

Theorem:

For every system $X$ with finitely many elements for which every element is defined, there is at least one element whose definition in $X$ is a tautology. If no element of $X$ is defined by tautology, then the definition of every element in $X$ belongs to a cyclic sequence of codependent definitions (the definition of $x$ is circular for all $xin X$).

Proof:

1. Let $X_n$ be the system containing the elements in $bigcup_{i=0}^n{x_i}$.


2. Let $Defleft(xmid Xright)$ be the definitions of $x$ in $X$.


3. From 1. and 2., it follows that $forall x_iin X_n.Defleft(x_imid X_nright)in Y:\ Y=left{{''x_i=x_i}''mid0leq ileq nright}cupleft{''x_i=f_i(textbf{x})''mid f_i:textbf{x}mapsto x_iland x_inotintextbf{x}right}$
   


4. If $exists x_iin X_n:Defleft(x_imid X_nright)={''x_i=x_i}''$ then the definition of $x_i$ is a tautology


5. If $nexists x_iin X_n:Defleft(x_imid X_nright)={''x_i=x_i}''$, then
   $Defleft(x_imid X_nright)in Z: Z=left{{''x_i=f_i(textbf{x})}''mid f_i:textbf{x}mapsto x_iland x_inotintextbf{x}right}$; Thus, the definition of $x_i$ is circular for all $x_iin X_n$.
   $square$
   

What this is supposed to mean:

It is impossible to have any system, including but not limited to groups, algebras, theories, and natural language, where every object is defined and no definitions are circular or tautological. As a system, mathematics itself cannot be closed without introducing tautologies or circular definitions.

So, is this correct and is there anything I can do to improve? Have I misapplied any terms? Is the notation (ignoring $x_iintextbf{x}$ which is a deliberate abuse of notation to save space) correct?