~=0

This is still a work in progress. It may be changed in the future.

~=0 is a failed attempt at a string based mathematical system created by User:Yayimhere. It came out to not work, however it's still interesting to read about. It uses a set of axioms.

Alphabet

~=0 uses an alphabet called $X$. Every element of $X$ is a set of brackets({ and }). Every element can be notated as $X$ with a numeric subscript, is equal as the same number of left and right bracket pairs, with one extra set wrapping around the whole object. These brackets can (as long as they match) be written. So for example, $X_{5}=\left\{\left\{\right\}\left\{\left\{\right\}\right\}\left\{\right\}\left\{\right\}\right\}$. This translation can of course be translated in the opposite order(from a bracket string to the subscript notation), however that direction is generally the only deterministic one. $\chi$ is the set of all possible concatenations of elements in $X$

Axioms

Alike cirt e mys, ~=0 is built on a notation for string modification. These can be defined as a sort of axioms:

• $A\bullet B=AB$(concatenation)
• $A*n=f^{n}(A),f(x)=x\bullet A$($n$ must be a number)

These operators can be used in the program. There is one special/uncommon operator:

• $X_{n}\times X_{k}=(X_{0}*k)\bullet X_{n}$

which has the single axiom:

• $X_{n}\times X_{k}=X_{k}\times X_{n}$

as well as an extra little axiom:

• $f^{0}(x)=x$

Which fully collected is:

• $A\bullet B=AB,A,B\in X,A\bullet B\in \chi$
• $A*n=f^{n}(A),f(x)=(x\bullet A),n\in \mathbb {Z} ^{+},A\in X$
• $X_{n}\times X_{k}=(X_{0}*k)\bullet X_{n},n,k\in \mathbb {Z} ^{+},X_{n}\times X_{k}\in \chi$
• $X_{n}\times X_{k}=X_{k}\times X_{n}$
• $f^{0}(x)=x$

Example

Here's an example of the failings of this system:
$\left\{\left\{\right\}\right\}\times \left\{\right\}=X_{1}\times X_{0}=\left\{\left\{\right\}\right\}=X_{0}\times X_{1}=\left\{\right\}\Rightarrow X_{1}=X_{0}$

Variation

There exists a variation, called 0*=~. It changes the axiom set to the following:

• $A\bullet B=AB,A,B\in X,A\bullet B\in \chi$
• $A*n=f^{n}(A),f(x)=(x\bullet A),n\in \mathbb {Z} ^{+},A\in X$
• $X_{n}\times X_{k}=(X_{0}*k)\bullet X_{n},n,k\in \mathbb {Z} ^{+},X_{n}\times X_{k}\in \chi$
• $X_{0}*n\bullet X_{c}*n=(X_{0}\bullet X_{c})*n,c,n\in \mathbb {Z} ^{+}$
• $f^{0}(x)=x$