${}^{\omega+1}\omega,{}^{\omega+2}\omega,{}^{\omega+3}\omega,...,{\omega+n}^n\omega,{}^{\omega+(n+1)}\omega,...$

that is, we don't allow any ordinals of the form ${}^{\lambda}\omega$ to have a series to start at terms $<{}^{\omega+1}\omega$.

Finally, to plug in the hole and complete the introduction of the discontinuity, I define ${}^{\omega+1}\omega$ as a fixed point of the exponenetial map $x\mapsto\omega^{x}$ and using the modified hyperopations, I can show it is not equal to $\epsilon_0$

> Formal construction of negative integers[edit]
See also: Integer § Construction
In a similar manner to rational numbers, we can extend the natural numbers N to the integers Z by defining integers as an ordered pair of natural numbers (a, b). We can extend addition and multiplication to these pairs with the following rules:

(a, b) + (c, d) = (a + c, b + d)
(a, b) × (c, d) = (a × c + b × d, a × d + b × c)
We define an equivalence relation ~ upon these pairs with the following rule:

(a, b) ~ (c, d) if and only if a + d = b + c.

$\textbf{Ver 7}$
define ordinal hyperoperation as follows:

Let $\alpha,\beta,\gamma,\delta,\lambda,m \in \textrm{On}$ where $\lambda$ is a limit ordinal and the rest are arbitrary. Then:

Associativity
\begin{align}
m=\{0,1,2\}: \alpha [m] (\beta [m] \gamma) & = (\alpha [m] \beta) [m] \gamma\\
\end{align}

Illustration:
\begin{align}
\alpha [0] \beta = \alpha^+\\
\alpha [1] \beta = \alpha+\beta\\
\alpha [2] \beta = \alpha\beta\\
\alpha [3] \beta = \alpha^{\beta}\\
\alpha [4] \beta = {}^{\beta}\alpha\\

[Cont.]
\begin{align}
\epsilon_0 & =\omega[4]\omega = \omega [3]^{\omega}\omega = a_{\omega,3}(\omega) = a_{0,4}(\omega)\\
\omega[4](\omega+1) & = \omega [3]^{\omega+1} \omega = \omega [3]^{\omega} (\omega [3] \omega) = a_{\omega,3}(\omega^{\omega}) = a_{0,4}(\omega^{\omega})\\
\omega [4] (\omega [2] 2) & = \omega [3]^{\omega [2] 2} \omega =a_{\omega 2,3} (\omega)\\
\omega [4] (\omega [3] \omega) & =\omega [3] ^{\omega [3] \omega} \omega = a_{\omega^\omega,3} (\omega)\\
\epsilon_0^{\omega} & =(\omega [4] \omega) [3] \omega = (\omega [3]^{\omega}) [3] \omega = a_{\omega,3}(\omega) [3] \omeg…

${}^{\omega+m}\omega=\sup ({}^{n+m}\omega|n \in \Bbb{N},m \in \Bbb{N}\textrm{ fixed})=\sup(\omega,{}^{1+m}\omega,{}^{2+m}\omega,{}^{3+m}\omega,...)=\epsilon_0$

${}^{m+\omega}\omega=\sup ({}^{m+n}\omega|n \in \Bbb{N},m \in \Bbb{N}\textrm{ fixed})=\sup(\omega,{}^{m+1}\omega,{}^{2+m}\omega,{}^{3+m}\omega,...)=\epsilon_0$

${}^{\omega 2}\omega=\sup ({}^{n 2}\omega|n \in \Bbb{N})=\sup(1,{}^2\omega,{}^4\omega,{}^6\omega,...)=\epsilon_0$

${}^{\omega^2}\omega=\sup ({}^{n^2}\omega|n \in \Bbb{N})=\sup(1,\omega,{}^4\omega,{}^9\omega,...)=\epsilon_0$