Phi function

Phi function is the ordinal collapsing function with two common variations: binary and linear. The first one has limit ordinal \(\Gamma_0\), while the second limits at \(\theta(\Omega^\omega)\), using the more powerful and extensible theta function. Essentially, the linear variant is an extension of binary, so all rules for linear works with binary too. Formal definition goes as follows:

Rule 1. Condition: arity = 1

$$\phi(\alpha) = \omega^\alpha$$

Rule 2. Condition: first entry = 0 and arity > 1

$$\phi(0,\#) = \phi(\#)$$

Rule 3. Condition: arity = 2, $$\alpha=1, \beta=S$$

$$\phi(1,\beta+1)[1] = \phi(1,\beta)$$

$$\phi(1,\beta+1)[n] = \phi(1,\beta)^{\phi(1,\beta+1)[n-1]}$$

Rule 4. Condition: $$\alpha=L,\beta=0$$

$$\phi(\#,\alpha,0,\cdots,0,0)[n] = \phi(\#,\alpha[n],0,\cdots,0,0)$$

Rule 5. Condition: $$\alpha=L,\beta=S$$

$$\phi(\#,\alpha,0,\cdots,0,\beta+1)[n] = \phi(\#,\alpha[n],\phi(\#,\alpha,0,\cdots,0,\beta)+1,\cdots,0,0)$$

Rule 6. Condition: $$\alpha=S,\beta=0$$

$$\phi(\#,\alpha+1,0,\cdots,0,0)[1] = \phi(\#,\alpha,0,\cdots,0,0)$$

$$\phi(\#,\alpha+1,0,\cdots,0,0)[n] = \phi(\#,\alpha,\phi(\#,\alpha+1,0,\cdots,0,0)[n-1],\cdots,0,0)$$

Rule 7. Condition: $$\alpha=S,\beta=S$$

$$\phi(\#,\alpha+1,0,\cdots,0,\beta+1)[1] = \phi(\#,\alpha,0,\cdots,0,\beta)+1$$

$$\phi(\#,\alpha+1,0,\cdots,0,\beta+1)[n] = \phi(\#,\alpha,\phi(\#,\alpha+1,0,\cdots,0,\beta+1)[n-1],\cdots,0,0)$$

Rule 8. Condition: $$\beta=L$$

$$\phi(\#,\alpha,0,\cdots,0,\beta)[n] = \phi(\#,\alpha,0,\cdots,0,\beta[n])$$