User blog:Edwin Shade/The Grand List Of Transfinite Ordinals

$$\omega$$
The first transfinite ordinal. It describes the order type of the natural numbers, and is the supremum of the sequence $$\{0,1,2,3,4,...\}$$. It also happens to be a limit ordinal, and is the second one, (the first is 0, which has no predecessor in the set of natural numbers).

$$\omega+1$$
The first transfinite ordinal that is a successor ordinal, or an ordinal which follows immediately after a previous ordinal.

$${\omega}2$$
The third limit ordinal, and the supremum of $$\{\omega,\omega+1,\omega+2,\omega+3,\omega+4,...\}$$.

$${\omega}^2$$
The proof-theoretic ordinal of RFA, or rudimentary function arithmetic. It's fundamental sequence is $$\{\omega,\omega 2,\omega 3,\omega 4,\omega 5,...\}$$.

$${\omega}^3$$
The proof-theoretic ordinal of EFA, or elementary function arithmetic. It's fundamental sequence is $$\{{\omega^2},{\omega^2}2,{\omega^2}3,{\omega^2}4,{\omega^2}5,...\}$$.

$${\omega^{\omega}}$$
The proof-theoretic ordinal of PRA, or primitive recursive arithmetic. It is the supremum of $$\{1,\omega,\omega^2,\omega^3,\omega^4,...\}$$. When the fast-growing hierarchy is indexed by this ordinal it represents the growth rate of Bower's Linear Array Notation.

$$\epsilon_0$$
The proof-theoretic ordinal of PA, or Peano arithmetic. It is the limit of the sequence $$\{1,\omega,\omega^{\omega},\omega^{\omega^{\omega}},\omega^{\omega^{\omega^{\omega}}},...\}$$, and when the fast-growing hierarchy is indexed by this ordinal it represents the limit of the growth rate of Saibian's Cascading-E Notation. It is also the first ordinal not in the set $$\{0,1,\omega\}$$ closed by the first three hyperoperators.

I will add at least one entry to this list a day, such that it will eventually supersede anything anyone has ever devised before, and be an accounting of every significant ordinal.

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