# set theory – How to compare three supremums of ordinals eventually writable by Ordinal Turing Machines?

This question implies that we have fixed: (i) a particular enumeration of Ordinal Turing machines; (ii) a particular way to encode an ordinal by an infinite binary sequence.

The class of $$(1)$$-machines is defined as the $$1$$st iteration of the strong jump operator for Ordinal Turing Machines. That is, a machine is equipped with an oracle able to answer any question of the following form (note that $$(0)$$-machines are Ordinal Turing Machines with no oracles):

Does an $$i$$-th $$(0)$$-machine halt given an infinite binary sequence $$x$$ as the input?

The ordinal $$alpha_1$$ is defined as the supremum of ordinals eventually writable by $$(1)$$-machines with empty input.

Let $$m_i(x)$$ denote a computation performed by an $$i$$-th $$(0)$$-machine, assuming that the input is $$x$$. If $$m_i(x)$$ eventually writes a countable ordinal $$alpha$$, then $$M_i(x) = alpha$$. Otherwise, $$M_i(x) = 0$$.

Then the function $$F(i)$$ is defined as follows: if the value of $$sup {M_i(x) | x in mathbb{R}}$$ is a countable ordinal, then $$F(i) = sup {M_i(x) | x in mathbb{R}};$$ otherwise, $$F(i) = 0$$. Here “$$x in mathbb{R}$$” implies that we take into account all infinite binary sequences.

The ordinal $$alpha_2$$ is defined as follows: $$alpha_2 = sup {F(i) | i in mathbb{N}}.$$.

The ordinal $$eta$$ is defined as the least ordinal $$gamma$$ such that $$L_gamma$$ and $$L$$ have the same $$Sigma_2$$-theory (see part 3 of Lemma 3.11 in the paper “Recognizable sets and Woodin cardinals: Computation beyond the constructible universe”). As far as I understand, $$eta$$ is equal to the supremum of ordinals eventually writable by Ordinal Turing Machines ($$(0)$$-machines) with empty input.

Question: which ordinal is larger, $$alpha_1$$ or $$alpha_2$$? Is any of these two ordinals larger than $$eta$$?