Running an infinite amount of Turing Machine steps in a finite amount of time: What consequences?

logic set-theory first-order-logic computability turing-machines

I don’t have time to write up a detailed answer right now, but the sets accepted by these generalized Turing machines are the hyperarithmetic (or $\Delta^1_1)$ sets. This has been studied extensively; look up Addison and the $\aleph_0$-mind in Hartley Rogers Jr., The Theory of Recursive Functions and Effective Computability.

Note that these generalized Turing machines are not physically realizable; this is an abstract model of computation.

The OP didn’t say, but I’ve assumed here that these “infinite Turing machines” can be used as blocks in other more complicated infinite Turing machines. Without that, it’s much less interesting.

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