[Math] word for an “infinite algorithm”

Question

According to Knuth's notes (see Slide 3), an algorithm, by definition, satisfies the following five properties:

1. Finiteness: Terminates after a finite number of steps.

2. Definiteness: Each step is precisely defined.

3. Input: Has zero or more inputs.
4. Output: Has one or more outputs, each of which has a specified relation to the inputs.
5. Effectiveness: All operations are sufficiently basic that they may be performed exactly and in finite length.

Is there a term for something that satisfies steps 2-5 and also 1 with the word finite replaced by countable?

Answer 1

You should check out

• Infinite Time Turing Machines by Joel David Hamkins and Andy Lewis,
• A Survey of Infinite Time Turing Machines by Joel David Hamkins,
• Infinite Time Turing Machines: Supertask Computation ( - "an expository account [...] for a philosophical audience" - ) by Joel David Hamkins,
• Wikipedia: Supertask, Hypercomputation.