Re: Concrete order types

On Aug 15, 6:53 pm, aatu.koskensi...@xxxxxxxxx wrote:
On 15 elo, 03:11, un student <un.stud...@xxxxxxxxx> wrote:

What does "We restrict ourselves to 'canonical' well-orderings of
concrete order types like \epsilon_0" mean? I couldn't find meaning of
"concrete order type" or "e_0".

Epsilon-0 is the least fixed point of the function alpha -->
alpha^omega, that is, epsilon-0 is the least ordinal such that
epsilon-0^omega = epsilon-0. It can also be described as the limit of
omega, omega^omega, omega^(omega^omega),
omega^(omega^(omega^omega), ... It is concrete in the sense that we
can easily visualise and explain the ordering.

Possibly you're reading something on proof theory or recursion theory.

Well, I'm sort of poking around different areas of logic and CS. Do
you happen to know any good introduction for proof or recursion
theory? (I didn't understand much about your answer...)

<..>
What, then, are the criteria for naturalness? At the moment we don't
really know how to explain or define this notion. It is, however,
almost always clear when an ordinal notation system or a (definition
of a) p.r. well-ordering is natural in the relevant sense. One obtains
the ability to recognize naturalness by osmosis, practice in proof
theory and recursion theory.

Naturalness is indeed one of those not-so-easy-to-define concepts.
Maybe one should talk about 'human order type' :)

.