Instants are usually said to be the infinitely small units of time which end an interval.

I think that these infinitesimals do away with the paradox of motion etc.

But for those who do ascribe to instants (everyone?) do they do so to every finite interval?

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    An open interval [0,1[ does not end with an instant. Not sure it's the question. – Quentin Ruyant Jun 4 '15 at 9:54
  • what's an open instant - ? – user6917 Jun 4 '15 at 15:28
  • An open interval, not instant – Quentin Ruyant Jun 4 '15 at 18:44
  • It depends on the type of interval. A wave (analog) would end at a trough which may be more than one point (A trough could be 2 lowest points of equal value). A pulse (digital) would seem to end at a single point but now that I think about it the only types of pulses that I know of are light, sound, etc which themselves are in fact waves. Hmmm... – Ronnie Royston Jun 5 '15 at 5:12

Sounds like you are thinking of non-standard analysis. On the hyperreal line, extension of the real line that contains infinitesimals, every number has an infinitesimal "monad" (the term goes back to Leibniz), a.k.a. "halo". This monad consists of all numbers infinitesimally close to it. Every interval with real endpoints will "end" with a "half" of their monads. The semi-open interval [0,1), for example, ends with all hyperreals of the form 1-ε, where ε is a positive infinitesimal. But monads do not contain a single "instant", there is a plethora of "instants" of different orders of smallness, a whole continuum of them.

Not sure how this helps with paradoxes of motion though.

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Instants, monads, closed and open intervals, etc. are terms that have been defined for different views of temporal “works.” These definitions are particular to the world-view being described and may, in fact, be different for different world-views and systems of thought. I ask, “Under whose world-view are you attempting to answer the question?”

I have a world-view that is described in Natural Logic of Space and Time (NL) which is a logic system and parallel-concurrent algebra suitable for DES (discrete event systems) process specification and control. For every condition that occurs there is an event that caused it. So, in my view and system, every meaningful interval (at which a condition significant to the process exists) has a causative event at either end. Meaningful, in this regard, is one that constructively participates in and is integral to the process.

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