The hyperfinite timeline
post by Alok Singh (OldManNick) · 2022-12-30T09:30:06.483Z · LW · GW · 6 commentsTake an interval. Cut it into H pieces, where H is hyperfinite. This serves as the index set of a stochastic process, among many other uses. Imagine that for each of the H steps, you flip a coin to get -1 or +1. Then move an infinitesimal distance left or right based on the sign. This is Brownian motion. Each infinitesimal piece of the timeline is profitably thought of as a Planck time.
Discrete events, such as sudden hard shocks, can be modeled on this line. They are appreciable over an infinitesimal fraction of the line.
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comment by Alok Singh (OldManNick) · 2022-12-30T22:30:20.391Z · LW(p) · GW(p)
Was being deliberately inaccurate here. Hard does mean more than a limited multiplier. Sudden means that there's an appreciable change over an infinitesimal variation aka discontinuous.
Lookup overflow, underflow, and "principle of permanence" in Goldblatt for why I'd do that. Also called overspill and underspill. The basic idea is "as above, so below" except this link is 2 way. Say some internal function has all infinitesimals in its range. Then it must have non infinitesimals too, since the set of all infinitesimals is known to be external, and images of internal functions over internal sets are internal. This is an example of overspill. Infinitesimal behavior has spilled over into the appreciable domain.
Replies from: Slidercomment by Slider · 2022-12-30T11:57:32.121Z · LW(p) · GW(p)
You can get to a second by a finite multiplier of planck time which is a property that this unit seems to lack. So in what dimensions is the analog meant?
"sudden" means only in a finite amount of H steps?
"hard" means more than a finite multiplier of +1?
Replies from: Mitchell_Porter↑ comment by Mitchell_Porter · 2022-12-30T12:40:49.611Z · LW(p) · GW(p)
Right, the Planck time is not infinitesimal. It's a finite interval of time whose duration is 1 in Planck units.
comment by Alexander Gietelink Oldenziel (alexander-gietelink-oldenziel) · 2023-01-02T11:17:55.905Z · LW(p) · GW(p)
Where can I learn more about hyperfinite Brownian motion?
Has this been developed deeply? (I am aware of Nelson's radically elementary probability book)
Replies from: OldManNick↑ comment by Alok Singh (OldManNick) · 2023-01-02T21:58:09.544Z · LW(p) · GW(p)
https://link.springer.com/book/10.1007/978-3-642-33149-7
Also includes Feynman path integral and a few other things. Note that you don't even need the full nonstandard theory.