Decision problem in nonstandard analysis

[phoenixthoth]

hello

given a random hyperreal number x, can one decide if it is limited or unlimited in a finite amount of time? this is equivalent to the question of whether it can be decided if 1/x is a nonzero infinitesimal in finite time.

here's a tutorial paper on the BSS machine which, i believe, involves basing machines on the continuous rather than the discrete. http://www.ulb.ac.be/assoc/bms/Bulletin/bul971/meer.pdf . i have a feeling that the BSS machine is what will lead to an answer to my question if it isn't trivial. if it is trivial, let me know. the only thing is that the BSS machine is based on R or C and not R*, the set of hyperreal numbers. hmm...

here's a tutorial paper on nonstandard analysis which gives the definition of the hyperreals along with limited, unlimited, and infinitesimal. it was written by a grad student, so excuse the shotty work of a hack: http://online.sfsu.edu/~brian271/nsa.pdf

i believe that the question has philosophical implications regarding the situation of a being claiming to be God (akin to "unlimited") and others trying to decide if it really is "unlimited." if this can't be done in finite time, that would be interesting, wouldn't it? and if it's undecidable, that would be interesting, wouldn't it? but if it can be done in finite time, that wouldn't really help us in the real world to actually prove that a being is "unlimited."

cheers,
phoenix

 

[Hurkyl]

How is the hyperreal number specified?

Anyways, terms like "limited" and "infintessimal" are relationships between the real numbers and the hyperreal numbers. If reality really is hyperrealistic (whatever that means!) then "limited" is an irrelevant term.

Besides, we would have hyperfinite time to solve problems.


Incidentally, I don't think it's possible to define "limited" et cetera strictly in terms of the hyperreals. The terms really only does have meaning in terms of the relationship between reals and hyperreals.

(Of course, I'm hardly an expert on these things)

 

[phoenixthoth]

Originally posted by Hurkyl
How is the hyperreal number specified?

Anyways, terms like "limited" and "infintessimal" are relationships between the real numbers and the hyperreal numbers. If reality really is hyperrealistic (whatever that means!) then "limited" is an irrelevant term.

Besides, we would have hyperfinite time to solve problems.


Incidentally, I don't think it's possible to define "limited" et cetera strictly in terms of the hyperreals. The terms really only does have meaning in terms of the relationship between reals and hyperreals.

(Of course, I'm hardly an expert on these things)

read the nsa.pdf file for your first question.

as for your second paragraph, why would "limited" be an irrelevant term? to me, that sounds as arbitary as saying "negative" is an irrelevant term. if the universe were infinite, for example, or if it was infinite dimensional, for example, (and neither of those things may be true), then terms like "unlimited" might be better than "infinite" for they are more precise. and rather than some physical quantities being zero, perhaps they are really infinitesimal. and instead of quantities being infinite, they could be considered unlimited. in nonstandard analysis, one can literally plug in an unlimited number and still get a result in the hyperreal set. limits go away and calculus becomes like algebra. who knows, this might help clear up divergence problems with infinite quantities in science. so i think the terms "limited" and "unlimited" may have quite a relevance.

hyperfinite is the same as finite.

the dependence of terms like "limited" on a relationship to reals is not a problem whatsoever.

cheers,
phoenix