# Decision problem in nonstandard analysis

## given a random hyperreal x, can it be decided in finite time whether x is limited?

• ### it's undecidable. (or is that the same as option2?)

• Total voters
2

#### 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 [Broken]

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

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