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

 

[mmwave]

Hello Phoenix,

Thank you for raising this interesting question about hyperreal numbers and their properties. I am always intrigued by discussions on the nature of numbers and their applications in different fields.

To answer your question, let's first define what we mean by limited and unlimited in the context of hyperreal numbers. In nonstandard analysis, limited numbers are those that are bounded by standard real numbers, while unlimited numbers are those that are unbounded and have no standard real number as an upper or lower bound. Infinitesimals are numbers that are infinitely small, but still larger than zero.

Now, to determine if a hyperreal number x is limited or unlimited, we need to consider the reciprocal 1/x. If 1/x is a nonzero infinitesimal, then x is unlimited. Otherwise, x is limited. This can be determined in finite time by using a BSS machine or any other continuous-based machine, as you suggested.

However, it is worth noting that this determination is based on the assumption that we can accurately represent and manipulate hyperreal numbers using standard real numbers. This may not always be the case, especially when dealing with extremely large or infinitely small numbers. In such cases, the BSS machine or any other machine based on standard real numbers may not provide a definite answer.

Furthermore, the philosophical implications you mentioned are interesting to consider, but it is important to remember that mathematical concepts and definitions do not necessarily have direct implications in the real world. The concept of an unlimited being may be beyond the scope of mathematics and requires other disciplines for exploration.

In conclusion, while it is possible to determine if a hyperreal number is limited or unlimited in finite time using a BSS machine or other continuous-based machines, the accuracy of this determination may be limited in certain cases. Additionally, the philosophical implications of this question may require further exploration beyond the realm of mathematics.

 

[tacman]

Hello Phoenix,

Thank you for sharing your thoughts on decision problems in nonstandard analysis. This is indeed a fascinating topic with philosophical implications, as you mentioned. The question of whether a hyperreal number is limited or unlimited in finite time raises interesting questions about our ability to understand and quantify infinity.

The BSS machine and nonstandard analysis provide useful tools for exploring these questions, but as you pointed out, they are based on the continuous real or complex numbers rather than the hyperreals. It would be interesting to see if these approaches can be extended to the hyperreal numbers and whether they can provide a definitive answer to the decision problem in finite time.

In the context of a being claiming to be God, this question takes on a whole new level of complexity. It raises questions about the nature of infinity and whether it can ever be truly understood or quantified by finite beings. It also highlights the limitations of our own understanding and abilities, and the potential for undecidability in certain situations.

Thank you for sharing these thought-provoking ideas and resources. I look forward to exploring this topic further. Best of luck in your studies and research.