Decision problem in nonstandard analysis

In summary, the conversation discusses the concept of hyperreal numbers and their relationship to real numbers. It is questioned whether it is possible to determine if a hyperreal number is limited or unlimited in a finite amount of time, and whether this has philosophical implications. Various resources are shared, including a tutorial paper on the BSS machine and nonstandard analysis. The relevance of terms such as "limited" and "unlimited" in understanding the hyperreal numbers is discussed.
  • #1
phoenixthoth
1,605
2
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
 
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  • #2
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)
 
  • #3
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
 
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1. What is the difference between standard and nonstandard analysis?

Standard analysis is based on the principles of classical calculus, while nonstandard analysis uses a different approach known as Robinson's framework. This framework allows for the use of infinitesimal and infinite numbers, which are not allowed in standard analysis.

2. What is a decision problem in nonstandard analysis?

A decision problem in nonstandard analysis is a mathematical question that requires a yes/no answer, and can be solved using the principles of nonstandard analysis. These problems often involve the use of infinitesimal and infinite numbers to solve them.

3. How is nonstandard analysis used in real-world applications?

Nonstandard analysis has been used in various fields such as physics, economics, and engineering to provide more accurate and efficient solutions to problems. For example, it has been used in the study of fluid dynamics and the optimization of financial portfolios.

4. What are some limitations of nonstandard analysis?

One limitation of nonstandard analysis is that it is not widely accepted or taught in mainstream mathematics, making it less accessible to many mathematicians and scientists. Additionally, some critics argue that the use of infinitesimal and infinite numbers is not rigorous enough to be considered a valid mathematical approach.

5. How does the use of infinitesimals in nonstandard analysis affect the concept of limit?

In nonstandard analysis, the concept of limit is extended to include infinitesimal values. This means that a function can have a limit at a point even if it is not continuous at that point. However, the use of infinitesimals does not change the fundamental properties of limits, such as the squeeze theorem and the algebraic limit theorem.

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