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Dragonfall
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I've been told that non-well-founded sets can be used to model nonstandard analysis, but I can't find any literature on the subject. Does anyone know where I can find it?
My best guess at what you mean is the following...Dragonfall said:I've been told that non-well-founded sets can be used to model nonstandard analysis, but I can't find any literature on the subject. Does anyone know where I can find it?
Dragonfall said:I've been told that non-well-founded sets can be used to model nonstandard analysis, but I can't find any literature on the subject. Does anyone know where I can find it?
Hurkyl said:My best guess at what you mean is the following...
First see this section of Wikipedia's article on NSA.
In order to do "ordinary" NSA, no non-well-foundedness is needed. Note that what they denote as V is actually [itex]V_\omega[/itex].
As I understand it, a problem occurs only when you consider unbounded formulae, or if you want to work in some [itex]V_\alpha[/itex] with [itex]\alpha > \omega[/itex]. In order to do either of these, you must assume some version of anti-foundation.
There's no problem with [itex]x_1[/itex] being untyped?gulliput said:Every nonstandard model IS non-wellfounded "in the standard sense" - if it is countably saturated ("a minimal requirement" for doing NSA) then there are infinitely descending 'membership chains' - [itex]x_1\ni x2 \ni \ldots[/itex]
Hurkyl said:There's no problem with [itex]x_1[/itex] being untyped?
Or, I suppose a more accurate question is whether requiring constants to be typed eliminates this issue.
Dragonfall said:I've read the section from Wikipedia and I can't see why we should evoke non-well-founded sets or use V_a for a greater than omega.
Nonstandard Analysis is a mathematical framework that extends the traditional methods of calculus and analysis to include infinitesimal and infinite numbers. It is based on the concept of non-well-founded sets, which allows for the existence of infinitesimal and infinite elements within a set.
Nonstandard Analysis differs from traditional analysis in that it allows for the inclusion of infinitesimal and infinite numbers in mathematical calculations. This allows for a more precise and intuitive understanding of mathematical concepts such as limits, continuity, and derivatives.
Non-well-founded sets are mathematical sets that contain elements which are not well-defined or do not follow the traditional rules of set theory. This includes infinitesimal and infinite elements, which are crucial in Nonstandard Analysis.
Nonstandard Analysis has applications in various fields, including physics, engineering, and economics. It allows for a more accurate and efficient analysis of systems that involve infinitesimal and infinite quantities, such as chaotic systems and optimization problems.
While Nonstandard Analysis offers many advantages, it also has some limitations. The use of infinitesimal and infinite elements can lead to inconsistencies if not used properly, and it may not always provide unique solutions to problems. Additionally, it is still a developing field and may not have as many established techniques and results as traditional analysis.