Nonstandard Analysis models using non-well-founded sets

[Dragonfall]

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]

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...

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 $V_\omega$.


As I understand it, a problem occurs only when you consider unbounded formulae, or if you want to work in some $V_\alpha$ with $\alpha > \omega$. In order to do either of these, you must assume some version of anti-foundation.

 

[gulliput]

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?

see section 8.3 'Non-well-founded set theories' in V.Kanovei and M.Reeken's book 'Nonstandard analysis, axiomatically' for a review.

you might also be interested in our Theory of Hyperfinite Sets where 'nonstandard analysis' is being developed within a non-well founded universe of hereditarily hyperfinite sets:
http://arxiv.org/abs/math/0502393

it is also interesting that (roughly speaking) what is needed for nonstandard analysis is infinitely descending 'membership chains', no 'cyclic memberships' are needed (although non-well-foundedness is usually associated with the possibility of cycles)

 

[gulliput]

Wikipedia article's description of logical foundations of NSA is quite limited and also outdated, to my opinion.

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' - $x_1\ni x2 \ni \ldots$

The nonstandard $V_\omega$ is "actually" a substructure of a saturated elementary extension of the standard $V_\omega$ - therefore it is non-wellfounded.

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 $V_\omega$.


As I understand it, a problem occurs only when you consider unbounded formulae, or if you want to work in some $V_\alpha$ with $\alpha > \omega$. In order to do either of these, you must assume some version of anti-foundation.

 

[Dragonfall]

Thanks for the replies. I'll check out the book.

 

[Hurkyl]

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' - $x_1\ni x2 \ni \ldots$

There's no problem with $x_1$ being untyped?

Or, I suppose a more accurate question is whether requiring constants to be typed eliminates this issue.

 

[gulliput]

There's no problem with $x_1$ being untyped?

Or, I suppose a more accurate question is whether requiring constants to be typed eliminates this issue.

I was inaccurate stating that in the "superstructure NSA" (the framework described in the Wikipedia article) there is non-wellfoundedness. The superstructure is indeed a substructure of an elementary saturated extension of some standard structure but this substructure is built so that it eliminates non-wellfoundedness of the extension. But this has its issues: for instance, you cannot have von Neumann natural numbers (every n is the set of smaller numbers) "on the ground floor" if you want no nonwellfoundedness whatsoever.

What is important for me about nonwellfoundedness is that it allows to consider the universe (of sets) being genuinely nonstandard (we were not aware of its nonstandardness before due to our limitations) - instead of seeing the nonstandard universe as some weird structure on top and inside of "the true universe".

 

[Dragonfall]

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.

 

[gulliput]

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.

as i said the contents of this section in Wikipedia is much narrower than it's title (it is not encyclopedic enough) since it does not speak bout more recent work on axiomatic foundations of nsa.

answering to your question - if we are satisfied with superstructures - we don't have to. but we might, and if we would we could feel better. we would have another picture, and a picture does make a difference. there is non-well foundedness in every non-standard 'full-scale' axiomatic set-theoretical framework i know of. but we don't have to use axiomatic frameworks! but we don't have to use nsa either! :)