Questions tagged [nonstandard-analysis]

Non-standard ordered fields are fields which have infinitesimals, that is, positive numbers which are smaller than any positive *real* number. Non-standard analysis is analysis done over such fields (e.g. hyperreal fields).

374 questions
117 views

Nonstandard Natural Numbers via Internal Set Theory in Coq

I'm trying to translate http://www.stat.umn.edu/geyer/nsa/o.pdf to Coq, but I'm stuck right at the start on the simple axioms for the predicate "standard" over natural numbers. The pdf linked here ...
325 views

Applications of Non-Standard Analysis to Number Theory and Topology?

S.E friends, I am wondering what might be some useful applications of the non-standard analysis to the number theory and topology? I am very interested in the set-theoretic topology and prime ...
645 views

Are there non-standard counterexamples to the Fermat Last Theorem?

This is another way to ask if Wiles's proof can be converted into a "purely number-theoretic" one. If there is no proof in Peano Arithmetic then there should be non-standard integers that satisfy the ...
382 views

What is the point of making dx an infinitesimal hyperreal?

It seems fairly common to describe $\mathrm{d}x$ in nonstandard analysis as an infinitesimal. But after thinking it through (and then skimming Keisler's text), I can't see the point and actually think ...
704 views

298 views

How to get an introduction to non-standard analysis?

I am a high school rising senior with an interest in mathematics, and I will be taking AP calculus AB next year. I have been doing research online, and recently came across hyperreal numbers, which I ...
247 views

What is the difference between hyperreal numbers and dual numbers

Wikipedia has two different but unconnected pages for Hyperreal and Dual numbers. https://en.wikipedia.org/wiki/Hyperreal_number and https://en.wikipedia.org/wiki/Dual_number I cannot stop seeing ...
372 views

Why do we need ultrafilter for construction of hyperreal numbers?

While trying to get some hold on the hyperreal numbers I found that their construction using the already existing real number system $\mathbb{R}$ requires the use of objects called free ultrafilters ...
3k views

What is the use of hyperreal numbers?

For sometime I have been trying to come to terms with the concept of hyperreal numbers. It appears that they were invented as an alternative to the $\epsilon-\delta$ definitions to put the processes ...
2k views

What's so different about limits compared to infinitesimals?

If you find the limit is 2 for a given function, wouldn't this be the same as $2 + \epsilon$ with $\epsilon$ being a negligible value? This different way of defining limit-like behavior seems rigorous ...
154 views

What properties do hyperreal extensions of real functions have?

If I have a function $f : \mathbb R \to \mathbb R$ and extend it to the hyperreal function $f^* : \mathbb R^* \to \mathbb R^*$, what are some of the properties that I know $f^*$ must have? ...
291 views

Why do the infinitely many infinitesimal errors from each term of an infinite Riemann sum still add up to only an infinitesimal error?

Ok, so after extensive research on the topic of how we deal with the idea of an infinitesimal amount of error, I learned about the standard part function as a way to deal with discarding this ...
254 views

On the HoTT Cauchy Reals

In the Homotopy Type Theory Book there is a construction given of a kind of Cauchy reals via higher inductive type and the authors remarked, that this construction is preferred to other notions (reals ...
72 views

digits of pi, the nature of its randomness, immunity to place selections (Von mises) reichenbach partial limits

The digits of $\pi$ uniform distribution non-standard analysis, and its partial limit I was wondering whether there has been any investigation into the distribution of the decimals of $\pi$ using non-...
846 views

What is… A Parsimonious History?

Interpreting historical mathematicians involves a recognition of the fact that most of them viewed the continuum as not being made out of points. Rather they viewed points as marking locations on a ...
104 views

Non standard complex analytic functions

I'm having trouble understanding what should mean analytic for hypercomplex functions. In deed I'm studying a book (Non standard analysis in practice by Diener) where they just say that the function ...
69 views

Is there a bijective map from $\mathbb{R}$ to ${}^{*}\mathbb{R}$? [duplicate]

Is there a one-to-one correspondance between the real numbers and the hyperreal numbers?
105 views

Proving that $\mathbb{Q}_p$ is not formally real.

I've been looking for a concrete proof without results. The only hint that I have found says: 1) $\mathbb{Q}_2$ contains a square root of $-7$. 2) $\mathbb{Q}_p$ ($p>2$) contains a square root of ...
40 views

Order in a quotient space of $\mathbb{R}^\mathbb{N}$ ($\mathbb{R}^\omega$)

Let $\mathcal{F}$ be a filter in $\mathbb{N}$ finer than Fréchet filter. In $\mathbb{R}^\mathbb{N}$ we define the equivalente relation : $(a_n) \equiv (b_n)$ iff $\{n | a_n = b_n\} \in \mathcal{F}$. ...
172 views

The meaning of “EXACT laws of large numbers”

I have come across various papers that consider a stronger form of probability-relative frequency convergence theorem called the 'exact law of large numbers". I note that in particular such theorems ...
149 views

Can the transfer principle apply to second-order logic if we transfer sets and relations to hyper-sets and hyper-relations?

The transfer principle doesn't apply to second-order logic. For example, if I take a standard statement. $$\text{A lower bounded set of Reals has a greatest lower bound}$$ Is false for the hyperreals:...
72 views

136 views

4k views

137 views

Mathematical descriptions of physical space

Bear with me as I'm a philosophy (not math) student. First some philosophical background, and then the math question. One philosophical view is that physical space is composed of infinitely many ...
233 views

Is there such a thing as “hypertopology” (analogous to the hyperreals)?

The hyperreal number system adds infinities and infinitesimals, allowing Calculus to be done using these things instead of limits (sort of like when calculus was originally invented, but with rigor)....