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# 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). Please specify the exact framework for non-standard analysis you are using in your question (e.g., what definition of "hyperreal number" you are using).

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### Is there an external reflection principle in IST?

Background The reflection principle is a theorem schema in ZF. Given a formula $φ$ and a set $M$ we obtain $φ$ relativized to $M$ by restricting all quantifiers of $φ$ to range over $M$. ...
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### How sensible is the limit from above against infinity in nonstandard analysis?

In nonstandard analysis, we have hyperreal numbers that are greater than any real number. As such, we can create a sequence of infinite, hyperfinite hyperreal numbers which grows ever smaller. More ...
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### Nonstandard Complex Analysis?

I recently discovered Nonstandard Analysis and am slowly working my way through Kelsier's textbook and Foundations companion. However while I have found plenty of stuff about real nonstandard ...
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### How to find infinitesimally small segments for graph $y = x^2$?

I read on Wiki: Intuitively, smooth infinitesimal analysis can be interpreted as describing a world in which lines are made out of infinitesimally small segments, not out of points. I tried to find ...
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### Uniqueness of standardization

So I'm reading Kanovei's and Reeken's book "Nonstandard Analysis, Axiomatically" and there is something which I'm not quite understanding. So, in page 15, it is stated that a "...
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### How to prove that 2 points $\varepsilon$ and 2$\varepsilon$ are indistinguishable not only in sole case? (Smooth infinitesimal analysis)

There are 2 points on the Real line: $A$ and $2A$. They are indistinguishable in sole case - if $A$ is $0$. But how to prove that $\varepsilon$ and 2$\varepsilon$ are indistinguishable not only in ...
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### Proposal for the number next to $0$ [duplicate]

What if we define the number next to $0$ on the real number line to be special like $0$, but not quite $0$. Is there some work done in this direction? Can someone point me towards it because I am ...
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### Nonstandard algebraic geometry: Fundamental Theorem of Algebra

I have been trying to study the basics of algebraic geometry using nonstandard analysis and I can't wrap my head around this issue. Let $^*\mathbb{C}$ be the extension of the complex numbers. Now ...
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### Arithmetic in a nonstandard model of Real numbers

A problem from Section 3.3 in "A Friendly Introduction to Mathematical Logic" (Christopher C. Leary, Lars Kristiansen; 2nd edition): (a) In the structure $\mathfrak{A}$ that was built in ...
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### Repeated transfer principle for transfinite induction.

In set theory, one can use a non-principal ultra filter, $U_0$, to construct the hyperreal numbers. $\mathbb{R}^*$ is constructed as $\mathbb{R}^{\mathbb{N}}/U_0$. The transfer principal means that ...
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The fields of the surreal and hyperreal numbers aren’t complete. I’ve noticed that $\mathbb{Q}^2$ isn’t rotationally complete as $(0,1)$ could be rotated to a point not in the rational plane (like $(1,... 0 votes 1 answer 65 views ### Can the hyperreal numbers have a property akin to completeness by considering hypernatural sequences? I’ve seen that$\mathbb{R}^*$isn’t complete as many Cauchy sequences won’t converge, and that includes power series. In other stack exchange posts, I’ve seen that even the exponential function won’t ... 2 votes 2 answers 169 views ### Can the hyper hyper real numbers be constructed? The hyperreal numbers can be constructed as$\mathbb{R}^{\mathbb{N}}/U$given some ultra filter and this allows first-order statements to be transferred over to$\mathbb{R}^*$. Can this be done again ... 1 vote 1 answer 92 views ### Approach to construction hyperreal number with sequence I read that hyperreal numbers can be constructed with sequences. For example,$\varepsilon = (1, 1/2, 1/3, ...)$and$\varepsilon$is infinitesimal. But there is no smallest number (infinitesimally ... 0 votes 0 answers 59 views ### Number of models of the naturals and reals with and without CH The Wikipedia page on true arithmetic says that it has$2^\kappa$models for each uncountable cardinal$\kappa$. This refers to the theory of all first-order statements of the naturals. I'm curious ... 1 vote 0 answers 33 views ### How is it possible that$dx$contains$dx^2N$and$N+1$times simultaneously? Let's assume that$N$is even infinite hyperinteger and$N=1/dx$,$N=dx/dx^2$. Let's assume that we have expression$dx+dx^2$. We can add up$dx^2$to$dx$and$dx$is obtained:$dx+dx^2=dx$(... 1 vote 1 answer 374 views ### In what sense is$\Bbb R(x)$an "instantiation" of the hyperreals? I'm teaching myself about hyperreal numbers. My main motivation for doing so is that they include infinite numbers, whose existence I hear disputed & doubted often as "quantifying the ... 6 votes 0 answers 68 views ### Transfinitely iterating the Puiseux, Levi-Civita, or Hahn series constructions There are many ways to take some real-closed field and generate a proper extension of it with elements that are infinite and infinitesimal relative to the original field. One well-known example is ... 4 votes 1 answer 138 views ### Why does this nonstandard model of Robinson Arithmetic fail to be a model of PA? As talked about in this question, there is a nonstandard model of Robinson arithmetic of the form:$z_0 + z_1\omega + z_2\omega^2 + z_3\omega^3 + ... + z_n\omega^n$in which$\omega$is a formal ... 2 votes 0 answers 63 views ### Levi-Civita field vs Puiseux series: why is Cauchy completeness important? The Levi-Civita field's main claim to fame is that it's the smallest real-closed field which is a proper extension of the reals and which is Cauchy-complete. That last bit is important, or else the ... 2 votes 2 answers 129 views ### "Real-closed" vs "transfer principle" The hyperreals are "real-closed," which means that any first-order statement that is true of the reals is true of the hyperreals. However, they also satisfy a "transfer principle," ... 0 votes 0 answers 68 views ### Formula for probabilistically selecting a good server There are$N$nodes, each has response time$t_i$(here and further I use$i$as an index). I want probabilistically choose a node with low response time; if there are several nodes with low response ... 4 votes 0 answers 490 views ### How do we naturally extend the average from the Hausdorff Measure for functions with a domain without a gauge function? Motivation: According to this question Some sets have a Hausdorff Dimension$\alpha$but have a zero-dimensional Hausdorff Measure. These sets may have another dimension function, i.e. a function$h:[... 1 vote
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### Nonstandard proof of Lebesgue Number Lemma

I am currently going through Munkres's Topology but trying to prove things with nonstandard methods. The one I am struggling with right now is the Lebesgue Number Lemma, not too sure if my proof is ...
$\frac{\partial x}{\partial y} \cdot \frac{\partial y}{\partial z} \cdot \frac{\partial z}{\partial x} = -1$ according to the triple product rule However, it would be 1, if derivatives behaved like ...