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Results tagged with nonstandard-analysis
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Nonstandard analysis is a way of doing calculus and analysis with infinitesimals. The historical approach of Leibniz, Euler, and others to infinitesimal calculus was gradually replaced by epsilon, delta techniques in the context of a real continuum, in the 19th century. It was not until the 1960s that Abraham Robinson developed a theory of a hyperreal continuum that allows for a development of analysis procedurally akin to that of its founders.
8
votes
In hyperreal field, can ln(ε) and ln(ω) be expressed as infinite sums?
To help avoid any misunderstanding that may arise for readers of this question, let me say that when understood in the usual sense, there are no nontrivial convergent sequences or series at all in the …
- 236k
45
votes
Accepted
Does every series of hyperreal numbers converge to some hyperreal number?
The answer is strongly negative.
Arbitrary extensions. The first thing to say is that whenever one extends $\newcommand\R{\mathbb{R}}\R$ to a larger ordered field $F$, one has immediately destroyed (e …
- 236k
31
votes
Why does CH imply that there is a unique ultrapower of $\mathbb{N}$?
The point is that the ultrapower of any structure $\mathcal{M}$ by
a nonprincipal ultrafilter $\mu$ on $\mathbb{N}$ is countably
saturated, that is, it realizes any finitely satisfiable $n$-type with …
- 236k
56
votes
How helpful is non-standard analysis?
The other answers are excellent, but let me add a few
points.
First, with a historical perspective, all the early
fundamental theorems of calculus were first proved via
methods using infinitesimals, r …
- 4,713
70
votes
Accepted
A remark of Connes on non-standard analysis
...as soon as you have a non-standard number, you get a non-measurable set.
Every nonstandard natural number $N$ gives rise to a nonprincipal ultrafilter $U$ on $\mathbb{N}$, by saying that a set $X …
- 2,821
4
votes
Turing degree of a turing machine with access to an (arbitrary) nonstandard integer
I shall give two different interpretations of the question. (The second interpretation using true arithmetic is modified in this update.)
Using arbitrary nonstandard models of PA. Let us say that a T …
- 236k
10
votes
Automorphisms of the hyperreals over the rationals and nontrivial automorphism groups
As a general principle, every first-order theory with infinite models, such as the theory of real-closed fields, will have models with rich automorphism groups.
The general reason is that one can ta …
- 236k
2
votes
Compactness and omega models
There is no property of $T$ alone that will ensure that $T+S$ always has an $\omega$ model in the circumstances you describe. In fact, there is no computably axiomatizable theory $T$ with the property …
- 236k
29
votes
Accepted
What are the advantages of the more abstract approaches to nonstandard analysis?
To my way of thinking, there are at least three distinct
perspectives one can naturally take on when undertaking work in
nonstandard analysis. In addition, each of these perspectives can
be varied on …
- 16.6k
3
votes
"Lebesgue-measurable" cardinals and real-closed fields
You say that $\kappa$ admits a $[0,1]$-valued measure, but actually, it is a $\{0,1\}$-valued measure; every subset of $\kappa$ has measure $0$ or $1$. If I am understanding your suggestion, you inten …
- 236k
12
votes
Are hyperreal numbers isomorphic to formal power series?
I would like to point out that it is not true that every every
hyperreal can be represented by a Laurent series in the way you
describe.
(Let me assume that by the term "hyperreals", you mean a
nonst …
- 236k
31
votes
Accepted
Is non-existence of the hyperreals consistent with ZF?
The answer is yes, provided ZF itself is consistent. The reason is that the existence of the hyperreals, in a context with the transfer principle, implies that there is a nonprincipal ultrafilter on $ …
CommunityBot
- 1
10
votes
Accepted
Surreal numbers, ultrapowers of $\Bbb R$, ordinal-valued functions and the slow-growing hier...
This kind of analysis is very well understood in ultrapowers, and one often sees this kind of thinking with ultrapowers, where one performs calculations with the representing function for an object. W …
- 236k
6
votes
tennenbaum phenomena for the reals?
This is a very interesting question!
One way to interpret the question is like this: we have the structure
$\langle\mathbb{R},{+},{\cdot},{\lt}\rangle$, which is a real-closed field, and Tarski prove …
- 236k
5
votes
Accepted
Star-transfer of powerset
$\cal{P}({}^\ast\mathbb{R})$ is the full standard power set of the nonstandard reals, the set of all subsets of ${}^\ast\mathbb{R}$. This power set includes the subsets consisting solely of infinites …
- 236k