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 ...

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83 views

Can there be a numerical system in which logarithms can be expressed in terms of exponentials in closed form?

The invention of complex numbers allowed to express trigonometric functions through hyperbolic ones in closed form. Is there possible an extension of real/complex numbers in which logarithms and ...
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173 views

Constructing black noise with non-standard analysis

With noise in the sense of i.i.d. random sequence, a noise is black if it is not isomorphic to standard Gaussian white noise. Tsirelson showed the existence of black noise through the scaling limit ...
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What's Reeb's take on naive integers?

Georges Reeb's "claim Q" is the statement that "naive integers don't fill up $\mathbb{N}$". To anyone familiar with model theory this could easily be interpreted as the existence of nonstandard models ...
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Berkovich Analytification of the transseries

I am looking for references to articles about the following subjects: Connections from the field of (real) transseries to the field of surreal numbers (mentioned very briefly in the introduction of ...
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More information on Kruskal's treatment of Surreal numbers as an asymptotic behavior of a real valued function

The only way that I could think about Surreal numbers is how Conway defined them inductively, with the two axioms and so on. I can't find any information about Kruskal's point of view and would very ...
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Deducing Skolem's nonstandard integers from downward Lowenheim-Skolem?

If one has a nonstandard model $\mathcal{N}$ of PA and adjoins to the first-order theory the countable list of axioms $1<H,\, 2<H,\, 3<H, \ldots$ (satisfied in $\mathcal{N}$) for all the ...