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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Leibnizian calculus textbook

Where can I find a calculus textbook that emphasizes differentials? Is there such a book that I could realistically require my calculus students to use? I want a textbook that supports me when I tell ...
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6answers
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A remark of Connes

In an interview (at http://www.alainconnes.org/docs/Inteng.pdf) Connes remarks that I had been working on non-standard analysis, but after a while I had found a catch in the theory.... The point ...
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1answer
343 views

Non Separability of the the Loeb Space

Let $X_i$, $i=1,2\ldots$ be finite sets of integers such that $|X_i|\rightarrow \infty$. Let $X$ be an ultraproduct of $X_i$. Let $\mu$ the Loeb measure associated with the the normalized counting ...
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1answer
625 views

Non-standard enlargements, $\zeta(s)$ and analytic continuation

Consider an extension of the Riemann zeta function $\zeta(s)$ where $s$ now runs over a non-standard enlargement of the complex plane. Observe that if $s=\sigma + it$ with $\sigma>1$ real and ...
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1answer
860 views

Uncountable nonstandard models of PA

Standard techniques (no pun intended) can be used to show that countable nonstandard models of Peano Arithmetic are order isomorphic to $\mathbb{N} + \mathbb{Z} \cdot \mathbb{Q}$. Once we have used ...
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1answer
602 views

Metrization of hyperreals

Hello, i was reading your article about non metrizability of *R. i was able to prove that the interval open topology is not metrizable by proving that the intersection of decreasing hyper-intervals ...
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9answers
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Would Euler's proofs get published in a modern math Journal, especially considering his treatment of the Infinite?

I was wondering how mathematicians of today would treat, for example, Euler's proof of zeta(2). In William Dunham's book 'Journey through Genius' ( ...
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10answers
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nonstandard analysis book recommendation

I wish to learn nonstandard analysis. Are there any good book recommendations? I'm familiar with the ZFC system, and learnt analysis the classical way. I've found some undergraduate texts, but they ...
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16answers
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How helpful is non-standard analysis?

So, I can understand how non-standard analysis is better than standard analysis in that some proofs become simplified, and infinitesimals are somehow more intuitive to grasp than epsilon-delta ...
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4answers
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Which topological spaces admit a nonstandard metric?

My question is about the concept of nonstandard metric space that would arise from a use of the nonstandard reals R* in place of the usual R-valued metric. That is, let us define that a topological ...
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5answers
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Isomorphism types or structure theory for nonstandard analysis

My question is about nonstandard analysis, and the diverse possibilities for the choice of the nonstandard model R*. Although one hears talk of the nonstandard reals R*, there are of course many ...