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Results tagged with nonstandard-analysis
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user 5903
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.
11
votes
Which topological spaces admit a nonstandard metric?
These are $\omega_\mu$-metrizable spaces, where $\omega_\mu$ is the cofinality of ${}^*\mathbb{R}$. Take a decreasing sequence $\langle x_\alpha:\alpha<\omega_\mu\rangle$ of positive elements that con …
- 11.4k
14
votes
Accepted
Is "All hyperreal fields $C(\mathbb{R})/M$ are isomorphic" independent of ZFC+$\lnot$CH?
In On ultra powers of Boolean algebras (Topology Proceedings 9 (1984) 269-291) Alan Dow proved (Corollary 2.3) that $\neg\mathsf{CH}$ implies there are two fields of the form $C(\mathbb{N})/M$ that ar …
- 11.4k