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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.
13
votes
Was the early calculus inconsistent?
I do not know whether the early calculus was consistent, but it surely can be made as consistent as modern mathematics, with practically no modifications of the basic setup. This goes under the name S …
- 48.8k
1
vote
What is the spectrum of possible cofinality types for cuts in an ordered field? Or in a mode...
This is not an answer, but it's too long for a comment. You can optimize away the distinction between filled and unfilled cuts as follows. Define a cut to be a pair of subsets $A$ and $B$ such that:
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- 48.8k
54
votes
Accepted
What is... a grossone?
I do not understand what the bounty on this question is for, as it seems to me that the other answers were already rather devastating. Here is a semi-reasoned technical answer.
According to G. Lolli …
- 48.8k
10
votes
Salvaging Leibnizian formalism?
You seem to think that synthetic differential geometry only handles squarenil infinitesimals, but this is not so. The generalized Kock-Lawvere axiom allows us to work with infinitesimals of any order, …
- 48.8k