![]() ![]() So in a sense we can always find "more infinitesimal" elements - we just have to go to larger and larger fields in order to do so. optional - can be a non-negative integer or the keyword all. standard part strips away the (negative) infinitesimal, resulting in the. (Alternatively, just use compactness! Keep in mind that compactness can be proved via ultraproducts, so it's not so much an alternative to the above as it is a repackaging of the basic idea so that we don't have to do it over and over again in other contexts.) optional - a list with the functional form of the infinitesimals of a symmetry generator. decimals add additional positive (infinitesimal) terms to the real value one. ![]() Now think about what happens with the identity function $id:\alpha\mapsto\alpha$ in the power structure $F^I$ after we "mod out by $U$". I wanted to know if by convention we take dx as the positive value, and specify otherwise if necessary, so that in this case, I could shift the dx without worry. Quibble: it's not really right to say that the hyperreals are a quotient of $\mathbb\in U$. I get what you mean that dy could be measured 'downwards-positive', and dx 'leftwards-negative'. ![]()
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