The real numbers are Dedekind complete. One exception is a recent reconstruction of infinitesimals — positive “numbers” smaller than every real number — devised by the logician Abraham Robinson and … Question from a couple of my students that I couldn't answer and can't find using google. Infinitesimals: “Do the math” in a different dimension, and bring it back to the “standard” one (just like taking the real part of a complex number; you take the “standard” part of a hyperreal number … But infinitesimals are, by definition, uncountably infinite. For instance, consider we want to integrate the expression \$3x(x^4+1)^3\$. There are also infinities of the nature presented previously as unlimited numbers. Infinitesimal, in mathematics, a quantity less than any finite quantity yet not zero. This last statement is easy to prove. There is also no smallest positive real number! Sometimes we think of this result as saying the real numbers are the points on a line with no gaps. Any number which is not Infinite is called a Finite number– including the Infinitesimals. (To be continued.) Infinitesimals are not real numbers, and therefore don't live on the real number line in the first place. Using Infinitesimals. The Standard Part Theorem says all the limited hyperreals are clustered around real numbers. Infinitesimals are the reciprocal of Infinite numbers, and as such, have an absolute value which is smaller than that of any Real number (except 0, which is considered to be Infinitesimal): . That's the whole point of the real numbers forming a continuum, it has a cardinality larger than the set of any countable number or sequence of numbers. During integration by substitution we normally treat infinitesimals as real numbers, though I have been made aware that they are not real numbers but merely symbolic, and yet we still can, apparently, treat them as real numbers. Not sure if this is the right place. They are not countable. When we take a line with no gaps and add lots of infinitesimals around each point, we create gaps! The multiplicative inverse of an infinity is an infinitesimal and vice versa. So if there are no such real numbers… Infinity and infinitesimals are both undefined quantities. Moreover, if , then , , etc… can be “much smaller than” itself (by many “orders of magnitude”). Given any real number , the number , but . Since positive infinitesimals are considered to be nonzero entities less than any positive real number the appropriate quantity for an multiplicative inverse would be an entity which is greater than any real number but not equal to infinity, a sort of infinitude. 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