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There are such transfinite numbers as $\aleph$ and $\omega$. Do they have infinitesimal analogues? I'm especially interested in numbers 'below' $0$ having one peculiarity: it should be non-negative. Is there chance there is/are infinitesimal number(s) which are non-negative, but still lower than zero? I'm not sure I should find what I need among infinitesimals, but your answer might help me to move forward with finding what I exactly need. So, are there such numbers that might be considered to be 'transfinitesimal'?

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1 Answer 1

Yes, though these 'transfinitesimal' numbers aren't below zero, they are above zero but smaller than every positive real number. What you are looking for is called non-standard analysis, http://en.wikipedia.org/wiki/Non-standard_analysis.

Proving the existance of such things is usually rather easy, BTW, if you apply the completeness and compactness theorems. It then amounts to

  1. Choosing a set of axioms that describes the structure you're interested in, like the natural numbers or the reals. Note that you must do this in first-order predicate logic, to have the completeness and compactness theorem available.
  2. Define a (usually infinite) set of axioms which, if taken all together, force the existance of "strange" objects. In the case of the reals, you could add a constant $c_\omega$ and take the set of axioms $\left\{0 < c_\omega < 1, 0 < c_\omega < \frac{1}{2}, 0 < c_\omega < \frac{1}{3}, \ldots\right\}$.
  3. Then, by compactness, if every finite subset of those new axioms is consistent with the original theory, the whole set is also.
  4. And by completeness, if the whole set is consistent, it has a model. That model must then include a "strange" object, since it must supply some value for $c_\omega$, and that value must fullfill the whole set of newly added axioms.
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