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Jan
19
comment Is $\textrm{d}x \in \mathbb{R}$?
This is only half correct. There are two main streams of nonstandard analysis. In the first, developed by Abraham Robinson from model-theoretic foundations, infinitesimals show up as new emements of an extension of the real numbers to the hyperreals. Infinitesimals are constructed; they are not real numbers. In the second main stream, Edward Nelson's internal set theory, new axioms give us a new ability to distinguish some real numbers from others. We acquire a sort of X-ray vision which allows us to see infinitesimals where before we could see only zero; infinitesimals are real numbers.
Jan
1
comment Is $0$ an Infinitesimal?
@DanielV, zero is indeed an infinitesimal by the definitions of all versions of NSA I'm familiar with. In internal set theory, zero is the only standard infinitesimal, which makes it easy to distinguish from the others. When necessary, we talk about positive or non-zero infinitesimals.
Dec
30
comment Transcendental a infinitely close to rationals?
@GEdgar, I rather disagree that the OP will not benefit from this answer. The only formalism in which the OP's question (and the whole class of questions tagged with infinitesimals) can even be given a precise meaning is a nonstandard one. Najib's answer is an on-the-one-hand/on-the-other-hand non-answer for this reason. ... @Tom Gatward, I suggest you pick up a book like Alain Robert's Nonstandard Analysis if you want to learn more about this.
Sep
15
awarded  Commentator
Sep
15
comment non-archimedean in lay terms
@user72694, sorry, I missed the "and only if" in your comment and just wanted to emphasize that the implication holds in both directions. I agree with what you've written.
Sep
13
comment non-archimedean in lay terms
@user72694, yes, that's a good way to put it. ... On the other hand, by the same reasoning one may argue that if there are no infinitesimals in IST, then there are no infinitesimals in Robinson-style NSA either. ;)
Sep
12
comment non-archimedean in lay terms
@user72694, by "the sense described above," you mean as in your comment, right? My point is that there are no infinitesimals in IST according to the definition of "Archimedean". Certainly IST has so-called infinitesimals in some sense, but not in that sense. This is significant because it differentiates IST from Robinson-style NSA and other "extensional" flavors of NSA. Even though it gets you the same place, IST is really fundamentally quite different.
Sep
11
comment non-archimedean in lay terms
@user72694, by "technically" I mean with respect to the usual definition of "Archimedean," which is given in the Wikipedia article linked above. Of course one can argue that this definition does not capture the right idea.
Sep
9
comment non-archimedean in lay terms
@Howard Pautz, you might take a look at Nelson's original paper. For getting practically acquainted with the axioms of IST, I recommend Alain Robert's Nonstandard Analysis. Note also that there is a "radically elementary" version of IST, which uses as axioms some facts that are theorems in full IST; you can get a feel for what it's like to use infinitesimals just looking at that—search for "radically elementary" nonstandard analysis.
Sep
7
comment non-archimedean in lay terms
Note also that internal set theory, one of the major versions of nonstandard analysis, reveals "really big" numbers in $\mathbb{N}$ without adding any new elements to it, so that $\mathbb{R}$ retains the Archimedean property. (Technically there are no infinitesimals in IST, but it certainly feels like there are.) This is done by introducing three new axioms governining the use of a new predicate, "standard," which gives us a richer vocabulary for talking about particularly big and small numbers. With it, we don't need to extend the number line to get the benefits of infinitesimals.
Jul
26
comment Surjectivity implies injectivity
Though technically correct, the claim that "the image of [an injective] $f$ has at least $n$ elements" is odd and misleading. It follows from the definition of a function that the image of any function has at most $n$ elements when its domain has $n$ elements. So proving the first part really just amounts to noticing that injectivity implies the image of $f$ has exactly $n$ elements, i.e., it coincides with $S$.
Jul
20
revised Law of iterated expectation in an algebraic axiomatization of probability theory
deleted 2 characters in body
Jul
20
revised Law of iterated expectation in an algebraic axiomatization of probability theory
deleted 20 characters in body
Jul
20
comment Law of iterated expectation in an algebraic axiomatization of probability theory
@Did, yes, everything's discrete. (The text proceeds to use non-standard analysis to extend discrete techniques to spaces that we would not be able to think of as discrete in a standard setting.)
Jul
19
revised Law of iterated expectation in an algebraic axiomatization of probability theory
Added $x$ in statement of law of iterated expectation in the closing parenthetical, clarified definition of conditional expectation
Jul
19
revised Law of iterated expectation in an algebraic axiomatization of probability theory
Added $x$ in statement of law of iterated expectation in the closing parenthetical, clarified definition of conditional expectation
Jul
19
revised Law of iterated expectation in an algebraic axiomatization of probability theory
Added $x$ in statement of law of iterated expectation in the closing parenthetical
Jul
19
asked Law of iterated expectation in an algebraic axiomatization of probability theory
Jul
19
comment Co-transitivity of the constructive order relation
@Carl Mummert, yes, that is my understanding, too. The caveat I gave was a poorly articulated attempt to clarify that, unlike in a classical setting, it is not the case in this field that only zero lacks a multiplicative inverse. ... And now I understand why constructivists had to come up with the notion of an apartness relation.
Jul
19
revised Co-transitivity of the constructive order relation
Clarification: multiplicative inverse; added 1 characters in body