If is -true, then . Conversely, if is not -true, then is -true, then . Arguing similarly using we see that is infinitesimal if and only if .

If you are interested in pursuing this topic further I would recommend reading a text on nonstandard analysis (e.g., Goldblatt’s “Lectures on the hyperreals”) rather than just working from this blog post. (One could also work through my lecture notes at https://terrytao.wordpress.com/2011/10/15/254a-notes-6-ultraproducts-as-a-bridge-between-hard-analysis-and-soft-analysis/ , which covers this topic in more depth than in this blog post.)

]]>Okay, according to the test, I believe being an infinitesmal means for any real number , is -true. But still I don't see how it relates to .

]]>If one knows that can be represented by a convergent sequence, say , then .

If is represented by a bounded but not convergent sequence,

say, , what is ? I guess the answer should depend on the choice of . In order to be consistent with the convergent case above, one may have . If this is true, then uniqueness follows (?) from that of limits and transfer principle.

If one can show that is an infinitesimal, then we are done. But this boils down to showing is an infinitesmal iff its p-limit is zero.

But I don’t see a clear definition of “infinitesmal”, only the example .

Moreover, is the same as for any real number , is p-true? I don't see how this follows from the axioms of -limit.

]]>It depends on the choice of ultrafilter , which is highly non-unique. Some ultrafilters contain the odd numbers (in which case the -limit of will be one), while the other ultrafilters contain the even numbers instead (in which case the -limit of will be zero). So it’s basically up to you whether you deem the odd numbers to be “large” or the even numbers; either choice is permissible in leading to a non-principal ultrafilter.

To create an non-principal ultrafilter one has to make an infinite number of arbitrary choices like this, which is why one needs (a weak version of) the axiom of choice to even be assured that such an ultrafilter exists. I like to think of an ultrafilter as a device that takes all the arbitrary choices one would potentially use in an analysis argument (e.g., when extracting a convergent subsequence from a bounded sequence) and makes them all upfront, so that one can then conduct the remainder of one’s analysis without making any further such choices.

]]>By property 3 in the limit laws, if has p-limit of 1, then it must have infinitely many terms equal to 1. Does this characterize the indicator sequence of ?

It is clear that if the limit of is 1, i.e., is eventually 1, then the p-limit of must 1 as well (since the p-limit extends the notion of limit for convergent sequences).

But how about the case when the indicator sequence has no limit but has infinitely many 1’s? For example, how can one calculate the p-limit for the sequence 1,0,1,0,…?

]]>1. Yes.

2. (a) and (b) determine the same subset of , namely the bounded nonstandard reals, while (c) determines the complement. (Exercise!)

3. I do not understand the queestion, but non-principal ultrafilters cannot be (provably) constructed explicitly since there are models of set theory without choice in which non-principal ultrafilters do not exist. This pretty much rules out any attempt to interpret ultrafilter-based objects in terms of more “constructible” notions, such as convergence of sequences.

4. Yes, bearing in mind that this equivalence class also contains sequences that do not converge to zero in the classical sense, only in the ultrafilter sense.

]]>If one is careful to only apply the notations to functions (as opposed to the mathematically equivalent, but philosophically slightly different, concept of a quantity that may potentially depend on a parameter ), then the set can indeed be constructed using the axiom schema of specification.

However if one wishes to manipulate functions instead of scalar quantities one has to be careful with abusing the transfer principle that assertions that are known to be true about scalars are also true about functions. For instance, addition of scalars is known to be associative, , and this automatically transfers to associativity of functions, for all functions . On the other hand, the only subrings of the integers are the integers themselves, however the ring of functions from the integers to the integers contains many non-trivial subrings, including the ring of bounded functions. In nonstandard analysis one can carefully retain a valid version of the transfer principle by making sure one always distinguishes “internal” concepts from “external” ones (with the property of being bounded being in the latter category, for instance). Without something like a nonstandard formalism one should either not identify scalars with functions (e.g., distinguish the constant 27 from the constant function ), or else take care not to abuse the transfer principle.

]]>2. In undergraduate real analysis, there are serval kinds of sequences of real numbers:

(a) convergent sequences;

(b) bounded sequences that do not have a limit

(c) unbounded sequences;

Do these together form a partition of so that the equivalent classes in is somehow a “finer” partition?

3. According to the definition, elements in are sequences of real numbers (modulo the p-a.s. equivalence). The notion of sequences is an “easy” one in undergraduate real analysis while that of “p-ture” is not. Can one phrase/develop the notion of “p-true”, and the notion of “non-principal ultrafilter” merely in terms of sequences of real numbers? (One then has a “cheap” yet full version of the non-standard reals that has the notion of ultrafilter.)

4. Is it correct to say that infinitesimal (non-standard real) is the p-true equivalent class of sequences of real numbers that converge to zero?

]]>So is it correct to define a set like ? Is this with the axiom schema of specification?

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