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Yes.


Aug
23
awarded  Autobiographer
Aug
23
comment Nonstandard extension of a function with a limit
Thanks for the clarification. I'm not sure why I didn't just write down the $\epsilon\mathrm{-}\delta$ definition and star things. That was very clear. I still am curious about general nonstandard convergence and limits (i.e., the obvious notion of the outputs being in the same halo for all inputs in a certain set but with laxer constraints on the input set than is needed for transfer (e.g., saying a hypersequence converges if there is any hypernatural index after which the values remain inf. close)). But that can wait for another time.
Aug
23
accepted Nonstandard extension of a function with a limit
Aug
23
comment Nonstandard extension of a function with a limit
@CarlMummert: Yes, I have noticed this and find it disappointing. I find the hyperreal space interesting in itself, but I am never able to find any literature investigating it. (If anyone knows of some examples, please share!)
Aug
22
comment Nonstandard extension of a function with a limit
@BenCrowell: I presumed that Hurkyl was talking about the extended hyperreals, i.e., $^*\infty$ is being considered as a member of an ultrapower of the extended reals. Are you saying that this makes no sense?
Aug
22
comment Does this proof resolve the Liar Paradox?
@DanChristensen: I believe you can read "A := B" as "A is defined to be B". So, e.g., in English, "monosyllabic" or "______ is monosyllabic" are normal predicates (depending on your idea of an English predicate) because "'Monosyllabic' is monosyllabic" is false.
Aug
22
comment Nonstandard extension of a function with a limit
Okay, but then $x$ (the variable approaching whatever) should be ranging over the hyperreals, yes? When extending $\lim_{x \to a}f(x)$, you are letting $a$ be nonstandard, yes? It seems that $x$ then should range over all hyperreals, or else the starring of $a$ is rather strange. I guess I should write out what the $\lim$ notation really says formally. (Also, hey, stranger! :^))
Aug
22
comment Nonstandard extension of a function with a limit
Apologies, I think my question was unclear. I tried to clarify just now in a comment to Ben's answer (which I know not how to link to). Like Ben, I also am curious about your starring of infinity. Is this because you are thinking of it as an argument of $\lim$? I had overlooked this, but I think that makes sense. And $^*\infty$ is then greater than every infinite hyperreal?
Aug
22
comment Nonstandard extension of a function with a limit
Apologies, I think I was not clear enough in my question that I don't mean $\mathrm{lim^*}$ to necessarily be the extension of a standard function. Perhaps putting the star on the opposite side was a bad idea. I want to think about "taking the limit" of a nonstandard function. I only mention the standard $\lim$ function because I am curious how its extension $^*\lim$ relates to this other thing $\mathrm{lim^*}$. The difference that I am insisting on is that, in $\mathrm{lim^*}_{x \to a}$, $x,a \in\,^*\mathbb{R}$, which apparently doesn't happen for $^*\lim$.
Aug
22
comment Nonstandard extension of a function with a limit
@BenCrowell: Yes, I just didn't think to simplify that bit when I generalized the question. It was originally related to a particular complex-valued function, but I suppose it is not essential. Brian is correct that $[\langle x_1, x_2,\ldots\rangle]$ denotes the equivalence class of a sequence mod an ultrafilter.
Aug
22
awarded  Necromancer
Aug
22
asked Nonstandard extension of a function with a limit
Aug
22
comment How to explain why this injection does what we want (basic math)
Yes, the proof that I wanted won't work. And yes to the second question, with the added bit that $f$ is Borel (there isn't always such a Borel $f$). Here, I am quite sure that if the cardinalities are all right (in the sense above), then there exists such a function since I have no constraints on $f$ above. But I want as constructive a proof as possible for this case, i.e, I want to postpone using AC until necessary so that I know more about what to do when I don't have it (when I add Borelness constraints).
Aug
22
comment How to explain why this injection does what we want (basic math)
For your example, you can send the point of size 2 to a point of size 2 (or whatever) and then make the assignments for the points of size 1. Perhaps you can always get around this type of example by making only finitely many assignments for each size at a time.
Aug
22
comment How to explain why this injection does what we want (basic math)
Right, the constraint in (2) is not on $|X|$ and $|Y|$ but on the sums of the sizes of their points. In case it is familiar to you, the $f$ that I am looking for is an injective reduction of Borel equivalence relations. So (re)think of $X$ and $Y$ as relations $E$ and $F$. Of course, I have also dropped all of the Borel constraints here.
Aug
22
comment How to explain why this injection does what we want (basic math)
Hm. This might be a problem indeed, but perhaps I was too careless with my generalization. Originally, $X$ and $Y$ were equivalence relations (on poss. different sets), and their "points" are their equivalence classes. So: (1) This isn't important for part of your example, but I should have excluded $0$ as a cardinal. (2) The underlying set of $X$ cannot have cardinality greater than the underlying set of $Y$ (orig. constraint). So when you sum up the sizes of the points, the sum for $X$ can't be greater than that for $Y$. I will rethink a modification of your example to see if it is avoided.
Aug
22
revised How to explain why this injection does what we want (basic math)
added 59 characters in body
Aug
22
comment How to explain why this injection does what we want (basic math)
@BrianM.Scott: In the motivating theorems that I have thought more about, they're continuum or smaller. But I would like to allow arbitrary sizes here if possible. This shouldn't cause proper-class complications for $f$ because $X$ and $Y$ are still sets.
Aug
21
revised How to explain why this injection does what we want (basic math)
added 184 characters in body
Aug
21
comment How to explain why this injection does what we want (basic math)
@BrianM.Scott: Ah, my explanation wasn't clear enough. I will edit.