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seen Mar 18 at 8:01

Yes.


Aug
11
comment What are authoritative publications regarding foundational mathematics?
+1 for a generic female mathematician. It helps! I'll just add that those standards exist so people can build things that work together without ad-hoc human communication/intervention, a solution unnecessary in math. Also, mathematicians are like implementations--how often are standards implemented exactly according to the specs? This variety/flexibility is good and efficient. If I want to know common or uncommon definitions, I sample the internet.
Sep
3
comment How does Borelness overlap with definability, computability, or constructiveness?
Thank you, that makes much sense. I also had not seen the Louveau paper, and it looks very interesting.
Sep
3
comment How does Borelness overlap with definability, computability, or constructiveness?
[...cont] My interest is logic, the paper is focused narrowly on Borel reductions, and the structures that we work with are orders, groups, and measures. I think it's interesting to ask how things change when you require everything to be Borel instead of allowing arbitrary sets. But I currently know no reason for Borel equivalence relations being a field of study (and populated by logicians). Surely it is not by pure happenstance that people write papers specifically about the classification of Borel equivalence relations. These objects and their classification must be special in some way. No?
Sep
3
comment How does Borelness overlap with definability, computability, or constructiveness?
I have read somewhere else that Borel sets were invented/defined because they are more appropriate than open sets for some questions in topology. This would be motivation enough for my introducing them except that, in ~20 pages, I mention topology only in the definition of Borel sets. When we actually go looking for Borel reductions, the topology and metric are completely useless. What matters is getting back choice in some limited way, e.g., through the group action, hyperfiniteness, or smoothness, or looking at the measures to see if what we want is possible. [cont...]
Sep
2
comment How does Borelness overlap with definability, computability, or constructiveness?
So are the Borel sets the ones with the simplest descriptions, e.g., using only countable quantification? The motivation that I give will be vague. I just want it to not be grossly inaccurate.
Aug
31
comment How many weak/strong limit cardinals exist under different assumptions?
I was thinking incorrectly. When an operation returns a subset/subclass of its input, the size of the set returned doesn't tell you how many elements were excluded. Thanks for your answers.
Aug
31
comment How many weak/strong limit cardinals exist under different assumptions?
Pardon the vagueness, but I am trying to get a general feel for these structures. So it's the limits that are responsible for the size of all of these classes (ordinals, cardinals, weak-limit cardinals, etc.)? That is, there are few objects between limits relative to the number of limits? You can keep taking limits (of limits of limits...) and always have a proper class?
Aug
31
comment How many weak/strong limit cardinals exist under different assumptions?
So among weak limit cardinals, there are limit weak limit cardinals that are analogous to limit ordinals in the order of On?
Aug
31
comment How many weak/strong limit cardinals exist under different assumptions?
Does it make sense to think about the "order type" of the class of weak limit cardinals? I really am interested in the idea of somehow iterating through this class, i.e., starting at the least one and reaching them all by repeated application of some operation.
Aug
29
comment What does the concept of computation actually mean?
To clarify for myself the distinction between your two types of knowledge, is it similar to the distinction between a formal language and the model that you interpret it to be talking about? That is, are there two separate structures here connected by a map? And the Tortoise's "failure" is in his map? That is, he has the premises as propositions in his theory and has interpreted them so that they are associated to objects in his model. However, his map is not such that the relations among the propositions in the theory are carried over to relations among the objects in the model?
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
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
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.