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 Aug21 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. Aug20 asked How to explain why this injection does what we want (basic math) Aug20 accepted Borel linear order cannot have uncountable increasing chain Aug19 comment Borel linear order cannot have uncountable increasing chain @William Thanks, that's good to know. :^) Aug19 comment Borel linear order cannot have uncountable increasing chain Thanks for the answer. Elsevier's site is not letting me access the original article (though it should). I emailed them, so now I am just waiting to double-check it. I suppose you are right about "increasing", but what would you say instead, for a linearly-ordered set without a greatest element? I would think "unbounded" might be confusing here. Aug19 comment Borel linear order cannot have uncountable increasing chain @FrançoisG.Dorais The paper is linked to in the question. projecteuclid.org/DPubS/Repository/1.0/… Aug19 asked Borel linear order cannot have uncountable increasing chain Jun23 comment Learning Model Theory I second Peter Smith's enthusiasm for Wilfrid Hodges. He's my favorite. If I am looking for exposition on a topic, I try to find something by Hodges first. He's knowledgeable and hysterical and will tell you lots of little useful things that most books won't mention. If you try his model theory book but don't feel ready for it, he wrote a little gem of a logic book, Logic. Jun8 awarded Constituent Jun8 awarded Caucus May29 comment Interpretation of nonlogical symbols in compactness arguments @mercio, Yes, sorry. $\mathbb{Q}$ is a model of $T$. May29 comment Interpretation of nonlogical symbols in compactness arguments @mercio, Regarding the new addition, I said in my question that a function won't satisfy all $r_n$. That is not what I need. I just need every $r_n$ to be satisfiable. The problem is that I don't know how to describe a reversing function without reference to the terms of the sequences, and for this, I need a second-order axiom, which means no compactness. Also, $\mathbb{Q}$ is not a model of $T$. $T$ is non-Archimedean. And compactness does tell you about your models because it gives you axiomatizations for them. It just doesn't give you a particular construction. May29 comment Interpretation of nonlogical symbols in compactness arguments @AndréNicolas, The reversing functions are all unary operations on a set of infinite sequences. Each reversing function reverses the first $n$ terms and leaves the rest unchanged. I might start a new question about the axioms for this class of function because the two that I have so far are problematic. (The worst: for finite cases, $r(x)$ should be eventually equal to $x$ (i.e., $r(x)$ and $x$ only disagree for a finite initial segment), but this equivalence does not intuitively hold for the infinite case since the entire sequence is (potentially) changed.) Thanks for the help. May29 comment Interpretation of nonlogical symbols in compactness arguments @QiaochuYuan, Yes, I just hadn't gotten that far yet. I was thinking intuitively about the models. But now I am concerned because I want to say that there exists a function with such-and-such properties, but I cannot say this directly. I think my original question has been answered, though. Thanks for the help. :^) May29 comment Interpretation of nonlogical symbols in compactness arguments Why does n have to be "high enough" (and what does this mean)? I am familiar with ultraproducts and transfer. Also, what is the cofinite ultrafilter? The cofinite (Frechet) filter on N is not an ultrafilter since it doesn't contain a member of every partition of N; e.g., it doesn't contain either the odd or even naturals, which more importantly are also relative complements. I also thought that the cardinality of a set does not matter when it comes to containment in an ultrafilter. Being countable is not enough to guarantee containment for every ultrafilter. May29 comment Interpretation of nonlogical symbols in compactness arguments @QiaochuYuan, If you are asking for axioms that a reversing function must satisfy, I don't have any yet. Is this what you mean? (I am familiar with ultraproducts, by the bye. An ultraproduct would be very nice.) May29 comment Interpretation of nonlogical symbols in compactness arguments @AndréNicolas, Okay, but then what can you say about the ultraproduct? For my problem, I want to say that there exists a function on it that reverses infinite sequences, by reasoning (I presume) similar to that that gives the existence of infinitesimals. That is, how do I know what union of all my $T_n$ actually says, taken together? It's not clear to me that I get the function that I want. May29 comment Interpretation of nonlogical symbols in compactness arguments @QiaochuYuan, I had in mind some appropriate homomorphism involving the relevant relation or operation. E.g., since you can extend the usual order on N to the usual order on R, I would consider these the same in the right way. May28 asked Interpretation of nonlogical symbols in compactness arguments May23 awarded Tumbleweed