Timeline for A better way to explain forcing?
Current License: CC BY-SA 4.0
9 events
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| Aug 21, 2020 at 20:22 | comment | added | Asaf Karagila♦ | @TimothyChow: Once you replace "field" by "ordered field", the abstract field is now the blueprint, the $\Bbb P$-names, and by choosing which rationals are smaller than $x$ you get a semblance of genericity. (And right here you can see how talking about this analogy made it better.) | |
| Aug 21, 2020 at 19:27 | comment | added | Timothy Chow | In number theory at least, people usually think about and talk about "embeddings" of $F$ into $K$ more than "subfields" $F\subseteq K$. That is, there is this abstract field $\mathbb{Q}(x)$ and it can be embedded in $\mathbb R$ in all kinds of ways, with $\mathbb{Q}(\pi)$ being one such embedding. | |
| Aug 21, 2020 at 18:53 | comment | added | Carl-Fredrik Nyberg Brodda | I should add that the idea that $\mathbb{Q}(\pi)$ is a subfield of $\mathbb{R}$ seemed very natural to me from the point of view of combinatorial (semi)group theory, where it's often very useful to makes analogous identifications. | |
| Aug 21, 2020 at 18:46 | comment | added | Carl-Fredrik Nyberg Brodda | "I was trying it out on some of the postdocs from representation theory". I remember you trying it out on some other unsuspecting victims too... | |
| Aug 21, 2020 at 12:53 | comment | added | Asaf Karagila♦ | If you want to appeal to theory builders you need to engage with other constructions that they might know, where truth is controlled "from below". Things like limits, or anything continuous really. We understand the truth of the limit as a limit of the truth of the sequence/diagram that led to it (in the infinite case, that is). Here it's the same. We want to understand the generic extension, so we need a blueprint, this is where the rational functions help. This motivates the need for names. Why are they complicated? Well, models of ZF are complicated. If you're a theory builder, you'll see. | |
| Aug 21, 2020 at 12:27 | comment | added | Timothy Chow | Maybe another way to think about it is that the audience is a theory builder who insists on asking annoyingly basic questions and isn't content with just using forcing as a black box. There may not be too many theory builders out there, but my feeling is that if the theory builder's questions can be answered then it could unlock benefits for other people too. | |
| Aug 21, 2020 at 12:25 | comment | added | Asaf Karagila♦ | I always had similar questions about localisation of rings, to be honest. Why do you need to drag me through these definitions. Just give me the $p$-adic numbers and get it over with. | |
| Aug 21, 2020 at 12:24 | comment | added | Timothy Chow | These are all very useful ideas and suggestions for explaining forcing, and I will add them to my arsenal of tricks. But as for what audience I have in mind for this specific question, I've had a number of people read my article and "get stuck" at roughly the same point, which to be honest is roughly the same point that I myself get stuck. Namely, why is all this machinery being dragged in? There is of course the a posteriori justification, "because it works." But is there an a priori justification for why it's needed? | |
| Aug 21, 2020 at 7:57 | history | answered | Asaf Karagila♦ | CC BY-SA 4.0 |