On Google+, Willie Wong posted a link to this interesting example, by Brian Gawalt: BART fares and the triangle inequality.

There is a natural way of measuring distance in a subway or train system, the “price between stations” metric. It turns out that when applied to BART, the Bay Area Rapid Transit system, this fails to be a metric, with the consequence that sometimes it is cheaper to take a detour, exiting and reentering an intermediate station, than going directly to one’s destination. As Gawalt points out:

It’s probably important to recognize the 15 cents you save by jumping out costs about 15 to 20 minutes of your life waiting for the next train to come pick you up.

Willie adds an interesting comment, that I reproduce here:

Heh, while BART fails to be a metric space (with the price between stations metric), it is interesting to note that the single-fare systems form ultrametric spaces.

The British Rail / PostOffice metrics, of course, reflect systems with concentric zones in rings for which to get from one place to another almost certainly require passing through the centre. Like London Underground for example.

The public transport in Lausanne does not form a metric space using the price-between-stations metric for another (somewhat strange) reason: the price-between-stations function is set valued: the same two stations can have different prices depending on which route the bus/train takes, even without you getting off. (This is the problem with a zone based system. For certain places there are two more or less identical routes but one goes through two or three more zones than the other: some of the zones looks like they are slightly gerrymandered.) Of course, in this case most sensible people would just buy the cheapest available fare and take the cheapest available route, showing that a zone-based system is much more like a Riemannian manifold (and commuters try to travel in geodesics)…

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qué interesantes esos casos ultramétricos – claro que si se tomara algo como el producto del precio por el tiempo usado, la cosa sería bien distinta en el caso del BART

Marginalia to a theorem of Silver (see also this link) by Keith I. Devlin and R. B. Jensen, 1975. A humble title and yet, undoubtedly, one of the most important papers of all time in set theory.

Given a positive integer $a$, the Ramsey number $R(a)$ is the least $n$ such that whenever the edges of the complete graph $K_n$ are colored using only two colors, we necessarily have a copy of $K_a$ with all its edges of the same color. For example, $R(3)= 6$, which is usually stated by saying that in a party of 6 people, necessarily there are 3 that know e […]

No, this is not consistent. Todorčević has shown in ZF that, in fact, there is no function $F\!:\mathcal W(S)\to S$ with the property you require. Here, $\mathcal W(S)$ is the collection of subsets of $S$ that are well-orderable. This is corollary 6 in MR0793235 (87d:03126). Todorčević, Stevo. Partition relations for partially ordered sets. Acta Math. 155 (1 […]

As suggested by Gerald, the notion was first introduced for groups. Given a directed system of groups, their direct limit was defined as a quotient of their direct product (which was referred to as their "weak product"). The general notion is a clear generalization, although the original reference only deals with groups. As mentioned by Cameron Zwa […]

A database of number fields, by Jürgen Klüners and Gunter Malle. (Note this is not the same as the one mentioned in this answer.) The site also provides links to similar databases.

You do not need much to recover the full ultrapower. In fact, the $\Sigma_1$-weak Skolem hull should suffice, where the latter is defined by using not all Skolem functions but only those for $\Sigma_1$-formulas, and not even that, but only those functions defined as follows: given a $\Sigma_1$ formula $\varphi(t,y_1,\dots,y_n)$, let $f_\varphi:{}^nN\to N$ be […]

I posted this originally as a comment to Alex's answer but, at his suggestion, I am expanding it into a proper answer. This situation actually occurs in practice in infinitary combinatorics: we use the axiom of choice to establish the existence of an object, but its uniqueness then follows without further appeals to choice. I point this out to emphasize […]

I think you may find interesting to browse the webpage of Jon Borwein, which I would call the standard reference for your question. In particular, take a look at the latest version of his talk on "The life of pi" (and its references!), which includes many of the fast converging algorithms and series used in practice for high precision computations […]

The reference you want is MR2768702. Koellner, Peter; Woodin, W. Hugh. Large cardinals from determinacy. Handbook of set theory. Vols. 1, 2, 3, 1951–2119, Springer, Dordrecht, 2010. Other sources (such as the final chapter of Kanamori's book) briefly discuss the result, but this is the only place where the details are given. More recent papers deal with […]

qué interesantes esos casos ultramétricos – claro que si se tomara algo como el producto del precio por el tiempo usado, la cosa sería bien distinta en el caso del BART