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

A classical reference is Hypothèse du Continu by Waclaw Sierpiński (1934), available through the Virtual Library of Science as part of the series Mathematical Monographs of the Institute of Mathematics of the Polish Academy of Sciences. Sierpiński discusses equivalences and consequences. The statements covered include examples from set theory, combinatorics, […]

There is a new journal of the European Mathematical Society that seems perfect for these articles: EMS Surveys in Mathematical Sciences. The description at the link reads: The EMS Surveys in Mathematical Sciences is dedicated to publishing authoritative surveys and high-level expositions in all areas of mathematical sciences. It is a peer-reviewed periodical […]

The answer is no, the statement that for every set $X$ we have $$X\not\to(\omega)^\omega_2$$ does not imply the axiom of choice. This was shown by Kleinberg and Seiferas in 1973, see MR0340025 (49 #4782) Kleinberg, E. M.; Seiferas, J. I. Infinite exponent partition relations and well-ordered choice. J. Symbolic Logic 38 (1973), 299–308. https://doi.org/10.23 […]

For positive integers $a_1,\dots,a_n$, recall that the multicolor Ramsey number $R(a_1,\dots,a_n)$ is the smallest integer $N$ such that if the edges of the complete graph $K_N$ are colored with the $n$ colors $1,\dots,n$, then there is some $i\le n$ and a set of $a_i$ vertices, all of whose edges received color $i$. A maximal Ramsey$(a_1,\dots,a_n)$-colorin […]

Georgii: Let me start with some brief remarks. In a series of three papers: a. Wacław Sierpiński, "Contribution à la théorie des séries divergentes", Comp. Rend. Soc. Sci. Varsovie 3 (1910) 89–93 (in Polish). b. Wacław Sierpiński, "Remarque sur la théorème de Riemann relatif aux séries semi-convergentes", Prac. Mat. Fiz. XXI (1910) 17–20 […]

Yes, this is a nice idea, and the approach is used in practice. I list four examples below, but there are many others. Any arithmetic statement, or any first order statement about $(\mathbb R,\mathbb N,+,\times,

Not necessarily. Consider the graph $G$ in ${\mathbb R}^2$ of the points $(x,y)$ such that $$ y^5+16y-32x^3+32x=0. $$ This example comes from the nice book "The implicit function theorem" by Krantz and Parks. Note that this is the graph of a function: Fix $x$, and let $F(y)=y^5+16y-32x^3+32x$. Then $F'(y)=5y^4+16>0$ so $F$ is strictly incre […]

Following Tomas's suggestion, I am posting this as an answer: I encountered this problem while directing a Master's thesis two years ago, and again (in a different setting) with another thesis last year. I seem to recall that I somehow got to this while reading slides of a talk by Paul Pollack. Anyway, I like to deduce the results asked in the prob […]

One way we formalize this "limitation" idea is via interpretative power. John Steel describes this approach carefully in several places, so you may want to read what he says, in particular at Solomon Feferman, Harvey M. Friedman, Penelope Maddy, and John R. Steel. Does mathematics need new axioms?, The Bulletin of Symbolic Logic, 6 (4), (2000), 401 […]

This is a transcendental number, in fact one of the best known ones, it is $6+$ Champernowne's number. Kurt Mahler was first to show that the number is transcendental, a proof can be found on his "Lectures on Diophantine approximations", available through Project Euclid. The argument (as typical in this area) consists in analyzing the rate at […]

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