Comments on Popularizing mathematical concepts

phi (2004-01-14T12:14:11Z)

ok merci (oui, oui, c'est une idée fixe…)

hé, tu es matinal aujourd'hui!
(enfin, pour l'heure GMT, tout de même!)

Ruxor (2004-01-14T11:59:24Z)

phi: No, I don't think what you want (if I understand it correctly) can be done in any reasonable way. On the other hand, the nonstandard reals as Robinson &al define them seem to correspond to what you want. But they do not compare easily with the Long Line (neither embeds in the other, for example).

phi (2004-01-14T01:00:03Z)

one more question:

could we say that the Long Line is to the limits of real sequences (finite or infinite) what the line is to the reals ?

The set of limits being defined as a quotient of the set of functions by a relation such as lim(f-g)<<lim f (which can be defined in such a manner that it is not circular)

Ruxor (2004-01-10T01:12:03Z)

Karl: Yes, both properties which you surmise are, in fact, true. And both are consequence of the fact (which I stated, although perhaps not clearly enough) that any countable subset of the Long Line is bounded (in the sense that there is a point on the Long Line that is to the right of all the countably many chosen points, and one which is to the left of all).

Karl (2004-01-09T14:37:56Z)

Perhaps it is impossible to have a set of points on the long line each numbered uniquely by in integer, such that every point on the long line is between two of these numbered points.

Karl (2004-01-09T14:22:40Z)

Talking about the long line. You said
--------------------------------------
There is no “distance” or “uniformity” on the Long Line. Let us be a little more precise than this. It makes no sense to say that A is closer to B than C is to D, because distance can be stretched or shrinked as one will. The only case in which it does makes sense is to say that A is closer to B than A is to C, when B and C are on the same side of A: then it means that B is between A and C, in other words that one must go through B to get from A to C (so certainly B is closer to A than C is).
---------------------------------------

You then later invoke the concept of distance, invalidated by the above statement in describing how the long line is longer than an ordinary line (such as the real numbers). This does not make sense to me. Can you describe how the long line differs from an ordinary line without invoking the concept of distance or any concept derived from distance such as speed.

Perhaps it is impossible to divide the long line into consecutive segments (each with two ends) such that one can number all the segments with integers (i.e. divide into a countable number of segments).

Ruxor (2004-01-02T04:18:51Z)

phi: There are no longer lines because they would not be varieties: at the point (omega_1,0) (the "end" of the Long Line) the system of neighborhoods would look qualitatively different from a line. Even as topological spaces, these longer lines are not as nice as the Long Line because they are not homogeneous (for the reason just stated: some points look different).

The reals of nonstandard analysis are very different from the Long Line. Actually, when seen as a topological space in the standard sense, the set of nonstandard real numbers is not even connected. Also, the nonstandard reals contain infinitely small quantities, not just infinitely large ones, whereas the Long Line differs only in length, not in refinement. And lastly, no addition and multiplication can be reasonably defined on the Long Line.

Yes, the exhaustion of the ordinary line presupposes a distance in the way I have stated it, but does not depend on the chosen distance.

phi (2004-01-01T17:03:47Z)

I finally took the time to read your text, which I enjoyed, as an ex- math student. Some questions/remarks:

It is not clear to me why there are no longer lines, with greater cardinals (would that be no variety? or would it be *homeo*morphic to the Long Line?).

Has it anything to do with the real line of non-standard analysis (cf. the local constancy of any continuously varying quantity)?

The exhaustion of the ordinary line by tranlating something présupposes distance doesn't it? Anyway, the overrun of any denumerable set of points gets the point.

Making sense of mathematics is not an easy task but a most interesting one…

Ruxor (2003-12-31T14:58:58Z)

Moui enfin ça c'est une propriété de omega_1, pas de la Longue Droite en elle-même : et c'est trompeur de le dire comme ça, parce que omega_1 n'est pas homogène (il y a des points successeurs et des points limite) alors que la Longue Droite, elle, elle l'est (c'est-à-dire que les points (alpha,0) ne sont pas canoniquement définis, tous les points de la Longue Droite se valent).

Mais la propriété sur le métro est en effet importante. Je l'ai utilisée (généralisée à un cardinal régulier quelconque) dans mon article commun avec Colliot-Thélène (à paraître au *J. Inst. Math. Jussieu*) sur les surfaces de Del Pezzo sur les corps de dim. coh. 1, pour rédiger proprement ce qui pour Merkur'ev est une pure trivialité. J'ai passé un bon moment à trouver la démonstration précise de ce qui me semblait une parfaite évidence, d'ailleurs…

denis f (2003-12-31T10:18:41Z)

Concernant la longue ligne, peu de gens savent qu'elle relie l'aéroport à l'hôtel de Hilbert, et qu'elle est parcourue par le Métro Transfini(TM), qui s'arrête à toutes les stations (alpha,0), ainsi qu'au terminus w1 (du compactifié, évidemment). Une propriété amusante de ce métro est que si, à chaque station, descend un passager (s'il en reste), puis monte une infinité dénombrable de nouveaux passagers, le métro arrive toujours vide à l'hôtel (voir la démonstration sur mon site : http://perso.wanadoo.fr/denis.feldmann/DVI/metro.dvi)


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