Thursday, October 14, 2010

Paradoxes

Have you ever wanted to know what a web page is? Well, here to satisfy your curiosity is a screen shot of part of a typical web page.


In fact, it's a picture of a Wikipedia web page that explains what a web page is. If you want to access the web page in question, just click the above picture. Now, the Wikipedia article also contains a picture of a typical web page, and you can click that picture to see the web page in question. Once again, that will provide you with a clickable picture of a typical web page. But the embedding seems to stop at that point… instead of going on to infinity (as I had hoped).

When I was a child, I was fascinated by the packet of breakfast cereals that displayed, on the front side of the packet, an image of itself. For years, that picture created a tempest in my mind, and screwed up the calm breakfast atmosphere at South Grafton.

In my previous article, I evoked modern logic. After the cereal packet featuring a picture of a cereal packet (which in turn featured a picture of a cereal packet, and so on), my next biggest mental shock (several years later) was the paradox of Bertrand Russell about sets that are not members of themselves. Consider the set of all possible ideas. Obviously, that set is itself an idea. So, the set of all possible ideas is a member of itself. On the other hand, it's clear that the set of all pipes is not a pipe. So, the set of all pipes is not a member of itself. Consider all possible sets of the latter kind: that's to say, the set of all sets that are not members of themselves. Is that gigantic set a member of itself? Good question. To be a member of itself, that set has to be a set that is not a member of itself.

That sounds like a lot of mere words. No, Russell's paradox was much more than mere words. Curiously, nobody ever bothered to inform members of the philosophy department at Sydney University that Russell had evoked this enigmatic set… and, in so doing, destroyed forever all the formulations of logic that had existed up until then.

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