A radically stronger logical structure than mathematics?
|August 1, 2011||Posted by News under Mathematics|
In “Ultimate logic: To infinity and beyond” ( New Scientist, 01 August 2011) Richard Elwes tells us,
This curiously intractable riddle is known as the continuum hypothesis, and it concerns that most enigmatic quantity, infinity. Now, 140 years after the problem was formulated, a respected US mathematician believes he has cracked it. What’s more, he claims to have arrived at the solution not by using mathematics as we know it, but by building a new, radically stronger logical structure: a structure he dubs “ultimate L”.
For most purposes, life within these structures is the same: most everyday mathematics does not differ between them, and nor do the laws of physics. But the existence of this mathematical “multiverse” also seemed to dash any notion of ever getting to grips with the continuum hypothesis. As Cohen was able to show, in some logically possible worlds the hypothesis is true and there is no intermediate level of infinity between the countable and the continuum; in others, there is one; in still others, there are infinitely many. With mathematical logic as we know it, there is simply no way of finding out which sort of world we occupy.
That’s where Hugh Woodin of the University of California, Berkeley, has a suggestion. The answer, he says, can be found by stepping outside our conventional mathematical world and moving on to a higher plane.
What is it about the expression “higher plane” that sets the horses bolting?
Before we bid farewell to meaningful numbers, thoughts?