http://www.telegraph.co.uk/science/8118823/Large-cardinals-maths-shaken-by-the-unprovable.htmlIn the esoteric world of mathematical logic, a dramatic discovery has been made. Previously unnoticed gaps have been found at the very heart of maths. What is more, the only way to repair these holes is with monstrous, mysterious infinities.
To understand them, we must understand what makes mathematics different from other sciences. The difference is proof.
Other scientists spend their time gathering evidence from the physical world and testing hypotheses against it. Pure maths is built using pure deduction.
But proofs have to start somewhere. For all its sophistication, mathematics is not alchemy: we cannot conjure facts from thin air. Every proof must be based on some underlying assumptions, or axioms.
And there we reach a thorny question. Even today, we do not fully understand the ordinary whole numbers 1,2,3,4,5… or the age-old ways to combine them: addition and multiplication.
Over the centuries, mathematicians have arrived at basic axioms which numbers must obey. Mostly these are simple, such as "a+b=b+a for any two numbers a and b". But when the Austrian logician Kurt Gödel turned his mind to this in 1931, he revealed a hole at the heart of our conception of numbers. His "incompleteness theorems" showed that arithmetic can never have truly solid foundations. Whatever axioms are used, there will always be gaps. There will always be facts about numbers which cannot be deduced from our chosen axioms.
Gödel's theorems showed that maths meant that mathematicians could not hope to prove every true statement: there would always be "unprovable theorems", which cannot be deduced from the usual axioms. Most known examples, it's true, will not change how you add up your shopping bill. For practical purposes, the laws of arithmetic seemed good enough.
However, as revealed in his forthcoming book, Boolean Relation Theory and Concrete Incompleteness, Harvey Friedman has discovered facts about numbers which are far more unsettling. Like Gödel's unprovable statements, they fall through the gaps between axioms. The difference is that these are no longer artificial curiosities. Friedman's theorems are "concrete", meaning they contain genuinely interesting information concerning patterns among the numbers, which must always appear once certain conditions are met. Yet, Friedman has shown, the fact that such patterns always appear does not follow from the usual laws of arithmetic.
These patterns are not yet affecting physicists or engineers, but mathematicians are having to take unprovability seriously. In the past, they just had to show whether an idea is true or false. Now, results such as Friedman's raise the awkward possibility that the standard laws of mathematics may not provide an answer.
There is one easy way to make an unprovable theorem provable: adding more axioms. But which axioms do we need? The new axioms require a hard look at one of the most contentious issues in mathematics: infinity.
For more than 100 years, mathematicians have known that there are different kinds, and sizes, of infinity. This was first shown by the 19th-century genius Georg Cantor. Cantor's discovery was that it makes sense to say that one infinite collection can be bigger than another. Infinity resembles a ladder, with the lowest rung corresponding to the most familiar level of infinity, that of the ordinary whole numbers: 1,2,3… On the next rung lives the collection of all possible infinite decimal strings, a larger uncountably infinite collection, and so on, forever.
This astonishing breakthrough raised new questions. For instance, are there even higher levels which can never be reached this way? Such enigmatic entities are known as "large cardinals". The trouble is that whether or not they exist is a question beyond the principles of mathematics. It is equally consistent that large cardinals exist and that they do not.
At least, so we thought. But, like gods descending to earth to walk among mortals, we now realise their effect can be felt among the ordinary finite numbers. In particular, the existence of large cardinals is the condition needed to tame Friedman's unprovable theorems. If their existence is assumed as an additional axiom, then it can indeed be proven that his numerical patterns must always appear when they should. But without large cardinals, no such proof is possible. Mathematicians of earlier eras would have been amazed by this invasion of arithmetic by infinite giants.
Dr Richard Elwes is the author of 'Maths 1001: Absolutely Everything That Matters in Mathematics' (Quercus Publishing)
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