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Large cardinals: maths shaken by the 'unprovable'

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Proud Mary

Tue Nov 09 2010, 09:15AM

Registered Member #543 Joined: Tue Feb 20 2007, 04:26PM
Location: UK
Posts: 4992

Large cardinals: maths shaken by the 'unprovable'
A shocking discovery has unsettled the world of numbers, says Richard Elwes.

By Richard Elwes
Daily Telegraph
Published: 7:50AM GMT 09 Nov 2010

In 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 1,2,3,4,5â€¦ 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 GÃ¶del 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.

GÃ¶del'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 GÃ¶del'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: 1,2,3â€¦ 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)

Sulaiman

Tue Nov 09 2010, 07:04PM

Registered Member #162 Joined: Mon Feb 13 2006, 10:25AM
Location: United Kingdom
Posts: 3141

WOW !!!

I wish I understood that.

I thought that Large Cardinals came about due to overeating .. d'oh!

Bjørn

Tue Nov 09 2010, 10:02PM

Registered Member #27 Joined: Fri Feb 03 2006, 02:20AM
Location: Hyperborea
Posts: 2058

Here is more information

with some nice .PDF files.

It does not seem to be very shocking, it is a more a development on some older ideas like all new discoveries in mathematics tend to be.

Proud Mary

Tue Nov 09 2010, 10:02PM

Registered Member #543 Joined: Tue Feb 20 2007, 04:26PM
Location: UK
Posts: 4992

Sulaiman wrote ...

WOW !!!

I wish I understood that.

I thought that Large Cardinals came about due to overeating .. d'oh!

Surely you're not having difficulty with the role of Zermeloâ€“Fraenkel set theory in the von Neumann Universe, Sulaiman? You've shaken my belief in 4HV to its very foundations!

Dr. Slack

Wed Nov 10 2010, 08:52PM

Registered Member #72 Joined: Thu Feb 09 2006, 08:29AM
Location: UK St. Albans
Posts: 1659

After reading through that lot, I'm not sure that I've seen anything fundamnetally different from my english language version of Goedell's, which paraphrases to
a) if a language is not sufficiently powerful enough to describe itself (eg C++) then it is complete, but cannot be proved within itself
b) if a language is powerful enough to refer to itself (eg I am lying) then self-contradictory statements can be constructed in it, and so cannot be proved within itself,
but then I'm not a mathematician, and my head hurts after reading the background material

It seems that as system complexity increases, never mind the detail of whether it's integers, or infinities, or sets of sets, or sets of sets (of sets ...), or axioms, or systems of systems of systems of axioms, Goedell is there pushing the limit up to keep it just beyond reach by, so to speak, his own bootstraps. That's possibly stated a little too vaguely for your jobbing mathematician, but it does for me.

Bjørn

Wed Nov 10 2010, 10:17PM

Registered Member #27 Joined: Fri Feb 03 2006, 02:20AM
Location: Hyperborea
Posts: 2058

Yes, it is the old Goedel / Turing thing, it is just that it is brought closer by finding an actual example of something (pattern in numbers) that requires something outside our normal framework to explain. I could not find any clear explanation about what this "pattern" thing was, so as usual when it comes to mathematics it is not easy to get much wiser. That their notation looks like line noise does not help much either.

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