Linked by Thom Holwerda on Fri 22nd Mar 2013 10:02 UTC

"But a powerful new type of computer that is about to be commercially deployed by a major American military contractor is taking computing into the strange, subatomic realm of quantum mechanics. In that infinitesimal neighborhood, common sense logic no longer seems to apply. A one can be a one, or it can be a one and a zero and everything in between - all at the same time. [...] Now, Lockheed Martin - which bought an early version of such a computer from the Canadian company D-Wave Systems two years ago - is confident enough in the technology to upgrade it to commercial scale, becoming the first company to use quantum computing as part of its business." I always get a bit skeptical whenever I hear the words 'quantum computing', but according to NewScientist, this is pretty legit.

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2009-01-29

Well, first of all, there are two different concepts of infinity that had been discussed in some of this threads. One is the idea of infinity as a number that is larger than all the real numbers. The other is the idea of a set having an infinite number of elements. They are

notthe same concept. Let me give a brief description of both to see the difference:1.- For certain applications, it is convenient to extend the real numbers to allow the values infinity and -infinity. (For example, this is a very common thing to do in measure theory and integration.) In this case infinity is (in a sense) a quantity that is bigger than all others quantities. The drawback of this approach is that infinity doesn't behave well under the usual arithmetic operations. For example infinity+1=infinity, while infinity-infinity is always undefined. In this setting you can't have anything bigger than infinity. Trying to manipulate divergent sequences as if they were the quantity infinity and doing algebraic manipulations with them is incorrect, and can quickly get you to absurd statements. For example, Riemann showed that if you have a conditionally convergent sequence, then you can rearrange its terms to make it have any limit you want.

2.- In set theory there is the notion of the "cardinality" of a set, which is a measure of how big it is. Two sets are said to be of the same cardinality if you can give a bijection between them. Of course, given any set you can always add one element to it to make it bigger. (This is based on the fact that there is no such a thing as "the set of all sets".) But having a set with more elements doesn't automatically means that it has a higher cardinality. For example the set of positive integers has the same cardinality as the set of even positive integers. (You just enumerate your positive even integers to get the bijection.) However, Cantor showed that the set of real numbers

doeshave a bigger cardinality than the set of integers. This is very shocking. It means that it doesn't matter how you try to enumerate the real numbers, at the end there is always an infinite number of real numbers that you left out. Given any set, the set of all its subsets always has a higher cardinality, so there is no biggest cardinal number. (Again this is related to the fact that there is no "set of all sets".)