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Arthur T Knackerbracket has found the following story:
With a surprising new proof, two young mathematicians have found a bridge across the finite-infinite divide, helping at the same time to map this strange boundary.
The boundary does not pass between some huge finite number and the next, infinitely large one. Rather, it separates two kinds of mathematical statements: "finitistic" ones, which can be proved without invoking the concept of infinity, and "infinitistic" ones, which rest on the assumption — not evident in nature — that infinite objects exist.
Mapping and understanding this division is "at the heart of mathematical logic," said Theodore Slaman, a professor of mathematics at the University of California, Berkeley. This endeavor leads directly to questions of mathematical objectivity, the meaning of infinity and the relationship between mathematics and physical reality.
More concretely, the new proof settles a question that has eluded top experts for two decades: the classification of a statement known as "Ramsey's theorem for pairs," or RT22. Whereas almost all theorems can be shown to be equivalent to one of a handful of major systems of logic — sets of starting assumptions that may or may not include infinity, and which span the finite-infinite divide — RT22 falls between these lines. "This is an extremely exceptional case," said Ulrich Kohlenbach, a professor of mathematics at the Technical University of Darmstadt in Germany. "That's why it's so interesting."
The abstract is available on arXiv — the full article is available as a pdf.
[Ed note: Not a new story but interesting and will hopefully spark some discussion.]
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(Score: 5, Interesting) by maxwell demon on Monday July 31 2017, @05:57PM
I'm not a mathematician, so I'm not sure if I understood correctly; however what I think I understood is:
The theorems known before basically fell into one of two classes:
Now it turned out that this specific theorem is "in between" in the following way:
That means, despite not being provable without infinite sets, those who don't believe in the existence of infinite sets can use it to prove stuff about finite sets, as anything they can prove that way could lso be proved without infinite sets (although those proofs may be much more complicated).
What I wonder is, however, whether the proof of the second property itself requires use of infinite sets (in which case those not believing in them won't accept it anyway).
The Tao of math: The numbers you can count are not the real numbers.