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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.]
-- submitted from IRC
(Score: 4, Interesting) by Anonymous Coward on Monday July 31 2017, @06:30PM (3 children)
I haven't read the paper, but basically this is the way mathematical systems work. You start with a bunch of facts (called axioms, which are true in the mathematical system by definition) and explore the consequences of these.
For a proposition P, we say that there is a proof of P in our mathematical system (alternately: we say our system proves P) if we can deduce P using a sequence of logical steps starting from the axioms. If our system proves two contradicting propositions, it is called inconsistent, otherwise it is called consistent. We want our systems to be consistent because otherwise they prove every proposition and are therefore useless. We assume our system is consistent (an interesting topic for another day).
In any sufficiently powerful axiom system (assuming it is consistent), there will be some propositions can be neither proved nor disproved. Such propositions are called independent of the axioms.
In the context of this article, these guys are presumably working in the axiom system known as Zermelo-Frankel Set Theory (ZFC), upon which most of modern mathematics is based. There are a lot of axioms in this system (actually an infinite number) but one is so important that it has a name: the axiom of infinity. This roughly says "There exists some set with an infinite number of elements". There are a couple different formal methods of expressing this which are basically all equivalent given the rest of the axioms of ZFC. From this one set we can then use the other axioms to prove the existence of many other infinite sets, such as the infinite ordinals and so on. One key fact is that the axiom of infinity is a proposition which is independent of the other axioms of ZFC -- that is, the existence (or non-existence) of infinite sets cannot be proved from the other axioms alone.
So now if we remove this axiom from ZFC then we get a new system (ZFC without the axiom of infinity) which cannot construct infinite sets so it has little to say about such objects. This system is less "powerful" than ZFC as it proves fewer propositions. A mathematician thus might be interested in asking what propositions can still be proved in this reduced system. In this case, there was a theorem (called "Ramsey's Theorem for Pairs") that is ostensibly a theorem about infinite sets but these guys managed to prove it in a system that doesn't presume the existence of any infinite sets. Put another way, this theorem is independent of the axiom of infinity. Neat!
(Score: 0) by Anonymous Coward on Monday July 31 2017, @07:23PM
(not GP)
Thanks. This was my understanding as well, but you put it in much simpler terms.
I also think the excitement comes from the possibility of now generating new proofs for other theorems, that are also independent of the axiom of infinity (unlike current proofs).
(Score: 2) by JoeMerchant on Tuesday August 01 2017, @12:40PM (1 child)
So, does that make the infinite sets in "Ramsey's Theorem for Pairs" a logical conclusion of ZFC without the axiom of infinity? Proving the existence of infinite sets without needing the axiom.
Would be fun to find another "provable axiom" to reduce out of the set of "required" ZFC axioms.
🌻🌻 [google.com]
(Score: 0) by Anonymous Coward on Tuesday August 01 2017, @04:20PM
Well, no, the axiom of infinity is known to be independent of the other axioms in ZFC so there is no way to prove the existence or non-existence of infinite sets in "ZFC without the axiom of infinity". (assuming that ZFC is consistent).
Anyway I decided to actually look at the paper and the system used is not in fact "ZFC without the axiom of infinity", but PRA (primitive recursive arithmetic), which is an axiom system for the natural numbers.