Could anyone explain this statement? I feel like it's key for understanding the rest! How exactly does RT22 fall between 'these lines'?
> Almost all of the thousands of theorems studied by Simpson and his followers over the past four decades have turned out (somewhat mysteriously) to be reducible to one of five systems of logic spanning both sides of the finite-infinite divide. For instance, Ramsey’s theorem for triples (and all ordered sets with more than three elements) was shown in 1972 to belong at the third level up in the hierarchy, which is infinitistic.
> A breakthrough came in 1995, when the British logician David Seetapun, working with Slaman at Berkeley, proved that RT^2_2 is logically weaker than RT^3_2 and thus below the third level in the hierarchy.
> “Since then, many seminal papers regarding RT^2_2 have been published,” said Weiermann — most importantly, a 2012 result by Jiayi Liu (paired with a result by Carl Jockusch from the 1960s) showed that RT^2_2 cannot prove, nor be proved by, the logical system located at the second level in the hierarchy, one rung below RT^3_2.
So: there are five nested axiomatic systems that have been in common use to classify how much a theorem relies on infinite concepts. RT^2_2 is weaker than the third of those, and hard-to-compare with the second of them. (The second system can't prove it, but it also can't prove the second system.)
The new result says that RT^2_2 is reducible to primitive recursive arithmetic, which should mean that PRA is capable of proving anything RT^2_2 can prove. The article mentions that Mysterious Classification Level 2 is also reducible to primitive recursive arithmetic, so, as far as I understand things, PRA was already a system that fell "between the lines" of the five Mysterious Classification Levels (since Mysterious Level 3 is infinitistic and PRA is not).
There are some mathematicians who doubt that there is an infinite object (we call such mathematicians "finitist"). For such mathematicians, there is a large chunk of the mathematical literature they just can't use, because it relies inherently on the existence of an infinite set. Ramsey's theorem for pairs looks like it relies on the existence of an infinite set; the surprising result described in this article is that while RT_2^2 talks about infinite objects, its proof doesn't actually rely on them. So finitists are free to use it.
Even more, according to that article, the introduction of R_2^2 into a proof apparently doesn't break the property that "there is an algorithm to compute the construction the proof is doing". Previously it was thought that introducing R_2^2 to an otherwise "computable" proof (that is, one which carries out some computable operation) might stop it from being computable (that is, would make it so that no computer program corresponded to what the proof was doing: basically making it less constructive). It is now known that that is not the case.
> One can make statements about π or any other explicitly defined real number, as theorems about a specific sequence of rational approximations
https://math.stackexchange.com/questions/501/if-all-sets-wer...
Unfortunately, while SOME such formulas worked (the ones you learned in calculus - stuff like the chain rule), other formulas generated using the same kind of reasoning ("just imagine that d becomes infinitely small") come up with answers that are nonsensical or just wrong. How can we know when it's OK to just say "let things get infinitely small" and when it gives bogus answers?
The solution, which is taught in high-school calculus, was the "epsilon-delta" formulation. Instead of saying "let d become infinitely small" (a statement that may just be nonsense), say "I will prove that for ANY small epsilon (greater than 0) I can find a delta > 0 such that for values of x closer than delta, the error will be less than epsilon". That statement doesn't require an infinity to exist anywhere -- it is just a statement about particular finite numbers. And we can build calculus on such principles.
This isn't an EXACT analogy to your question about real numbers, but it uses the same type of reasoning, and I'm hoping the analogy is in terms of mathematical reasoning you are already familiar with.
Color the pair blue if its elements are 1 and 0. Color the pair red otherwise.
Color the triplet blue if its elements are 1, -1, and 0. Color the triplet red otherwise.
> When this is done, RT22 states that there will exist an infinite monochromatic subset: a set consisting of infinitely many numbers, such that all the pairs they make with all other numbers are the same color.
I read this as saying "There exists and infinite number of x's which satisfy the relation f(x, y) = blue for all values of y over some arbitrary function f()". What am I missing?
Because it's not an axiom? I feel like I'm not understanding your question completely
You may enjoy "Meta Math" by Gregory Chaitin (jump to Chapter 5 for the impatient):
http://arxiv.org/abs/math/0404335
"Finally, and perhaps even more devastatingly, it turns out that the set of all reals that can be individually named or specified or even defined or referred to—constructively or not—within a formal language or within an individual FAS, has probability zero. Summary: reals are unnameable with probability one.
So the set of real numbers, while natural—indeed, immediately given—geometrically, nevertheless remains quite elusive:
Why should I believe in a real number if I can’t calculate it, if I can’t prove what its bits are, and if I can’t even refer to it? And each of these things happens with probability one!"
You might also want to take a look at some of Norman Wildbergers work, like:
"Set Theory: Should You Believe?"
web.maths.unsw.edu.au/~norman/papers/SetTheory.pdf
It's commonly debated whether math is discovered, or invented. I think it depends on which part of math you're talking about. I think 1 + 1 = 2 is way over on the "discovered" end of the spectrum. It's very strongly grounded in nature, reality, everyday experience, etc. The mainstream treatment of infinity, including infinite sets, "different sized" infinities, bijections, etc., rely more on definitions and consensus about what passes as an acceptable "proof". For example, the tangent function is cited as a mapping between [-pi/2, pi/2] and [-inf, +inf], which is supposed to show that a subset of the reals is the same "size" as the whole set. But this requires defining division by zero as infinity in this context, whereas that's commonly considered undefined. I also have a big argument with the use of bijections (mappings) to compare the supposed "sizes" of infinities, but I can't fit it in a comment. The summary is that I think the ideas of cardinality and infinity are inherently contradictory, and putting them together creates nonsense. The idea of different sized infinities is actually created by (rather than proved by) the conventional restrictions on the style of bijections / mappings that are proposed and considered. But that's just another way of saying I think this area of math is a lot more on the "invented by definitions" end of the spectrum, and that whole philosophical question is another area on which people are going to differ in their attitudes.
On equivocation: Infinity is neither a number nor a quantity. It's not a number, because you can't get there by counting. It's not a quantity, because it can't be measured. But in the mainstream treatment of infinity, all the common intuitions about number and quantity get mixed in, for example the idea of different sized infinities. Infinity means "in this place where a number belongs, the value is unlimited". It's not itself an unlimited number, because as soon as it becomes unlimited, it no longer refers to any number. Similarly, I believe a more reasonable treatment of sets would say that "infinite" and "set" are incompatible, and attempting to force the concept of infiniteness onto a set makes it no longer a set, but rather something else like an abstract category. I think it was a mistake to generalize sets to include infinite sets.
As to how I treat infinity, it simply means something is boundless, inexhaustible, unlimited. I have no problem saying there are infinite reals, while minding that "infinity" is not a "count" of the reals. There are also infinite natural numbers. It doesn't make any sense to say there are more reals than naturals, in spite of "bijection theory". Reals, integers, natural numbers, etc. can all be considered different ways of naming items plucked from an infinite bag. When laid on a number line, reals and integers acquire one difference - integers can be adjacent, and reals can't.
One last note: I'm fully aware that my whole argument can be easily refuted by saying that math is made of definitions. That's fine. My position is simply that when people say things like "Hey, did you know there are actually different sized infinities? Isn't that cool?" they should be mindful that they're talking about a convention within a theory based on conventions, and not a natural or logical fact.
That's an oddly narrow implied definition of "number" which, as well as infinity, would exclude everything other than the natural numbers. Maybe it extends to the integers, if you use an unusually generous definition of "counting". But it certainly excludes non-integral rationals, and, a fortiori, all irrationals from the set of "numbers".