Two Mathmatician Solve a Decade Old Problem About infinity With a Breakthrough Proof.

Nothing will make you feel smaller or more mind-blown than the concept of infinity. Well, buckle up, because it only gets mind-blowier from there. Not only are there different kinds of infinity, they come in different sizes too. In 2016, two mathematicians blew the lids off their peers' heads by solving a decades-old problem about comparing infinities. Here is the 60-page proof summed up in just a few characters: p = t. Let us explain...

Uncomfortable Apples And Oranges.

Believe us when we say this article could be infinitely long, but ain't nobody got time for that. Let's start with infinity, a number that goes on and on and on forever. Because you could count 1, 2, 3... forever, there are infinite whole numbers. But, wait, you could do the same thing with just prime numbers. And even numbers too. Oh whoa. We just proved there are different kinds of infinity. Nice!

The numbers we were just talking about are called natural numbers, and they're just a tiny little branch on the overall number tree. If numbers are all one big umbrella, the category at the top that encompasses everything below is real numbers. A real number can be 3, or it can be √3, and it can even be 0.37846577246230456 — they're all real.

Comparing the infinite sets of natural numbers (whole numbers and zero, basically) is easy — there's a one-to-one relationship there. These are called countable sets because, uh, you can count them. Uncountable sets is what we're dealing with for real numbers. If you start at 1.0, is the next number in the infinite set 1.00001? Or is it 1.00000001? There are no spaces between numbers on the real number line, so they're uncountable, thus uncomparable to countable sets. Apples and oranges, people.

Every real number is essentially an infinity within itself, because you can have infinitely many decimal points. That being said, it's pretty clear to see that these overwhelmingly massive uncountable sets are larger than countable ones. This knowledge led mathematicians to wonder: if there are big and small infinite sets, can we have medium infinities too? Voilà! This question is the continuum hypothesis and it is literally one of the biggest (no pun intended) unsolved problems. In 1900, the German mathematician David Hilbert made a list of 23 of the most important problems in mathematics. He put the continuum hypothesis at the top.

Disproving the continuum hypothesis would mean that there are medium-sized infinities; proving it means there are only the bigs and the smalls. In 1940, mathematician Kurt Gödel showed that it couldn't be disproved within the usual axioms of mathematics, a.k.a. set theory. In the 1960s, mathematician Paul Cohen showed that the continuum hypothesis can't be proved by set theory. Ding ding ding! This won Cohen the Fields Medal, the highest honor in mathematics. And so, we inched just a tiny little bit closer to solving the continuum hypothesis.

Eureka.

One particular infinity-related question has persisted since the 1940s, even after Gödel's and Cohen's work: The problem of p and t. Mathematicians believed that if we could crack this problem, we could once and for all solve that darn continuum hypothesis. And, phew, in 2016, the p and t problem was finally solved.

Enter our heroes: Maryanthe Malliaris, of the University of Chicago, and Saharon Shelah, of the Hebrew University of Jerusalem and Rutgers University. The two mathematicians published a proof to this problem in the Journal of the American Mathematical Society and were honored in July 2017 with one of the top prizes in the field of set theory. (Here's a much shorter summary of the proof by Cornell University's Justin Moore, by the way.) But what'd they solve?

The question at hand asks whether p (one variant of infinity) is equal to t (another variant of infinity). Both p and t quantify the minimum size of collections of subsets of the natural numbers in precise (and probably unique) ways. The details of p and t aren't important; just know this: Both sets are larger than the infinite set of natural numbers, and p is always less than or equal to t. If p is less than t, then p would be a medium infinity and the continuum hypothesis would be false. Pretty major.

In 2011, Malliaris and Shelah started working on a totally different problem. (It was about ordering problems based on complexity, building off Keisler's order, in case you were wondering.) In the process, they realized they were also, kind of, accidentally making headway with the p and t dilemma. So they went with it. The two published a 60-page paper that solved their initial problem and the famous p and t problem at the same time. By proving that p and t are equally complex, they concluded that p equals t.

They proved it by carving out their own lane between two branches of mathematics: set theory and model theory. Their work is already opening new frontiers of research in both fields. Why does that matter? The more we know about math, the more we can understand the mysterious ways of the world around us. Thanks, Malliaris and Shelah!

(Psst — There's an unsatisfying end to this breakthrough mathematics story, too: Their work didn't solve the continuum hypothesis like mathematicians thought it would. Oh well! However, experts are pretty sure there have to be medium-sized infinities. Infinity is so weird that even the weirdest theories could be true, probably, maybe. Why not?)

So that it for now friend. Meet you guys in another mind blowing article. Have any questions ask me in the comments. If you like this blog make sure to subscribe it for daily science article. If you want us to make a article on your favorite topics mail us, Thank you ,Have a great day.

The Invention Of Zero.

When you were a little kid, even before you ever dealt with your first word problem in math class, you probably had to solve a problem something like this. You have four Starbursts, and you eat four Starbursts. What are you left with? That's right: sadness. And also no candy. But though even small kids can understand "nothing," the concept of "zero" is actually a bit more advanced; so advanced, in fact, that by the year 1200 C.E., it had only just barely reached the brightest mathematicians in Europe. This is the story of the invention of zero, and how a whole lot of nothing ended up changing the world.

Making Something Out Of Nothing.

It almost sounds impossible that ancient people wouldn't have the concept of "zero." Even animals can understand nothingness — just let your cat's dish go empty if you don't believe us. But there's a big difference between nothing as a tangible emptiness and zero as a mathematical concept. One forerunner of the mathematical zero can be seen in the earliest known counting system, devised by the Sumerians. At first, they'd use a blank space to indicate a nothing value, and when that grew confusing, they began using a pair of angled wedges as a placeholder for a blank space. But in a sense, that symbol indicated a lack of a number, not a number in and of itself.

Similar placeholders for an empty value can be found in other counting systems, including those of the Mayans and the Babylonians. But most scholars agree that zero as a mathematical concept originated in India. The earliest use of the round symbol that would become the universal zero comes from the Bakhshali manuscript, a merchant's document explaining mathematical equations for various transactions. It also included a placeholder zero in the form of a little black dot, and was in common parlance in India in the 3rd or 4th centuries C.E. Just a couple of centuries later, the symbol was used by legendary mathematical scholar Brahmagupta. In the 7th century, he wrote the earliest surviving explanation of how, exactly zero works: "When zero is added to a number or subtracted from a number, the number remains unchanged. A number multiplied by zero becomes zero."

He also worked out that subtracting a positive number from zero gave you a negative number, and that subtracting a negative number from zero gave you a positive. That's the first known account of knowing how zero works in relation to other numbers, and we can only assume he went on to coin the phrase, "Ditch the zero, get with the hero."

Arabic Zero.

Zero Goes Abroad.

After zero caught on in the Indian subcontinent, it was only a matter of time before other cultures began to recognize its significance. China and the Arabian peninsula were first (although it's worth noting that some historians believe the Arabic zero was a direct descendant of the zero precursors of Sumeria and Babylon), and it was in the Arabic numeral system that it first took the form of an empty oval. Muslim mathematicians called the symbol "sifr" (anglicized as "cipher"), and with it, invented both algebra and algorithms. And as Islam spread to Africa, zero came along for the ride.

But after that, it ran into some issues. Namely, Europeans. When the Moors conquered Spain, they brought their math along with them, and from there, zero made it to Italy. Where it was promptly outlawed. Yes, religious leaders of Europe saw the devil in that little blank circle, which they strongly associated with Islam. But the number didn't stop being useful, and merchants knew that very well. So when they'd include zeroes on their ledgers, they did so in secret — and the word "cipher" came to be synonymous with "code" in the process.

Fortunately for European mathematics, the taboo didn't last. Without zero, Newton and Leibniz wouldn't have been able to come up with calculus, Descartes couldn't have figured out how to graph points, and car dealers wouldn't be able to dazzle customers with the mysterious phrase "0% APR."

So that it for now friend. Meet you guys in another mind blowing article. Have any questions ask me in the comments. If you like this blog make sure to subscribe it for daily science article. If you want us to make a article on your favorite topics mail us, Thank you ,Have a great day.