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For 50 years, mathematicians have believed that the total number of real numbers is unknowable. A new proof suggests otherwise.There are an infinite number of infinities. Which one corresponds to the real numbers?An Infinity of InfinitiesYes, infinity comes in many sizes. In 1873, the German mathematician Georg Cantor shook math to the core when he discovered that the “real” numbers that fill the number line — most with never-ending digits, like 3.14159… — outnumber “natural” numbers like 1, 2 and 3, even though there are infinitely many of both.Infinite sets of numbers mess with our intuition about size, so as a warmup, compare the natural numbers {1, 2, 3, …} with the odd numbers {1, 3, 5, …}. You might think the first set is bigger, since only half its elements appear in the second set. Cantor realized, though, that the elements of the two sets can be put in a one-to-one correspondence. You can pair off the first elements of each set (1 and 1), then pair off their second elements (2 and 3), then their third (3 and 5), and so on forever, covering all elements of both sets. In this sense, the two infinite sets have the same size, or what Cantor called “cardinality.” He designated their size with the cardinal number _{0} (“aleph-zero”).But Cantor discovered that natural numbers can’t be put into one-to-one correspondence with the continuum of real numbers. For instance, try to pair 1 with 1.00000… and 2 with 1.00001…, and you’ll have skipped over infinitely many real numbers (like 1.000000001…). You can’t possibly count them all; their cardinality is greater than that of the natural numbers.Sizes of infinity don’t stop there. Cantor discovered that any infinite set’s power set — the set of all subsets of its elements — has larger cardinality than it does. Every power set itself has a power set, so that cardinal numbers form an infinitely tall tower of infinities.Standing at the foot of this forbidding edifice, Cantor focused on the first couple of floors. He managed to prove that the set formed from different ways of ordering natural numbers (from smallest to largest, for example, or with all odd numbers first) has cardinality _{1}, one level up from the natural numbers. Moreover, each of these “order types” encodes a real number.His continuum hypothesis asserts that this is exactly the size of the continuum — that there are precisely _{1} real numbers. In other words, the cardinality of the continuum immediately follow _{0}, the cardinality of the natural numbers, with no sizes of infinity in between.But to Cantor’s immense distress, he couldn’t prove it.In 1900, the mathematician David Hilbert put the continuum hypothesis first on his famous list of 23 math problems to solve in the 20th century. Hilbert was enthralled by the nascent mathematics of infinity — “Cantor’s paradise,” as he called it — and the continuum hypothesis seemed like its lowest-hanging fruit.To the contrary, shocking revelations last century turned Cantor’s question into a deep epistemological conundrum.The trouble arose in 1931, when the Austrian-born logician Kurt Gödel discovered that any set of axioms that you might posit as a foundation for mathematics will inevitably be incomplete. There will always be questions that your list of ground rules can’t settle, true mathematical facts that they can’t prove.As Gödel suspected right away, the continuum hypothesis is such a case: a problem that’s independent of the standard axioms of mathematics.These axioms, 10 in all, are known as ZFC (for “Zermelo-Fraenkel axioms with the axiom of choice”), and they undergird almost all of modern math. The axioms describe basic properties of collections of objects, or sets. Since virtually everything mathematical can be built out of sets (the empty set {} denotes 0, for instance; {{}} denotes 1; {{},{{}}} denotes 2, and so on), the rules of sets suffice for constructing proofs throughout math.In 1940, Gödel showed that you can’t use the ZFC axioms to disprove the continuum hypothesis. Then in 1963, the American mathematician Paul Cohen showed the opposite —you can’t use them to prove it, either. Cohen’s proof, together with Gödel’s, means the continuum hypothesis is independent of the ZFC axioms; they can have it either way.

Not everything that is true can be proven. This discovery transformed infinity, changed the course of a world war and led to the modern computer.

https://brilliant.org/wiki/infinity/Infinity is the concept of an object that is larger than any number. When used in the context "...infinitely small," it can also describe an object that is smaller than any number. It is important to take special note that infinity is not a number; rather, it exists only as an abstract concept. Attempting to treat infinity as a number, without special care, can lead to a number of paradoxes.Infinity is not a number!

Because infinity is not a number, it requires special rules of arithmetic in order to remain consistent. This is a useful view in fields such as measure theory, which extends the real numbers by adding two numbers ∞ and −∞, that satisfy some additional rules:a+∞ = ∞+a = ∞ (for any a besides −∞)a−∞ = −∞+a = −∞ (for any a besides ∞)a⋅∞ = ∞⋅a = −∞ (for positive a)a⋅∞ = ∞⋅a = −∞ (for negative a)a/∞ = a/-∞ = 0 (for real a)∞/a = ∞ (for positive a)∞/a = -∞ (for negative a)It is worth noting that 1/0 is not ∞. Additionally, operations involving multiple infinities (such as ∞−∞ and ∞/∞) are not generally well-defined.Importantly, note that this is an extension of the real numbers chosen specifically to allow infinity to be treated as a number; without this context, infinity remains a concept rather than a number. For instance, the following sections continue to treat infinity as a concept rather than a number, because the measure theory context above no longer applies

IMO, the paradox and confusion comes from how we treat infinity.

Infinity can throw up some interesting paradoxes, from filling Hilbert's Hotel to painting Gabriel's Trumpet... Mark Jago is a philosophy lecturer with a background in computer science.

Isn't " Infinity " Not a value, but just a never ending process.

Hi again.Of the videos you've posted, which ones would you actually recommend and why?There's 15 minutes x 3 videos = nearly an hour of stuff.Best Wishes.

I recommend all of them, otherwise I wouldn't put them here in the first place.The last video is the continuation of the second, so it's not recommended to go straight to the last one.

IMO, the continuum hypothesis conundrum occurs when we accept the validity of Cantor's method to define cardinality of number set.

IMO, the continuum hypothesis conundrum occurs when we accept the validity of Cantor's method to define cardinality...........So, if we reject Cantor's method, the conundrum would disappear.

OK. I'm not able to watch an hour of videos at the moment, sorry. Those videos don't have subtitles and when I can't see the presenter I have little idea what they are saying half the time so I can't engage with those videos very much.

Cantor did lots of things. What are you calling "the Cantor method"?

Why is the Continuum hypothesis a conundrum?

To the contrary, shocking revelations last century turned Cantor’s question into a deep epistemological conundrum.

ES: What are you calling "the Cantor method"?HY: You can read in the opening post.

if we reject Cantor's method = ?? bijections to define cardinality ??, the conundrum would disappear.

Can't find "Cantor method" as a phrase in the OP. Best guess ---> it's the idea of finding bijections between sets.

I can't find "Cantor's question" in the article except in this one sentence.Best Guess ----> the question was supposed to be "is the continuum hypothesis provable?"

But to Cantor’s immense distress, he couldn’t prove it.

This was discussed below. You're absolutely right, we could just define the size or cardinality of all infinte sets to be ∞. It's a definition of cardinality that is of limited use but it does show that infinite sets are bigger than finite sets, which is enough information in many situations.

Math Has a Fatal FlawIMO, the paradox and confusion comes from how we treat infinity.

How to "Prove" That 2 = 1Let's begin our journey into the bizarre world of apparently correct, yet obviously absurd, mathematical proofs by convincing ourselves that 1 + 1 = 1. And therefore that 2 = 1. I know this sounds crazy, but if you follow the logic (and don't already know the trick), I think you'll find that the "proof" is pretty convincing.Here's how it works:Assume that we have two variables a and b, and that: a = bMultiply both sides by a to get: a^{2 }= abSubtract b2 from both sides to get: a^{2 } - b^{2 } = ab - b^{2 }This is the tricky part: Factor the left side (using FOIL from algebra) to get (a + b)(a - b) and factor out b from the right side to get b(a - b). If you're not sure how FOIL or factoring works, don't worry—you can check that this all works by multiplying everything out to see that it matches. The end result is that our equation has become: (a + b)(a - b) = b(a - b)Since (a - b) appears on both sides, we can cancel it to get: a + b = bSince a = b (that's the assumption we started with), we can substitute b in for a to get: b + b = bCombining the two terms on the left gives us: 2b = bSince b appears on both sides, we can divide through by b to get: 2 = 1Wait, what?! Everything we did there looked totally reasonable. How in the world did we end up proving that 2 = 1?https://www.quickanddirtytips.com/education/math/how-to-prove-that-1-2

Instead of blaming math,....

Math Has a Fatal Flaw

I'm lost. Who was blaming Maths? Only the person who wrote the original article in the first post, or was it someone else? I'll bet it was Alancalverd, it's always him.