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Assume that we have two variables a and b, and that: a = bMultiply both sides by a to get: a2 = abSubtract b2 from both sides to get: a2 - b2 = ab - b2This 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 = b

I'm not sure what the relevance of the last posts were. I'm not sure what you were trying to say with it.

I was thinking that the problem introduced by the article in the op can't just go away because it contains a mathematical fallacy, like what I quoted in my latter post.

The Natural numbers, for example, is not a finite set. Given any Natural number, N, we can obviously find more than N elements in the set of Natural numbers. The subset { 1, 2, 3, 4, ......, N, N+1 } is contained in the Naturals and it has size N+1 which is clearly greater than N. So the Natural numbers cannot have a finite size. However the Natural numbers are the obvious example of a set that is infinite but can be enumerated. We can put the set of Naturals into a 1-to1 correspondence with the set of Naturals ---> The identify mapping will do it (just map 1 → 1; 2 → 2 and 3 → 3 ..... etc....... ).

Let's check if countability has a binary value.

Assume that set of natural numbers is countable, while set of real numbers is uncountable.

Then find a set more diluted than real numbers, and determine if it's still countable.

Let's start with set of rational numbers.

If you think it's countable, find another set that's denser. Otherwise, find another set that's more diluted. Repeat the process until threshold point is found.

https://en.wikipedia.org/wiki/Power_set#PropertiesCantor's diagonal argument shows that the power set of a set (whether infinite or not) always has strictly higher cardinality than the set itself (or informally, the power set must be larger than the original set). In particular, Cantor's theorem shows that the power set of a countably infinite set is uncountably infinite. The power set of the set of natural numbers can be put in a one-to-one correspondence with the set of real numbers (see Cardinality of the continuum).

https://en.wikipedia.org/wiki/Cardinality_of_the_continuumIn set theory, the cardinality of the continuum is the cardinality or "size" of the set of real numbers R , sometimes called the continuum. It is an infinite cardinal number and is denoted by c (lowercase fraktur "c") or |R|.The real numbers R are more numerous than the natural numbers N. Moreover, R has the same number of elements as the power set of N. Symbolically, if the cardinality of N is denoted as _{0}, the cardinality of the continuum isThis was proven by Georg Cantor in his uncountability proof of 1874, part of his groundbreaking study of different infinities. The inequality was later stated more simply in his diagonal argument in 1891. Cantor defined cardinality in terms of bijective functions: two sets have the same cardinality if, and only if, there exists a bijective function between them.The smallest infinite cardinal number is _{0} (aleph-null). The second smallest is _{1} (aleph-one). The continuum hypothesis, which asserts that there are no sets whose cardinality is strictly between_{0} and c, means that c=_{1}.[3] The truth or falsity of this hypothesis is undecidable and cannot be proven within the widely used Zermelo–Fraenkel set theory with axiom of choice (ZFC).

Let's assume that the power set of the set of natural numbers is uncountable.

How do we determine the countability of less dense set of numbers, such as the power set of the set of prime numbers? or twin primes?

SideNote - i am a self proclaimed & self diagnosed entity who has a self proclaimed Myself unfit for the Subject.(OP)

Different Individuals with different backgrounds & identities...moving forward towards a common goal.

How Many Numbers Exist? Infinity Proof Moves Math Closer to an Answer.QuoteFor 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.https://www.quantamagazine.org/how-many-numbers-exist-infinity-proof-moves-math-closer-to-an-answer-20210715/What do you think about this continuum hypothesis?

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.

There are no infinities in nature.

If mathematicians claim infinity holds answers to calculable infinities,

That means philosophically that infinity itself is not an observable and there is no finite machine capable of counting to infinity.

In physics, we've been working meticulously to get rid of infinities from over a dozen crucial models, including those which we once thought existed inside of black holes or the beginning of the universe.

Quote from: BilboGrabbins on 12/10/2021 02:24:32There are no infinities in nature.There ae a infinite number of ways in which I can place a coffee table in a room.

No you can't. Unless of course the room is infinite? But then how'd you know it's a room with walls?

Quote from: BilboGrabbins on 12/10/2021 16:44:56No you can't. Unless of course the room is infinite? But then how'd you know it's a room with walls?Yes, I can.I can align the table North/ South.And there are an infinite number of angles through which I can then rotate it.So that's an infinite number of "ways in which I can place a coffee table in a room."