[Eternal Student]

Hi.
   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 also don't think it would be helpful if I just repeat some of what I've said before.
   Perhaps it would just be better to explain what it is you were or are hoping to do now or where you think there is a problem now.

Best Wishes.

[Bored chemist]

There are lots of puzzles like this one

Assume that we have two variables a and b, and that: a = b

Multiply both sides by a to get: a2 = ab

Subtract b2 from both sides to get: a2 - b2 = ab - b2

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 = b

As you say,
 what you call "cancelling" is actually dividing each side by a-b
However, you start by saying that a = b
so a-b is zero.
so, that "cancelling" step is actually division by zero; it is forbidden.

More importantly, there are only 60 numbers- you can verify this by looking at a railway timetable.

[hamdani yusuf (OP)]

   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.

[Eternal Student]

Hi.

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.

   This sounds right.
The article isn't completely thorough about proving anything but the results they were talking about are widely recognised and there doesn't seem to be any mathematical fallacy in the more formal proofs of those results.

Best Wishes.

[hamdani yusuf (OP)]

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 uncountable. 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.

[Eternal Student]

Hi.

Let's check if countability has a binary value.

   Not a phrase that is commonly used, so I can only guess what was wanted by reading the rest of the post.

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

   OK.  This is correct anyway.

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

   Dilution is another phrase that isn't commonly used.  I'm going to assume that "dilution" means finding a subset.

Let's start with set of rational numbers.

   This is countable.   This proof isn't too difficult.  There are many methods to available for this.   One easy approach is to note that any rational can be written as  a/b   where a and b are co-prime, this just means write the fraction in it's simplest form.     Then   map  a/b  to   the integer  2^a.3^b.   By the fundamental theorem of arithmetic, this identifies a unique integer to each rational.
   There are some alternative proofs to be found here:  https://proofwiki.org/wiki/Rational_Numbers_are_Countably_Infinite

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.

    Warning:  The phrase "dense" isn't used in the way you're trying to use it here.   
A set, S, is said to be "dense" in the Reals     if and only if   
For any real number r, we can always find a sequence of terms in S that converge to r             <=>
∀r∈ℜ and ∀ε>0  ,  ∃s∈S   such that  | s - r | < ε
    So, for example, the Rationals are countable but they are dense in the Reals.

   However it seems that you are asking a slightly different question:   Can we find a subset  ℜ    such that every proper subset of S is countable but  S is uncountable.
    The answer is no, you can't. 
Proof:   Assume (for contradiction) that we can find such a set S.
    Then S is not empty because S is uncountable.
    So there exists an element s∈S   and  then consider   S \ {s}  which is a proper subset of S.
    Then  S \ {s}  is  countable.     Then    S = S \ {s}    U   {s}    is a countable union of countable sets.
    So S is countable.     →←     (Contradiction obtained).
- - - - - - - - -

    So there isn't any subset, S,  of the Reals which would act as a "threshold", with all subsets of S being countable while all supersets of S are uncountable.

Best Wishes.

[hamdani yusuf (OP)]

https://en.wikipedia.org/wiki/Power_set#Properties

Cantor'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_continuum
In 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 is


This 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 between₀ 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?

[Eternal Student]

Hi.

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

   Yes, good call.
   

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?

     The set of primes is an infinite set.   See Wikipedia for a proof and discussion:   https://en.wikipedia.org/wiki/Euclid%27s_theorem
     The set of all primes, P  is obviously contained in the Natural numbers.   So the identity mapping  I:  P →ℵ  is an injective mapping.   So we have  |P|  ≤  |ℵ|   (where |S| denotes the cardinality of the set S.     But P is inifnite so |P|  ≥  |ℵ|.    Hence,  |P| = |ℵ| = ℵ₀.    This assumes results about the ordering of infinite cardinal numbers, which you can find discussed elsewhere    (for example, https://en.wikipedia.org/wiki/Cardinal_number).    As I recall we would require the axiom of choice to show that ℵ₀ is the smallest infinite cardinal.  If you're interested, we can check to see if this can be established without the axiom of choice and just in the ZF axiom system.
      An alternative approach would be to use the fact that the primes are ordered,   the order relation <  defined on the Reals will be an order relation on the primes.   So we should be able to order the primes from smallest to largest and then find a bijection  from  P → ℵ.   (I haven't checked the results about well-ordered sets and I suspect we will still need the axiom of choice - but that's where I'd be looking first, if indeed you're worried about using the axiom of choice at all).
    Anyway, the Power set of the primes must have a greater cardinality.  If you wish to accept the continuum hypothesis, this would be the same as the cardinality of ℜ which, in turn, is the same as cardinality of the power set of Naturals.   If you don't accept the continuum hypothesis then the Power set of the primes has a cardinality that can be written as 2^|P|.   I think we can still show that  2^|P| = 2^N₀  if that's useful. It's just that, without the continuum hypothesis, we would know only that this is greater than ℵ₀  but not necessarily equal to the cardinality of the continuum.

    The twin primes are harder to work with, since we don't know if this is an infinite set.  If it's a finite set, then finding the it's power set and the cardinality of it's power set is easy.  If it's an infinite set then it has cardinality equal to that of Natural numbers (and for the same reasons as that shown above) and it's power set follows the same rules.   I can tell you with certainty that the set of all twin primes is countable (it's either finite or countably infinite).  However, we can't determine the cardinality of the Power set without knowing if the twin prime conjecture is true, i.e. if the set of all twin primes is infinite.
    Minor note:  Just because we can't (or I can't) identify the cardinality of the Power set of the twin primes, does not mean that it doesn't exist or that it will never be determined.  It's just that I, personally, can't identify it.
     If indeed the twin prime conjecture was ever proven, then the cardinality of the power set follows easily.   Conversely, if the cardinality of the power set was known then the truth or falsehood of the twin prime conjecture follows easily.   So, if the twin prime conjecture was shown to be undecideable  (neither it's truth or falsehood can be established from the ZFC axioms) then the cardianlity of the power set of the set of twin primes remains undecideable.

Best Wishes.

[Zer0]

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

But i still follow this OP.
Coz i Really Like what you both are doing in here.
👍

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

Not everybody can Collaborate soo Harmoniously.
Good Job!
Please keep up the Good Work.
🙏

Ps - There are Instances of " Aim & Hit & Miss "...But Real Failure is when One Stops Trying.
🖖

[Eternal Student]

Hi @Zer0 ,

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

  Self-diagnosed?  I don't know where you live but if it becomes useful or relevant you could get a formal test and diagnosis.  It could be that something simple can be done.   Maybe all you need are some coloured glasses and some classical music playing in the background,  or less of something you're normally exposed to (like caffeine or some other substance).

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

   Hopefully, that's what should be happening in most forum threads.

   Anyway, I wish you well, Zer0.   Bye for now.

[BilboGrabbins]

How Many Numbers Exist? Infinity Proof Moves Math Closer to an Answer.

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 Infinities
Yes, 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?

Oh dear, the mathematicians in love with infinity pretending to be physicists. I'll put a damper on this quickly.

Infinity is a concept. If mathematicians claim infinity holds answers to calculable infinities, this had two massive errors. Not only is it a concept, it's not a number, its an array of numbers which are fundamentally uncountable. There are no infinities in nature.

Even if we made a machine that could count infinitely onwards, any observer going to the machine, at any point will never be infinite, no matter what number it will read on its screen. That means philosophically that infinity itself is not an observable and there is no finite machine capable of counting to infinity.

Infinity is a much abused concept and often while most physicists will agree with what I said, there's still a handful of physicists who hold onto it like it was the ring of Mordor.

[BilboGrabbins]

We can work well in calculus as well without them and physics hates them to add. In calculus we just replace infinities with ultra large boundary conditions which are still finite. 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.

[Bored chemist]

There are no infinities in nature.

There ae a infinite number of ways in which I can place a coffee table in a room.

[Eternal Student]

Hi.

If mathematicians claim infinity holds answers to calculable infinities,

   These are words put together in a way that sounds poetical.  I don't know what it means but it sounds cool.

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

    The last bit seems reasonable, there's no finite machine capable of counting to infinity.  However, this means only that we can't count to infinity.    Can you still observe an infinite set anyway?   Well it probably depends on whether space and time is continuous or discrete.  If, for example, there was something like a fundamental Planck length and the location of a particle can be only one of a finite set of choices then you might be right.   Otherwise the sort of thing mentioned by @Bored chemist  seems to apply.

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.

    You might be worrying yourself needlessly.  I'm not sure if the OP (Hamdani) was really concerned about using infinity in some piece of Physics,  you'd have to ask him.   There isn't a Mathematics section in this forum.  It seemed that the OP was asking something about Mathematics without any reference to some application in Physics.

Best Wishes.

[BilboGrabbins]

There 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?

And even if you could state that correctly, that would be a concept. You wouldn't be able to arrange a table an infinite amount of times. Such a concept is nonphysical. In fact infinities themselves are best defined in physics as unphysical concepts.

[Eternal Student]

Hi @BilboGrabbins ,

  Consider a race between the tortoise and the hare.
The hare travels at twice the speed of the tortoise.  However, the hare allows the tortoise to get half-way along the race track before he starts.
  They should both cross the finish line at the same time.   Check this yourself.

Anyway,  it is possible to consider how many times the distance between the hare and the tortoise has been halved.
They started  with a distance   D/2     between them  when the hare began to run.    (Where D = distance of the whole track).
When the hare was half way up the track....  the tortoise was only a quarter of the way from the finish.... the distance between them was  then  D/4
.....  there was a time when they were D/8  apart.....
.... D/16  apart....     then D/32.......  etc......
... we know that at the finish line they were 0 distance apart  (they crossed the finish line together).
So the distance between them was halved "infinitely many" times.... and this all happened within a finite amount of time.    The hare reached the finish line in an amount of time =    D / (Speed of hare).

To phrase this another way:   It seems perfectly realistic that you could have watched this race.  It would only have taken a finite amount of time but during the race you would have seen the distance between the hare and tortoise being halved "infinitely many" times.

Best Wishes.

[BilboGrabbins]

Eternal Student,

I'm aware of this thought experiment... And no. You cannot divide infinitely. There is a physical stop sign in physics . Again, infinities are not physical things.

[Bored chemist]

No 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."

[BilboGrabbins]

No 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."

Lol. Are you being serious? A room has a finite boundary condition. You might get a hell of a lot of orientations but certainly not infinite. I don't like people misleading others into such baloney!

[BilboGrabbins]

Nevertheless, infinity remains a good concept even if not applicable in principle. A good example are Penrose tiles. In principle, if not in concept, there are an infinite amount of non repeating patterns. But to physically show this is impossible.