[hamdani yusuf (OP)]

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?

[hamdani yusuf (OP)]

Math Has a Fatal Flaw

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.

The video title sounds bombastic, perhaps deliberately to increase views.

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

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

[hamdani yusuf (OP)]

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

Here are some other rules commonly used for infinity, which are not included in previous post:
1/0 is not ∞ because the sign can't be defined.
∞.∞ = ∞
∞.-∞ = -∞
∞^a = ∞ (for real a>0)
∞^∞ = ∞

[Eternal Student]

Hi.

   I'm not sure what the question is.

How Many Numbers Exist?       Many, much as described in the article.  Unless you want to get philosophical and argue that numbers do not exist at all, they are just abstract.

What do you think about this continuum hypothesis?      It can be an axiom in set theory but it doesn't have to be.  It is independant of the ZF(C) axiom system (that's the Zermelo-Fraenkel axioms of set theory with or without the axiom of Choice).  These are the axiom systems most commonly used.   This just means that you can include the continuum hypothesis as an axiom if you want to, it doesn't invalidate the rest of the axioms.  It's also not required for most things, for example you can still construct the Real Numbers without it, it's just that you wouldn't know if there was a set with cardinality between    ℵ   and  ℜ.    This wouldn't matter as far as, say for example Applied Mathematicians are concerned, you could still construct tools like calculus with the axioms in ZFC and do everything else that you might want to do.

If you want a more general discussion:   The rules of arithmetic you are presenting in the later posts are for an object called the Extended Real Numbers.   This is just a partial extension of the field of Real numbers, in particular it isn't a field.   Not all of the field operations are defined for this ∞.   For example  0 x ∞   is undefined.
    From the point of view of the Pure mathematician's, this makes it fairly uninteresting and it does nothing more than act as shortcut for Applied Mathematicians rather than calculating limits more formally.   
    A more interesting object is the Hyperreal Numbers   which is a proper field extension of the Reals.

    There is a uniqueness theorem for the Real Numbers that holds in the ZFC axiom system.  It states that the Real Numbers are the the only complete, ordered and Archimdean field.  This means that a proper field extension has to lose at least one of those properties.   The Hyperreal Numbers, for example, are totally ordered but they are not Archimedean.  It doesn't matter too much, they are loads more fun anyway.

Best Wishes.

[hamdani yusuf (OP)]

Here are some videos discussing interesting things about infinity.

Infinity Paradoxes - Numberphile

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.

The following videos are from Zach Star.

Things get weird at infinity

What happens at infinity? - The Cantor set

[Eternal Student]

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.

[Zer0]

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

If i was " Eternal "...
Then i would simply just keep counting until Eternity.

To escape immortal boredom, i might count randomly..
Say 1,2,3,7,8,9,6,5,4...on n on so that none of the numbers were ever repeated.

Noone has ever reached the end point of Infinity, Right?
Just like giggles & laughter.
🤭

Ps - Say if i had the last laugh or the last word, n die abruptly, then did i really have the last laugh or the last word?
Pretty much seems like it, Who quarrels or debates with the dead anyway.
♾️

[Eternal Student]

Hi Zer0,

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

  Yes, sometimes.   Overall this would be a better way of thinking about it.  Well done.

Best Wishes.

[hamdani yusuf (OP)]

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.

[hamdani yusuf (OP)]

IMO, the continuum hypothesis conundrum occurs when we accept the validity of Cantor's method to define cardinality of number set. But many arithmetic rules that work on real numbers don't work on infinity.
Moreover,
∞^∞ = ∞
And we can stack those infinity power as many times as we want, and the answer is the same.
So, if we reject Cantor's method, the conundrum would disappear. 

[Eternal Student]

Hi.

I hope you are well and I'm delighted that someone has posted something about Mathematics.

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.

  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.

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

   Cantor did lots of things.  What are you calling "the Cantor method"?
Why is the Continuum hypothesis a conundrum?
....either the continuum hypothesis or its negation can be added as an axiom to ZFC set theory, with the resulting theory being consistent if and only if ZFC is consistent....      [From Wikipedia]
    The continuum hypothesis is independent of the ZFC axiom system, you can accept or reject it or leave it indeterminable.  It's interesting to Pure Mathematicians but it makes no difference to the majority of Mathematics that is in current use.

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. 

   The continuum hypothesis is the assumption that there isn't a set with cardinality between  the set of Naturals (ℵ) and the set of the Real Numbers (ℜ).
    If you don't have a definition of "cardinality" or decide to use a different one from every one else, then yes, the hypothesis changes it's meaning and/or becomes meaningless.
    For example, we could define the cardinality of every infinite set to be  C    (whatever you want C to be, perhaps it's a symbol like ∞),  then obviously every infinite set has cardinality C and so there isn't an infinite set with cardinality between  ℵ and  ℜ.
    However, it should still make sense to talk about bijections and then the continuum hypothesis can be rephrased to restore it's original meaning:   Is there a set   S   such that  ℵ    but no bijections can be found from  ℵ →S   and  S→ℜ.

    We consider bijections to be important because, IMO, that is how human beings compare things.  We tend to put things into 1-to-1 correspondence with other things.   This is how we can count a number of cows in a field, we find a correspondence between the first n counting numbers and the cows.  We point to each cow but only once and don't stop until all cows have been pointed at.  We're careful to start at the number 1 and increment by 1 each time,  so we are establishing a bijection from  {Cows}  → {first n counting numbers}.   We say that there are n cows but this is just shorthand for saying  "I have found a bijection between the set of cows and the first n counting numbers".  Bijections are the most natural way of comparing the size of two sets.   We can compare the number of cows to a number of sheep in a different field.... we go and count the sheep by the same process.... establish a bijection from {Sheep} to the {first m counting numbers}.       Now if   m = n   then we can combine the two bijections
  {Cows} → {first n counting numbers}  =>   {first m counting numbers} → {Sheep}
  The bijection labeled => from {first n counting numbers} to {first m counting numbers} exists provided m=n,  the identity mapping will do,  i.e. the two sets are identical anyway.   We obtain an overall bijection from  {Cows} → {Sheep}.  We say there are the same number of cows and sheep but this is just shorthand for saying "there is a bijection between the set of cows and the set of sheep".   If m≠n then no bijection between the {cows} and {sheep} can be found.
     Anyway.... spend a moment and think about what you've actually always been doing when you count things, you've always been finding bijections.    The definition of "cardinality" where two sets have the same cardinality  if and only if   there exists a bijection between them   wasn't arbitrarily decided upon.  It is one of the most natural ways to compare the size of two sets.

Best Wishes.

[hamdani yusuf (OP)]

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.

I've added the titles of each videos to help referring which is which in discussion.

[hamdani yusuf (OP)]

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

You can read in the opening post.

Why is the Continuum hypothesis a conundrum?

Please read again the opening post.

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

[Eternal Student]

Hi again.

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

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

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

   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?"
   If you were just after an answer to that  --->  "No it can't be proved in the ZFC axiom system but neither can it be falsified in the ZFC system".

   It's also a PopSci article,  they say it's "a deep epistemological conundrum" - but they are not likely to say anyting else.

Going back to your earlier statement:

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

     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.

Best Wishes.

[hamdani yusuf (OP)]

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

You guess it right. In the article, it's shown in picture format.

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?"

From the article,

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

[Eternal Student]

Hi again.

   Like the many short articles for the general public, they miss out several fine details.
Here's one example:
   Suppose you had an enumeration of the real numbers that looked like this:

1  →  0.099999.......     (recurring digit 9)
2  →  0.19378652...
3  →  0.4793656.....
4  →  0.39592478...
............
The first number has recurring  digit 9,   it is 0.0999999...
The next numbers in the enumeration need to have the digit 9 in the n^th  decimal place  BUT all other digits are arbitrary. 
   Now build your test number as directed in the article (add 1 to the digit and note that 9 gets taken to 0), I've made the appropriate diagonal digits glow,
  Our test number = 0.1000000....  =  0.1
    However,  you might have seen the mathematical result that shows 

So that our test number, 0.1  is precisely equivalent to  0.0999999999.....  which was the first number in the enumeration.  So in this enumeration, the construction has failed to produce a number that wasn't on the list.  We would be unable to conclude that this enumeration wasn't a total enumeration of the real numbers.

   The requirement that there is a 9 in the (n)^th decimal place isn't as restrictive as you might think.  We could still argue that all real numbers between 0 and 1 might lie in that enumeration.   Consider, for example, the number   0.1239321.   It might seem that it has only has one 9 digit in the 4th decimal place and so it would have to be the number 4th on our list but this isn't true.   It's the same number as  0.123932099999999999999999....     (replace the last digit 1 with infinitely recurring 9) - so that it could be anywhere on the list from the 8th entry on the list through to the ∞ entry on the list.

    There's several places where the article has rushed or glossed over some complications that arise but it does what it set out to do well enough.  It's an interesting article that appeals to the general public, keeps them on the edge of their seat and more importantly makes the reader feel that they have understood something.  We all tend to learn more and feel more motivated by successfully understanding something.  It should help to get people interested in Mathematics, so thanks for presenting it here.

Best Wishes.

{Late editing:  Fixed some mistakes over n^th  and (n+1) ^th  decimal place.}

[hamdani yusuf (OP)]

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.

It's mind boggling that many brilliant mathematicians over a century failed to see this, while a few random forum members can solve it in just a few posts. Perhaps we can collaborate to write a proper research paper on this.

[hamdani yusuf (OP)]

Math Has a Fatal Flaw

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

Instead of blaming math, we should do introspection to check on the validity of our method/derivation involving infinity or zero. When used incorrectly, we would be able to "proof" that 1=2, like an example below.

How to "Prove" That 2 = 1
Let'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 = 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

Since a = b (that's the assumption we started with), we can substitute b in for a to get: b + b = b

Combining the two terms on the left gives us: 2b = b

Since b appears on both sides, we can divide through by b to get: 2 = 1

Wait, 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

[Eternal Student]

Hi.

Instead of blaming math,....

     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.

Math Has a Fatal Flaw

    It's incomplete where the phrase is used in the sense of formal Mathematical logic and axiom systems.   It's interesting but it's not fatal.
    I don't think Mathematics has enough general interest so words like "fatal flaw" and "free food" are sometimes used when promoting lectures to the general public.

Best Wishes.

[hamdani yusuf (OP)]

     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.

It's from the title of Veritasium's video, which I quoted. You can click on the hyperlink to go to my post containing the video.