1

New Theories / Re: How Many Numbers Exist?

« on: 14/10/2021 19:08:11 »

@BilboGrabbins

Hi there!
🙋

I have a Query, Especially for You.
Interested?

A_______B

A simple line.
Start point A.
End point B.

I suppose i can keep dividing it into Halves.

But considering Planck's Length, Can i Divide it until Infinity?

If i Know the Line's point of Origin(A) & am Aware of the Line's point of Cessation(B)...Does that sound like Infinity?

Or if i didn't know the Origin, n was clueless bout the End.
Infinity!

Ps - Thanks E_S for the suggestions.
I've Realized my capacity to understand & learn is Not infinite.
Hence i do not bother myself, & also spare Others of goin thru the troubles of explaining me things which i Firmly Believe i shall never understand.
(Reason i Request for short & brief answers, i Hate it when Teachers waste their Precious Time on Futile Things)
: )

No, to divide through the Planck length infinitely many times would be equivelent to a singularity. That is itself equivelent to a breakdown in physics.

2

New Theories / Re: How Many Numbers Exist?

« on: 13/10/2021 22:27:22 »

Hi again.

I'm aware of this thought experiment... And no. You cannot divide infinitely. There is a physical stop sign in physics .

What physical stop sign in physics?   Are you talking about an indivisible planck length?  This was mentioned in reply #53.  Otherwise lengths such as those in the tortoise and hare experiment or angles such as in Bored Chemist's example can be divided more times than any finite number of times (which is fair description of infintely many times) and it can often all be done within a finite amount of time.

Best Wishes.

Yes the Planck length. This is the ground rule that prevents  continuous divisibility.

3

New Theories / Re: How Many Numbers Exist?

« on: 13/10/2021 18:40:57 »

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.

4

New Theories / Re: How Many Numbers Exist?

« on: 12/10/2021 16:44:56 »

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.

5

New Theories / Re: How Many Numbers Exist?

« on: 12/10/2021 02:24:32 »

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.