Anonymous at Wed, 22 May 2024 18:36:39 UTC No. 16188269

>>16188206
good job, you did it

🗑️ Anonymous at Wed, 22 May 2024 18:48:18 UTC No. 16188289

Cantor’s diagonal argument is extremely flawed.
Anyone who is unable to spot the problem with such absurd statement is not fit to be a mathematician but a parrot, only able to memorize and repeat.

Anonymous at Wed, 22 May 2024 18:52:28 UTC No. 16188297

>>16188206
>1) A real number can be represented by a finite string of characters that uniquely defines that number. For example, there are numerous formulas for pi.
The definable numbers are a countable strict subset of the reals. You are retarded.

Anonymous at Wed, 22 May 2024 19:02:39 UTC No. 16188315

Your heresy is:
Predicativism
https://plato.stanford.edu/entries/philosophy-mathematics/#Pre

Anonymous at Wed, 22 May 2024 19:03:45 UTC No. 16188319

Cantor’s diagonal argument is extremely flawed.
Anyone who is unable to spot the problem with such absurd statement is not fit to be a mathematician but a parrot, only able to memorize and repeat.

Destroyer at Wed, 22 May 2024 19:08:47 UTC No. 16188329

>>16188297
What makes you so sure that there exists a real number that can’t be defined?

Anonymous at Wed, 22 May 2024 19:16:15 UTC No. 16188338

>Cantor le bad and le wrong because he was an old white man
>let's arbitrarily redefine real numbers to match our DEI ideology
>let's ignore logic and proofs
So heckin woke, OP. Thanks for making math more feminist.

Destroyer at Wed, 22 May 2024 19:21:21 UTC No. 16188352

>>16188338
Your magical undefinable numbers have nothing to do with reality. Like gender identity, it only exists in your mind.

Anonymous at Wed, 22 May 2024 19:23:36 UTC No. 16188354

>>16188352
I identify as a alirn goatee

Anonymous at Wed, 22 May 2024 19:41:20 UTC No. 16188377

>>16188352
Uncomputable numbers can be instantiated physically almost trivially.

>measure spin of a particle first along the z axis and then measure spin of the same particle along the y axis
>repeat this procedure with the same particle infinitely many times
>the resulting sequence of zeros and ones defines the decimal places of an uncomputable number almost certainly
>to do this in finite time, send an observer through the event horizon of a black hole and make use of time dilation effects

Destroyer at Wed, 22 May 2024 19:44:54 UTC No. 16188382

>>16188377
Even assuming that you could take this process to infinity, you still haven’t proven that such a number could not coincidentally align with a number that has a finite definition. Math is supposed to be rigorous, remember?

Anonymous at Wed, 22 May 2024 19:46:30 UTC No. 16188383

>>16188382
Which part of "almost certainly" did you fail to understand? The computable numbers are a set of measure zero in the reals.

Destroyer at Wed, 22 May 2024 19:49:57 UTC No. 16188386

>>16188383
Still no proof. By the way, how could you “instantiate physically” a number that requires an infinite definition? Where are you writing all the digits?

Anonymous at Wed, 22 May 2024 19:55:55 UTC No. 16188400

>>16188386
>Where are you writing all the digits?
Yeah, where do you write them? Where do you write the digits of an undefinable number, huh? You're so close to understanding, but still tripping over your own legs. Looks like your brain isn't ready yet.

Anonymous at Wed, 22 May 2024 19:56:44 UTC No. 16188401

>>16188377
Two obvious objections:
1. Pilot-wave theory could be true, making that "random" process completely deterministic.
2. That's not how black holes work. Infalling observers do not get to see arbitrarily far future of the external universe.

Destroyer at Wed, 22 May 2024 19:59:30 UTC No. 16188406

>>16188400
“undefinable” simply means that the definition is infinite and cannot be finite. The definition in this case would be the number itself, with every digit being written out. You might wanna sit this one out little boy

Anonymous at Wed, 22 May 2024 19:59:49 UTC No. 16188407

>>16188401
>1. Pilot-wave theory could be true, making that "random" process completely deterministic.
Not a valid objection. Computability and determinism are different categories. A deterministic process doesn't need to be computable and can produce an uncomputable result.
>2. That's not how black holes work. Infalling observers do not get to see arbitrarily far future of the external universe.
Depends on the spacetime metric. There are models for hyperturing computation using black holes.

Anonymous at Wed, 22 May 2024 20:01:42 UTC No. 16188411

>>16188406
It's amazing how you almost correctly explain these things to yourself but still don't understand. Almost LLM-like.

Anonymous at Wed, 22 May 2024 20:03:08 UTC No. 16188415

>>16188206
1. No guarantee that all real numbers can be represented unambiguously in a single numbering format.
This is the real cause of cantor's proof. Decimal strings are insufficient to describe every real.
In fact, for any radix < infinity, cantor's proof will hold. Bounded strings are insufficient to describe every real.

Anonymous at Wed, 22 May 2024 20:03:55 UTC No. 16188418

>>16188329
OP said it can be done using a finite sequence of characters. That is a gross mistake because is not hard to find a number whose representation, even a symbolic one, requires one extra character and so on, and so on.

Destroyer at Wed, 22 May 2024 20:05:04 UTC No. 16188420

>>16188411
I’m simply pointing out to you that cannot represent undefinable numbers in any way, so if they “exist” in any meaningful sense, then it would require infinite space/time, which may not be the case for our universe. And all of this is still assuming that you’ve established that a number can in fact be undefinable, that is, that finite definitions do not encompass all possible decimal strings.

Anonymous at Wed, 22 May 2024 20:05:12 UTC No. 16188422

>>16188407
>A deterministic process doesn't need to be computable and can produce an uncomputable result.
Yes, but your entire argument hinges on the number being almost certainly uncomputable because it is random.
>Depends on the spacetime metric. There are models for hyperturing computation using black holes.
Write down whatever metric you want, and you can get metrics with closed timelike curves and all sorts of other bullshit. Show me a model for hyperturing computation with a physically realistic metric.

Anonymous at Wed, 22 May 2024 20:07:38 UTC No. 16188428

>>16188415
>1. No guarantee that all real numbers can be represented unambiguously in a single numbering format.
0.999...

Anonymous at Wed, 22 May 2024 20:09:58 UTC No. 16188432

>>16188420
>I’m simply pointing out to you that cannot represent undefinable numbers in any way
Wow, what an insight, you heckin genius. An undefinable number cannot be given a finite definition? How did you figure that out?

Lmao, you're retarded.

Anonymous at Wed, 22 May 2024 20:13:11 UTC No. 16188436

>>16188428
0.999... is an uncomputable number. No Turing machine can print all its digits in finite time.

Anonymous at Wed, 22 May 2024 20:15:57 UTC No. 16188441

>>16188436
What about 1.000...?

Destroyer at Wed, 22 May 2024 20:16:39 UTC No. 16188442

>>16188432
The insight is that such numbers are useless and maybe even non-existent in *this* universe. But even if you had an infinite universe, you would need to be able to apprehend the whole universe at once to truly comprehend the number’s definition. And THEN you would still have to prove that the number is NOT definable, which is much harder said than done, as proven by your unwillingness to even attempt that it’s even possible for a number to be undefinable. You can’t just say that the Collatz conjecture is “almost certainly true” for most numbers and call it a proof

Anonymous at Wed, 22 May 2024 20:16:54 UTC No. 16188443

>>16188422
>Yes, but your entire argument hinges on the number being almost certainly uncomputable because it is random.
The randomness of a spin measurement outcome isn't challenged by Bohmian mechanics. Whether at its core it is deterministic pseudorandomness doesn't matter, for the statistical distribution of measurement outcomes passes all tests and matches all definitions for randomness.

>Write down whatever metric you want, and you can get metrics with closed timelike curves and all sorts of other bullshit. Show me a model for hyperturing computation with a physically realistic metric.
Who is the judge on what's a "physically realistic metric"? How many black holes have you observed so far?

Anonymous at Wed, 22 May 2024 20:21:29 UTC No. 16188453

>>16188436
>>16188441
If only we had hyperturing computation that could compute both of these numbers to their end, then we could add up the place values of their digits and check once and for all whether they are equal.

Anonymous at Wed, 22 May 2024 20:23:28 UTC No. 16188456

>>16188442
1. The real numbers are uncountable.
2. The definable numbers are countable.
Both of these statements are trivial to anyone who attended a first year math class. If you do not understand them I will not explain them to you.

Anonymous at Wed, 22 May 2024 20:24:23 UTC No. 16188458

>>16188415
Consider an infinite base, where each place holder, α, can range from 0 - 999...
Cantor's diagonal argument no longer works as any shifting of any range of numbers is covered under 0 - ααα...

Destroyer at Wed, 22 May 2024 20:25:39 UTC No. 16188460

>>16188456
>the real numbers are uncountable
This assumes that a number cannot have a finite definition. Until you prove this, you’re just religiously following what you’ve been taught.

Anonymous at Wed, 22 May 2024 22:55:34 UTC No. 16188654

>>16188206
just swap out real number with algorithm and you got it

Anonymous at Wed, 22 May 2024 22:56:35 UTC No. 16188656

>>16188456
learn to count dumbass

Anonymous at Thu, 23 May 2024 02:02:28 UTC No. 16188840

>>16188206
Congratulation, you gave a single example of the well-ordering theorem.

Anonymous at Thu, 23 May 2024 02:16:56 UTC No. 16188856

>>16188319
Cantor's diagonalization argument is a natural consequence of the infinite geometric series with parameter (1/2).

Can you actually elaborate the problem with it? Do you deny that there are numbers between 0 and 1 which take an infinite number of (1/2)^k terms summed together for different indices of k to produce? If so, can you explain what the problem is?

Anonymous at Thu, 23 May 2024 02:53:28 UTC No. 16188892

>>16188206
forget the reals, convince me the power set of the naturals is countable. otherwise, the existence of undefinable things is conceded

Anonymous at Thu, 23 May 2024 03:05:34 UTC No. 16188904

>>16188892
I posted >>16188458
I think this shows that it isn't countable because it maps to uncountables, cantor's reals. Not sure how someone would prove it. I think the primary dilemma is showing that alpha doesn't in fact contain any uncountable numbers, and is not just merely inaccessible from cantor's approach to the problem.

Anonymous at Thu, 23 May 2024 03:31:45 UTC No. 16188913

>>16188892
First prove that there exists an infinite subset of the power set that is undefinable. That is, the set cannot be represented by a finite definition. For any given set, it may be the case that there is some mind-boggling complex formula for generating each number in the set. I don’t think this is something that we can properly discuss. This is an important math problem and no one has ever even mentioned it. They focus on the Continuum Hypothesis but never the Undefinability Hypothesis. How DO you prove that a number is definable or undefinable, given the digits? Well that’s the problem: since the digits are infinite, we could never prove such a thing because we wouldn’t even be able to identify the number in the first place. And even if we did, how on earth would we show that it’s not definable? You get the sense that this is far too out of reach and impractical, but it is assumed for the sake of declaring the reals and the power set of the naturals to be uncountable.

Anonymous at Thu, 23 May 2024 03:55:46 UTC No. 16188940

>>16188206
Step (1) is wrong. You've essentially given the definition of a definable number, which is a countable set, so of course the diagonalization argument is not going to apply.
>>16188460
That a real number must almost certainly be undefinable follows from cantor's diagonalization argument, so you're effectively asking for a proof of the uncountability of the reals without the diagonalization argument. I tried searching around a bit, and there are alternative proofs, but it's not clear if those proofs are diagonalization arguments in disguise, nor is it clear that you could ever claim a proof is not another proof in disguise.
I don't think you can just reject the diagonalization argument without breaking a ton of other things, but you're welcome to rework set theory without it and see what happens. I doubt you'll get very far. I also doubt you won't end up using a diagonalization argument in disguise, making you a hypocrite.

Anonymous at Thu, 23 May 2024 04:01:43 UTC No. 16188946

>>16188940
Nothing in the real number axioms imply undefinable real numbers. There is no proof that there exists an undefinable real number. Cantor’s proof uses a method of construction that is not rigorously defined. Computers can’t verify the proof because it’s not even using axioms, it’s just imagination taken to infinity. The proof actually ASSUMES that the reals are uncountable because the proof relies on the diagonal number construction being valid. But what if we assumed that the list actually contains all real numbers? Then that would mean that the diagonal number itself is the problem, and not actually a real number, since it would implicitly be defining itself to be different itself (since all numbers are already in the list). So the diagonal number is not a number at all, undefinable numbers haven’t been proven to exist, and we can wake up from fantasyland and continue doing mathematics that can actually be proven and tested in the real world.

Anonymous at Thu, 23 May 2024 04:08:18 UTC No. 16188952

>>16188656
Learn to count like a man, woman. Math follows rigorous definitions and not shallow gossip like you.

Anonymous at Thu, 23 May 2024 04:22:21 UTC No. 16188961

>>16188206
finitism trolling should be punishable with pegging till you cum

Anonymous at Thu, 23 May 2024 04:36:51 UTC No. 16188966

Where do the retards in these threads come from? To not know what "countable" or "real number" means is one thing, but to appear to understand definitions and then disagree with logically proven conclusions simple enough for any first-semester analysis student to understand is another. I have never seen a person like this in real life, only on the internet.

Anonymous at Thu, 23 May 2024 04:43:48 UTC No. 16188973

>>16188913
>How DO you prove that a number is definable or undefinable, given the digits?
ah, a halting problem type deal, funny thing is that the mechanism for turing's proof of it relies on the same mechanism as cantor's diagonalization, funny stuff eh?

Anonymous at Thu, 23 May 2024 04:46:11 UTC No. 16188975

>>16188946
>Computers can’t verify the proof
why would they? the halting problem relies on the same mechanism as cantor's proof
https://www.youtube.com/watch?v=dwNxVpbEVcc

Anonymous at Thu, 23 May 2024 04:48:53 UTC No. 16188976

>>16188966
they are called finitist's and they are the mathematical equivalent of a flat earther, that is therefore the reason you only seeing them on the internet

Anonymous at Thu, 23 May 2024 04:59:22 UTC No. 16188981

>>16188966
It seems to be a socially inept nerd version of trolling. There are some topics like politics or ethics where trolling is naturally easy and effective by arguing an emotional or controversial opinion. But instead people like OP can only troll by "pretending to be retarded". Denying obvious logical facts while addressing an audience of mathematically educated people is literally the dumbest and most ineffective method of trolling. The only answers OP gets are people calling him stupid. What a sad waste of time.

Anonymous at Thu, 23 May 2024 11:02:08 UTC No. 16189316

>>16188966
>I have never seen a person like this in real life, only on the internet.
They are bots. Finite state bots. They cannot comprehend the infinite.

Anonymous at Thu, 23 May 2024 11:35:13 UTC No. 16189351

>>16188297
>undefinable numbers
new addition to religious dogma of the real number cult just dropped

Anonymous at Fri, 24 May 2024 00:16:52 UTC No. 16190430

>>16188329
diagonalization nigga

Anonymous at Fri, 24 May 2024 17:50:28 UTC No. 16191346

>>16188946
>Computers can’t verify the proof because it’s not even using axioms
https://us.metamath.org/mpeuni/ruc.html

Anonymous at Fri, 24 May 2024 18:42:14 UTC No. 16191416

>>16189351
Do you have a problem with the halting problem as well or is it the reals primarily which trip you up? I.e., do you think that all sequences of binary numbers must either terminate (meaning ending in all zeros) or repeat after some finite number of bits?

DoctorGreen !DRgReeNusk at Sat, 25 May 2024 00:47:38 UTC No. 16191895

>implying the concept of a number is logically true

Anonymous at Sat, 25 May 2024 01:17:27 UTC No. 16191918

>>16191346
>https://us.metamath.org/mpeuni/ruc.html
Can we do it just from the real number axioms, without pulling in all of set theory?

Anonymous at Sat, 25 May 2024 01:29:55 UTC No. 16191932

>there exists a collection of non-empty sets whose Cartesian product is empty
Do you really believe that?

Anonymous at Sat, 25 May 2024 01:42:27 UTC No. 16191941

>>16191918
you can do it in RCA0 with intervals, see Subsystems of Second Order Arithmetic

Anonymous at Sat, 25 May 2024 01:47:58 UTC No. 16191943

>>16189351
>Undefinable numbers
When can we admit that mathfags are schizos

Anonymous at Sat, 25 May 2024 08:54:06 UTC No. 16192375

>>16188961
>Aggressive homosexual fantasies emerge from those who reject Finitism
Unsurprising