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.

2) It follows that the real numbers can be arranged in order of their smallest possible definitions, from smallest to largest

3) If the real numbers can be arranged in the order of the lengths of their definitions, then Cantor’s diagonal proof is a faulty construction, since the diagonal number’s definition MUST be in the list and both out of the list at the same time.

4) Therefore, Cantor’s proof merely proves that one cannot define a number to be different than all other real numbers, and that this is the real cause of the contradiction.

Q.E.D.

good job, you did it

>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 moronic.

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

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.

diagonalization homie

>undefinable numbers

new addition to religious dogma of the real number cult just dropped

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?

>Undefinable numbers

When can we admit that mathgays are schizos

Your heresy is:

Predicativism

https://plato.stanford.edu/entries/philosophy-mathematics/#Pre

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.

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?

>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.

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

I identify as a alirn goatee

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

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?

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

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?

>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.

“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

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

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.

>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 moronic.

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

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.

>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.

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.

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.

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.

>Computers can’t verify the proof

why would they? the halting problem relies on the same mechanism as cantor's proof

>Computers can’t verify the proof because it’s not even using axioms

https://us.metamath.org/mpeuni/ruc.html

>https://us.metamath.org/mpeuni/ruc.html

Can we do it just from the real number axioms, without pulling in all of set theory?

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

learn to count dumbass

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

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.

>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.

>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.

>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?

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.

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

0.999...

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

What about 1.000...?

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.

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...

just swap out real number with algorithm and you got it

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

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

I posted

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.

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.

>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?

finitism trolling should be punishable with pegging till you cum

>Aggressive homosexual fantasies emerge from those who reject Finitism

Unsurprising

Where do the morons 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.

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

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 moronic". 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.

>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.

>implying the concept of a number is logically true

>there exists a collection of non-empty sets whose Cartesian product is empty

Do you really believe that?