/sci/ - Science & Math

>real numbers

You're in the math club doing some set theory and this guy slaps your fake number system in the ass. What do you do?

>>12379480
that the guy whos name starts with el?

>>12379480
>Nobody is able to refute him.
Yeah well since he rejects axiomatics, there's not much ground for refutation in a formal mathematical sense.
Nobody should waste time discussing "are there finite amounts of naturals in the universe" for more than 15 minutes with anyone.
Nobody will really drop [math]{\mathbb N}[/math] and I think there's barely a practical reason to assume the class of Dedekind cuts for a set (unless you really need to work with [math]hom(-, {\mathbb N})[/math] or [math]hom(-, U(1))[/math] and the like), so yeah - sure if he demands more realizable math be taught, that's a good enterprise I guess. His geometry and algebraic number theory content is good too, so he gets a pass from me.

But there's already a thread on the Burger.

>>12379623
He is a Wild Egg

>>12379480
>Nobody is able to refute him.
Yeah well since he rejects axiomatics, there's not much ground for refutation in a formal mathematical sense.
Nobody should waste time discussing "are there finite amounts of naturals in the universe" for more than 15 minutes with anyone.

Nobody will really drop [math]{\mathbb N}[/math]. Then again, I think there's barely a practical reason to assume the class of Dedekind cuts for a set (unless you really need to work with [math]hom(-, {\mathbb N})[/math] or [math]hom(-, U(1))[/math] and the like), so yeah - sure if he demands more realizable math be taught, that's a good enterprise I guess. His geometry and algebraic number theory content is good too, so he gets a pass from me.

>>12379627
>there's barely a practical reason to assume the class of Dedekind cuts for a set
working with sets of reals

>>12379643
Sorry, deleted a typo.

I don't fully understand your response though.
Sure, it's convenient if the reals are a set, but mostly just to prove stuff about the reals themselves. Individual Dedekind cuts will be sets be Infinity and Separation and you can work with them - if you really need them - just fine.

Dunno, on the other video he said that there is no algorithm to sum two real numbers, but that's as wrong as the hollow earth.

>>12379480
You can't refute or affirm axioms. That's why they're axioms.

Yeah, he proved the goldbach conjecture and somehow people didn't recognize yet.

>>12379690
I meant to say he solved it. He proved to be false.

>>12379675
Yes, you can accept them or reject. Just like the word of Jesus Christ.

>>12379697
That's different from refuting. His math is is perfectly fine for the axioms he uses. He's called a crank because he rejects infinities.

>>12379668
Well given that there formally exist reals that are undefinable (necessarily, at least if you accept LEM and set up "the reals" in a standard way), it's fair to say that not all reals can be added - some can't even be named.

>>12379707
Real numbers all have a sequence of rational numbers that converge to them. That being said, the sum of the two numbers is the limit of the sum of two sequences, which also converges to a number. This limit is also a real number. In other words, if you accept infinities, you accept closure of real numbers under addition.

>>12379762
>>12379707
>>12379668
He was talking about real numbers as infinite decimal expansions and what he said is true. Adding two such real numbers given by algorithm reduces to solving the halting problem, for which there is no algorithm.
Obviously when viewed as Cauchy sequences of rational numbers it's easy to add them: you just add termwise.
You, like the typical critic, misinterpret what he says, ignore the context completely and use this as an excuse to label him as a crank, even though he was completely right and you are wrong.

>>12379789
His conundrum arises from the fact that he discards infinities. My interpretation leads to the conclusion that all reals are closed under addition.
If you knew what you were talking about, you'd understand he doesn't even use real numbers. His number system has a completely different structure. Different axioms yield different mathematical structures. I literally just said his math is fine, given his axioms.

>>12379803
>My interpretation leads to the conclusion that all reals are closed under addition
What is a real number, according to your interpetation?

>>12379808
From an analytical standpoint, the closure of the rationals. However, the union of cuts is also fine.

>>12379815
I mean from your standpoint.
What is a the closure of the rationals?
What is a "cut"?

>>12379762
>Real numbers all have a sequence of rational numbers that converge to them.
Well for some sense of "has" anyway.
If, for a non-describable real, you can't describe any such sequence, then it's arguably a stretch to say it "has" a sequence of rationals converging to it.
I mean that's basically a main point in the whole debate - how far do we want to stretch existence, as claimed in a formal math theorem.

>>12379818
The closure of a set S is the union of S and the limit points of S. The limit points are those points that are the limit of a convergent sequence in S. I am not going to spend time proving that every real number has a sequence of rationals that converge to it, since it's been done a million times, and I expect you to already understand Analysis. Plus, if you reject infinities, that proof is useless information to you anyways. This fact, though, means that every point in R is a limit point of Q, so the closure of Q is just R.
>what is a "cut"?
https://en.wikipedia.org/wiki/Dedekind_cut
I'm not going to even attempt to reinvent the wheel on that one.

>>12379836
Do you have examples of a non-describable real number?

>>12379843
>The closure of a set S is the union of S and the limit points of S.
In that case the closure of the rationals is just the rationals, so you don't get anything new.
>https://en.wikipedia.org/wiki/Dedekind_cut
>I'm not going to even attempt to reinvent the wheel on that one.
Ok I opened the link and it says dedekind cuts are sets with certain properties listed. Can you explain to me what a set it? At least what it means in your interpretation. What's the boundary between a set and not a set?

>>12379847
Are you trying to make a rhetoric point?

It's a discussion bound to fail on an informal 4chan post level, especially since "describe" has no description in the object language, akin to Tarki's undefinability result.
The informal tea table argument, of course is that if D are the describable reals, then R\D is a nonempty set. And then, classically, there exist an x such that x in R\D.

>>12379866
D=R.

>>12379855
A set appears on the left hand side of \in in some valid ZF theorem

>>12379877
I have nothing against R being subcountable, but that only works in some topos that only CS people care about

>>12379885
>A set appears on the left hand side of \in in some valid ZF theorem
Sure, but how is that a definition of what a set is?
>I have nothing against R being subcountable
D=R doesn't imply R being subcountable. Actually, D is uncountable.

>>12379890
>Actually, D is uncountable
I realize this may seem confusing to you, Here's the proof.
Let L be some enumeration of elements of D, viewed as a function of n. Then describe a new number Q in base 3 as 0.q_1q_2... such that q_1 is 1 if the first digit of L(1) is not 1 and 0 otherwise. In this way, we have described a new number that's not enumerated by our enumeration. This means our enumeration was incomplete.
Hence there is no enumeration of D, i.e. D is uncountable.

>>12379916
I didn't say R is enumerable, but that's more or less another topic anyway

>>12379932
What's the difference between being subcountable and being enumerable?

>>12379789
>Adding two such real numbers given by algorithm reduces to solving the halting problem, for which there is no algorithm.
That's literally not a problem. It doesn't matter that the machine will not stop (in fact we know it won't for two infinite numbers), what matters is that it is a finite algorithm.

>>12380047
There is no such algorithm.

>>12380055
For simplicity let's say the sum will be less than 1, so I don't need to write the part that does the sum on the left of the integer part. Also that both numbers have no ending, so the algorithm will never stop. Let's say we have two infinite tapes with those numbers starting on the middle and going to infinity to the right. We will always run with both tapes syncronized.

@1: Read both tapes, sum those numbers and rewrite them on the top tape
@2: If the top tape has a number less than 9 jump to @8
@3: Write top tape number less 10 on the bottom tape
@4: Write 0 on the top tape
@5: Go one symbol to the left
@6: Write 1 on the top tape
@7: Go to @1
@8: Write top number on bottom tape
@9: Write 0 on the top tape
@10: Go one symbol to the right
@11: Go to @1

>>12380092
The idea is to have an algorithm that when you ask for a digit at some position it would always stop and return the digit for any position.
With such algorithms, there is no algorithm that for two algorithms for real numbers, produces an algorithm for their sum.

>>12380100
Algorithm for what? Transfinite turing machines can do that.

why is he too stupid to just make something like a "SuperCauchy" sequence that converge faster and be satisfied with that
a sequence [math]a_n[/math] is super cauchy if for any [math]N[/math], we have [math]i,j > N \implies |a_i - a_j| \leq 2^{-N}[/math]
which is nicely behaved in all the ways he wants, and is clearly equivalent to the base2 representation and the normal cauchy sequences
but knowing him he'd still complain about the sequences being infinite because hes a retard

>>12379855
>The closure of the rationals is just the rationals
Union the irrationals. Unless you're trying to tell me that number such as sqrt(2) and pi are irrational, to which I would respond that you have rejected infinity as an axiom, and so we have a conflict on constructions of real numbers, and we are back where we started.
tl;dr we're going in circles!

>>12379480
wildberger is legit. you aren't going to prove his math wrong. it's just his approach doesn't rely on some common tools that most mathematicians don't mind using and to great success. it's good to have guys like him around for many reasons

Assume there is a proper class of Woodin cardinals.

so is math fake or

>>12379480
Is there a natural number which answers the question "how many prime numbers are there"?

>>12380137
iirc that's called modulated Cauchy and is what Bishop used

>>12380033
Enumerable: X injects into N.
Countable: N maps onto X
Subcountable: A subset I of N maps onto X.

If your axiom prove that all non-finite subsets of N are in bijection with N, then the last two are the same. If it doesn't, then you have the chance of more constructive semantics of "countable".

E.g. all strings X are countable, say via a bijection b.
Some strings Y correspond to compiling C++ programs that halt when executed with main(). It's not decidable for all x in X (i.e. not decidable in general) whether a given x is in Y.
There is an infinite index set I subset N such that b(I)=Y. Of course n in N being in I is not decidable either. I is subcountable.

(And I is not actually countable (in the sense that there's no counting algorithm). Of course, ZF proves I is in bijection with N, but that's just because of LEM. It's not "ackshually true" that I is "countable".)

>>12380137
iirc that's called modulated Cauchy and is what Bishop used

>>12380033
Enumerable: X injects into N.
Countable: N maps onto X
Subcountable: A subset I of N maps onto X.

If your axiom prove that all non-finite subsets of N are in bijection with N, then the last two are the same. If it doesn't, then you have the chance of more constructive semantics of "countable".

E.g. all strings X are countable, say via a bijection b:N->X.
Some strings Y subset X correspond to compiling C++ programs that halt when executed with main().
It's not decidable for all x in X (i.e. not decidable in general) whether a given x is in Y.
There is now an infinite index set I subset N such that b(I)=Y, basically the preimage for b of Y.
Of course n in N being in I is not decidable either, otherwise we could decide b(n) in Y.
So Y is subcountable, via I.

And so I or Y are not "actually countable", in the sense that there's no counting algorithm.
Of course, ZF proves I is in bijection with N, but that's just because of LEM. It's not "ackshually true" that Y is countable, you can't count Y. It's just a ZF theorem that Y is countable.

>>12380137
iirc that's called modulated Cauchy and is what Bishop used

>>12380033
Enumerable: X injects into N.
Countable: N maps onto X
Subcountable: A subset I of N maps onto X.

If your axiom prove that all non-finite subsets of N are in bijection with N, then the last two are the same. If it doesn't, then you have the chance of more constructive semantics of "countable".

E.g. all strings X are countable, say via a bijection b:N->X.
Some strings Y subset X correspond to compiling C++ programs that halt when executed with main().
It's not decidable for all x in X (i.e. not decidable in general) whether a given x is in Y.
There is now an infinite index set I subset N such that b(I)=Y, basically the preimage for b of Y.
Of course n in N being in I is not decidable either, otherwise we could decide b(n) in Y.
So Y is subcountable, via I.

And so I or Y are not "actually countable", in the sense that there's no counting algorithm. I is an uncountable subset of N. As in, nobody will ever find an algorithm that maps n in N to just all the elements in i.

Of course, ZF proves there is a bijection function from I N, but that's just because of LEM. It's not "ackshually true" that Y is countable, you can't count Y. It's just a ZF theorem that Y is countable.
(Apart from godly bijection functions, ZF+Choice also proves there's choice functions for all non-empty sets and ZF+CH proves there's orderings functions between 2^N and omega_1, but all those functions are arguably just axiomatized into the theory for convenience.)

>>12380137
iirc that's called modulated Cauchy and is what Bishop used

>>12380137
How do you multiply such sequences?

>>12380288
>Union the irrationals
What are the "irrationals"? I already explained that you dont get any new objects when you take the closure of the rationals since the rationals are closed in themselves.

>"sets are not rigorous, absolutely unacceptable"
>rigorously defines numbers literally as strokes on a blackboard
yep, confirmed for schizo

>>12381126
How is that schizo? Hes completely right.

>>12380137
You're flattering yourself too much. Sure, by insisting that the sequences you're considering have a certain nice property then checking that property becomes very easy. However, insisting on such a property you lose generality and make other things much harder.
Insisting on fast convergence doesn't help your cause at all. All such and similar attempts at "defining" real numbers suffer from needing infinite amount of work to perform even the simplest arithmetic operations, i.e. there being no algorithm to do so.
You thinking your new "definition" solves anything just goes to show how clueless you are when it comes to the issue. I suggest you watch more Wildberger videos.

>>12379480
I used to laugh at him but then I saw this ad.

>>12381164
define stroke
define blackboard

>>12381195
Do you seriously not know what those words mean? Maybe you need to go back to 1st grade.

>>12381183
>All such and similar attempts at "defining" real numbers suffer from needing infinite amount of work to perform even the simplest arithmetic operations
So what?

> i.e. there being no algorithm to do so.
Doesn't follow.

>>12381198
>do you seriously not know what the word "sequence" means? maybe you should read a math book.

>>12381215
>So what?
So it means you can't actually compute with them because we humans are incapable of doing infinite amount of work.
>Doesn't follow.
By algorithm I mean a finite, deterministic procedure, the same thing as was meant by Hilbert and Turing when they formulated and solved the Entscheidungsproblem.

>>12379668
Name a single real number that can be added algorithmically

>>12381218
A string of strokes on a blackboard is completely clear, there is no ambiguity involved, unlike with the notion of a "sequence".
What type of sequence is meant? Must the sequence be given by an algorithm of some sort? If not, then must at least the terms in the sequence each be describable by an algorithm? Of not, must at least the sequence be describable in set theory? Or must all of the numbers be describable in set theory?
Answering any of these questions (an even more questions that I cba to write down) in a different way leads you to very different notions, and none of these questions have obvious answers when you just talk about a general sequence, which shows just how vague the notion of a "sequence" is.
Ask a computer scientist what a sequence is and you will most likely get a very different answer than you would get if you asked a mathematician.
However, you can ask what a string of strokes on a blackboard is and everyone would agree.
You pretending you don't know what a string of strokes is obviously just a cope. You're trying to argue that a string of strokes is as vague of a notion as that of a sequence which is obviously not the case.

>>12381126
not really different from saying you start with 0 and prepend S's

>>12381235
0.
That said, the VAST majority of "real" numbers are unaddable in most "constructions".

>>12381226
>So it means you can't actually compute with them
No it doesn't.

>By algorithm I mean a finite, deterministic procedure
Yes, a finite algorithm can repeat infinitely.

If you don't even know what the terms you're using mean, how are you going to show they determine what axioms we use?

>>12381241
no to all questions
sequence of rational numbers = one rational number for every natural number
you pretending to not understand this trivial concept is obviously just a cope

>>12381254
What good is an algorithm that doesn't halt?
I have come up with an algorithm that solves any math problem you want, with just one caveat: it doesn't halt!

>>12381278
>What good is an algorithm that doesn't halt?
I don't know, you're the only one that demanded a finite deterministic algorithm. I don't care that you don't like it. I find it especially odd that you keep talking about adding and multiplying even though pi+e is just as computable as pi itself. So why not rail against pi as "non-computable?" Is it because tons of advanced work has been done on computing pi and you would look like a fool to complain about it?

>>12381306
>pi+e is just as computable as pi itself
Of course, in the orthodoxy of modern mathematics that's true.
The problem asks you what it IS, not whether or not it's computable.
>So why not rail against pi as "non-computable?"
Because there are algorithms that generate as many of the digits of "pi" as you want.

>>12381322
>The problem asks you what it IS, not whether or not it's computable.
LOL, that's what you've been complaining about this entire thread.

>Because there are algorithms that generate as many of the digits of "pi" as you want.
And there is for pi+e as well. So does that mean you accept pi as real?

>>12381258
>LOL, that's what you've been complaining about this entire thread.
That was just one of the questions. It asks what IS pi+e. Can you answer it? So far it seems like you can't.
>So does that mean you accept pi as real?
I accept an algorithm that generates a Cauchy sequence of rational numbers that get very close to the ratio of the circles circumference to its diameter doesn't. To answer whether or not I accept pi as real I'll need you to explain to me what this "pi" IS.

>>12381322
>The problem asks you what it IS
point on a number line

>>12381390
What is a "number line"?

>>12381123
that anon doesnt want to explain a dedekind cut to you, probably because he (correctly) believes you are retarded.

I'll explain it to you though.

There is a simple proof that sqrt2 is irrational. I'm going to make some generalizations here.
You can approximate sqrt2 with a rational number A, and a rational number B,
where A<sqrt2<B.
The dedekind cut is determined by separating all the infinite number of rationals less than sqrt2, and all the infinite number of rationals greater than sqrt2.

In other words, if we take a set X, we can fill it with all the rationals less than sqrt2. In which case the "limit" of that set is sqrt2, but that set only contains rationals.

That's as much as im going to explain about rationals, since the other anon told you to read up on dedekind cuts, and it doesnt seem like you have; and im not going to spoonfeed you

>>12381390
its like a stroke on a blackboard but longer

>>12381342
>That was just one of the questions
Why ask it?

>It asks what IS pi+e.
Too vague to answer. And you didn't even ask that, but whatever. Nice to see you've given up the algorithm nonsense.

I don't know if he's correct, (still learning lol), but he's pretty entertaining and his educational videos are solid.
I lost my sides once watching a video of his on group theory where he was talking about matrix groups and he aggressively said "with rational numbered entries"

funny how this >>12381258 went unanswered

>>12381399
>hat anon doesnt want to explain a dedekind cut to you, probably because he (correctly) believes you are retarded.
Read his post. He's attempting to define the real numbers as the closure of the rationals, which is retarded because the closure of a topological space is the space itself, unless you specify a larger space in which you're taking the closure. But the goal is to construct the larger space in the first place.
If you're honestly defending him you are as retarded as him. Go pick up a textbook on basic topology.
>There is a simple proof that sqrt2 is irrational
In the context of this discussion we haven't constructed the "real numbers" yet so sqrt2 is undefined, and by extension proving that sqrt2 is irrational is meaningless.
There is a simple proof that no rational number squares to 2, due to Euclid though, which is a much more elegant and correct way to phrase what you probably have in mind.
>In other words, if we take a set X, we can fill it with all the rationals less than sqrt2
Very interesting choice of words here. You're picking a set and then filling it with rational numbers. Curious.
Again you haven't proved shit and your construction doesn't work because it relies on sqrt(2) already being defined which it's not, in the context of this discussion.
I can just as well prove that the number xauee is irrational by copy pasting your "proof" and replacing sqrt(2) with xauee.
>>12381402
I feel I'm going to have a stroke if I keep talking to you.
>>12381403
>Why ask it?
Because it's an interesting question and a good illustration of how the real numbers are fake.
>Too vague to answer
How is it too vague? According to you pi and e should be real numbers, and real numbers should be closed under addition. So my question asks what is the real number pi+e, in your understanding of these "real" numbers. Seems like a simple, straightforward question.

>>12379480

Thunderf00t deboonked him

>>12381469
so you have no problem with the notion of topological space?

>>12381403
>Nice to see you've given up the algorithm nonsense
Everything I've said about algorithms is still true, I haven't given up the issue. These are just distinct issues.
>>12381442
>sequence of rational numbers = one rational number for every natural number
The issue is that there are infinitely many natural numbers which is where the issue arises.

>>12381486
>The issue is that there are infinitely many natural numbers which is where the issue arises.
that's not an issue. it may be an issue for you, but not for me.

>>12381475
I have no problem with the notion of a finite topological space.
Also I understand topological spaces from the orthodox viewpoint of mathematics, which anon clearly has no clue about.

>>12381493
You seem to have no problem with taking all choice sequences of rational numbers as a completed whole, so please explain to me your perspective on the continuum hypothesis. Is it a definite question? What does the proof that it's independent of ZFC tell you?

>>12381502
>please explain to me your perspective on the continuum hypothesis
I don't care

>>12381514
It decisively shows that your notion of a "rational number for every natural number" is too vague. Wouldn't it be nice if we could just ignore all the facts that contradict our beliefs like you do?

>>12381521
it's as vague as "stroke on a blackboard"

>>12381530
I can show you strokes on a blackboard, do explicit computations with them, compare them etc.
Nothing of the sort is possible with "sets" or your correspondences one rational number for every natural.

>>12381536
I can show you sequences, do explicit computations with them, compare them etc.

>>12381541
Allow me to define the sequence of prime numbers
(2,3,5,7,11,13,...)
Now let me define another sequence whose terms are the differences of the successive terms in the previous sequence, so it's
(1,2,2,4,2,...)
Now please calculate for me how many 2's there are in this resulting sequence. Don't worry, I'll wait.

>>12381559
I don't know and I don't see how is that an issue

>>12381572
The issue is that you can't do computations with these "sequences" in general, just like you can't compute the answer to the question I asked.
These "sequences" are nothing like the strokes on a blackboard.

>>12381469
>Because it's an interesting question and a good illustration of how the real numbers are fake.
How does it show the real numbers are fake? Weird how you can calculate any digit of a "fake" number.

>How is it too vague?
Because it has many possible answers depending on what you mean. For example, pi+e is a real number. Is that the answer you wanted? It's also the sum of pi and e. Do you need me to explain what each of those words mean?

>>12381486
>Everything I've said about algorithms is still true
And still irrelevant.

>>12381579
>The issue is that you can't do computations with these "sequences" in general,
true, so what? how is that an issue?
>just like you can't compute the answer to the question I asked.
again, so what? plus it might be possible to find the answer by other methods.
>These "sequences" are nothing like the strokes on a blackboard.
of course they're very different, but they're defined just as vaguely

>mathmatician realises most math is pseudo physics

the only Real numbers are physics constants, everything else is ape-brain cope

>>12381559
>how many natural numbers with the given property are there?
how come you don't insist that this question is computable in your math?

>>12381606
>Weird how you can calculate any digit of a "fake" number.
You can't in general. Read the thread before posting.
>Do you need me to explain what each of those words mean?
Exactly.
>>12381616
>true, so what? how is that an issue?
It's an example of how the notion of a sequence is much less concrete than that of a natural number.
It's also an issue because (the equivalence classes of) these sequences are viewed as "real number" and yet you can't do even perform the most basic arithmetic with them to get an answer.
> but they're defined just as vaguely
Cope. I can show you a string of strokes on a blackboard. You can't show me an "infinite" sequence. The best that you can do is to provide me a finite algorithm or description of an "infinite sequence" which is not at all representative of what a general sequence according to you looks like.
>>12381631
Because in my math such a sequence is not a completed whole and you wouldn't expect to be able to answer such a question.

>>12381653
>Because in my math such a sequence is not a completed whole and you wouldn't expect to be able to answer such a question.
who talks about sequences. how many prime numbers are there?

>>12381653
>You can't in general.
We're not taking in general, we're taking about pi.

>Exactly.
Look it up.

>>12381653
>It's an example of how the notion of a sequence is much less concrete than that of a natural number.
agreed, it's more abstract. so what?
>It's also an issue because (the equivalence classes of) these sequences are viewed as "real number" and yet you can't do even perform the most basic arithmetic with them to get an answer.
you can up to arbitrary precision. which should be a satisfactory answer, considering how math is tied to the real world and all.
>>12381653
>I can show you a string of strokes on a blackboard. You can't show me an "infinite" sequence. The best that you can do is to provide me a finite algorithm or description of an "infinite sequence" which is not at all representative of what a general sequence according to you looks like.
you also can't provide a general representative of a string of strokes on a blackboard. actually I think you can't even show me a string of strokes on a blackboard.

>>12381579
>you can't do computations with these sequences in general
Of course. And why does that matter?

>>12381685
>We're not taking in general, we're taking about pi.
Obviously there will be special cases where the calculations are possible and easy. You don't even need to go to pi for that, just look at the rationals as a subset of the reals. You can do arithmetic easily with the rationals, that means you can do arithmetic with general reals? Not so! Arithmetic is impossible with the vast majority of real numbers, nor can you describe them, nor can you have any access to them whatsoever. They don't even deserve to be called "numbers".
>Look it up.
I already explained multiple times why the "definitions" of real numbers and their arithmetic as described by wikipedia and books like Rudin's Principles of Mathematical Analysis are completely unsatsifactory and essentially boil down to simply moving the goalposts.
If you have your own explanation that you think makes sense, please provide it. I'm interested in reading it.
>>12381663
>who talks about sequences
We are.
>how many prime numbers are there?
Not sure. If you believe in the basic axioms of arithmetic with naturals, then the prime numbers are unending (you can write down arbitrarily many of them). But in reality that's impossible, which is why I'm not sure about the answer.
>>12381687
>agreed, it's more abstract. so what?
It's not just the difference between more concrete and more abstract. That's vastly trivializing the issue.
The difference is like that between a chair and an infinitely large gnome. One is an actual thing that you can observe, interact with and tinker with. The other is an imaginary concept for which there is no evidence that it even exists or that the words that make it up are even meaningful when put together the way that they are.
>you can up to arbitrary precision
Explain what you mean by this.

>>12381700
>Here is my number system! These are completely real and true numbers. That said, there is no way to add, multiply and exponentiate them. Also majority of these numbers are completely inaccessible by anyone. But of course, none of these are actual problems. This number system is as real as the natural numbers, I swear! It's in the name, they're real numbers!

Lmao, who's the retard that keeps asking about pi+e together as if it's any different than pi or e alone?

Just take [math]\frac{4(-1)^x}{2x+1}+\frac{1}{x!}[/math] from x=0

If the real number are indeed no well defined what should we do with the totality of mathematics that rely on real number? In particular what happens to analysis, especially the useful parts like harmonic analysis?

>>12381792
>Obviously there will be special cases where the calculations are possible and easy.
Then how does your criticism invalidate the reals? Your criticism doesn't apply to all reals yet you act like it does.

>They don't even deserve to be called "numbers".
Your argument aren't math.

>I already explained multiple times why the "definitions" of real numbers and their arithmetic as described by wikipedia and books like Rudin's Principles of Mathematical Analysis are completely unsatsifactory and essentially boil down to simply moving the goalposts.
No you didn't.

>>12381183
>All such and similar attempts at "defining" real numbers suffer from needing infinite amount of work to perform even the simplest arithmetic operations, i.e. there being no algorithm to do so.
>>12380137
>but knowing him he'd still complain about the sequences being infinite because hes a retard
nice job reading to the last line of my post you fucking moron
with these you can get error bars that are guaranteed to shrink by 1/2 with every step
so to whatever accuracy you want you can calculate it in a determined number of steps
no one cares that i cant do things an infinite number of times
even in the real world you cant measure any thing exactly, you have to measure to a certain accuracy and then improve it
no one cares what the exact value is

the real numbers are symbols so that i can take two and add, subtract, multiply, divide, order and take cauchy sequences of them

>>12382385
Explain to me please how you multiply and divide these "SuperCauchy" numbers.

>>12381617
>the only Real numbers are physics constants
>the only Real number is 1
kinda based

>>12382390
admit your loss first
im not letting you change the goalposts before you admit that youre a retard

>>12382414
What loss? I was right about everything.

>>12379803
nigga what axioms does norm use? his whole point is that his system isn't axiomatic

>>12381792
>tossing dirt on Rudin
Awesome, I'm excited. Have you found any examples of non-computable numbers that aren't the limit of a sequence of rational numbers yet? After all, he proved every real number was. If you can't, I will continue to suppose that addition of non-computables still works, since you can add the sequences of the two numbers, and that sum is still convergent (Rudin also proved this). It would help if you've read his book, but it's not necessary. After all, to disprove something, you only need 1 counterexample.

Lol, the same anime pleb (no catgrils) from the other thread is here.
Convergence ≠ Completeness

>>12382920
>Have you found any examples of non-computable numbers
Lol. I wouldn't claim such foolishness. It's you who is claiming such numbers exist, and that in fact they are the vast majority of numbers. To me, a "non-computable number" is a meaningless phrase. It doesn't refer to anything, so using the term is pointless.
>I will continue to suppose that addition of non-computables still works, since you can add the sequences of the two numbers, and that sum is still convergent (Rudin also proved this).
You can continue to suppose whatever you want. You can continue to suppose that every infinitely large gnome is made up of a sequence of donuts, if that appeals to you. Just don't pretend any of it is real or even makes sense.
Also Rudin defined the reals in terms of Dedekind cuts, not Cauchy sequences. And for Dedekind cuts there is no way to add them.
>It would help if you've read his book, but it's not necessary.
Oh I have, and more advanced analysis texts as well. It's a good book.

>>12383612
You must be the same retard from before. Two different people already explained to you multiple times how you're retarded, one would think you would learn to shut up, pay attention and listen by now.

>>12383623
>still no catgirls
Lol, I enjoyed watching you and the other Anon flail around and drown in undergrad topology. I was sad that the thread was prematurely 401 when I woke up this morning.
Convergence ≠ Completeness

>>12383656
>Convergence ≠ Completeness
Of course. Convergence is a property of a sequence in a specified metric space and completeness is a property of a metric space.

>>12383664
No, convergence has nothing to do with a metric space.
Convergence is a concept of getting closer.
Metric spaces are transformations.
>still no catgirls
You are a brown belt in math and you are a brown belt in anime.

>>12383700
Convergence is a notion that applies to a sequence in a general topological space. A sequence x_n is convergent if there is a point x in the topological space such that for every open set containing x, the sequence x_n is eventually in that set.
Metric spaces are not transformations. That's doesn't even make any sense. A metric space is a set X together with a function d: XxX->R such that for all x,y,z in X
d(x,y)>=0 and d(x,y)=0 if and only if x=y.
d(x,z)<= d(x,y)+d(y,z)
That's it.
A metric space is complete if every Cauchy sequence in the space is convergent. Not every metric space is complete.

>>12383720
Nonsense. Convergence is simply getting closer forever.
And metric spaces are absolutely transformations. Even your own description literally describes a transformation. Try to describe even one metric space without insinuating a function that transforms one set to another set.
Convergence doesn’t rely on metric spaces; metric spaces rely on convergence.
Your anime is as backward as your math

>>12383803
You will never be a real mathematician.

>>12383819
>t. Anonymous
I’m so sad

>>12379480
Stop lying, asshole.

>>12383848
Why would I lie? Theres no point in lying.

Still waiting for a non identical metric space that doesn’t involve a transformation. Surely the brave mathematician won’t simply choke up and die on his or her own sword here.

>>12383898
Are you french? What is your native language?

>>12383906
I‘m Cayugan but I also speak fluent St. Laurent. Feel free to write your math niaiseries in French if you want. I beat you there, too.
But if you write your math in Korean, maybe you can “win.” I don’t speak Korean.

>>12383993
Youre genuinely retarded.

>>12384018
I’m a retard, and you don’t understand convergence is. And you post shit anime.
I win by two miles.
Learn to think, it will help you in life

>>12384018
I’m a retard, and you don’t understand what convergence is lmao. And you post shit anime.
I win by two miles.
Learn to think, it will help you in life silly

>>12383617
>non-computable number" is a meaningless phrase
You’re not just ignorant, you’re an ignoramus.

>>12379789
>Adding two such real numbers given by algorithm reduces to solving the halting problem, for which there is no algorithm
t. retard

>>12380988
>Enumerable: X injects into N.
>Countable: N maps onto X
Lol more retardation

>>12381235
Lmfao

>>12381322
>there are algorithms that generate as many of the digits of "pi" as you want.
and “pi”+”e” in the same way, imbecile

>>12381521
>ZFC
>Q
>N
lmao imbecile

>>12381559
>plz calculate the number of 3s in the decimal 1/3
imbecile

>>12379617
I literally saw this guy at my uni (UNSW). Got memestruck.

>>12383700
>convergence has nothing to do with a metric space
LOL suck my BALL(p,ε)S.

>>12384446
> (p,ε)
This describes closeness, not completeness. Try again?

>>12384464
>try again?
No. Learn 2 metric space. Not my responsibility.

>>12381469
>Again you haven't proved shit and your construction doesn't work because it relies on sqrt(2) already being defined which it's not, in the context of this discussion.
I can just define the set to be rationals whose square is less than 2, no problem, you pedantic retard. kys desu

>>12384918
undergrad brainlet cope lmfao

>>12385359
>undergrad brain let cope lmfao
t. high schooler
Probably took an advanced course where you got an introductory analysis lesson, and now you think you're the topology wizard. https://en.wikipedia.org/wiki/Limit_point
https://en.wikipedia.org/wiki/Metric_space
Play with the bouncy balls.

>this thread

https://youtu.be/CTV2JzTBgBs

Mathematics which does not obey or conform to thermodynamics is invalid. What is sad is that Wildberger isn't smart enough to find the solutions to the problems he observes.

>>12379480
>>12379789
>>12380047
>>12380055
>>12380100
>>12381114
>>12381226
>>12381235
>>12381278
Alright you stupid niggers, analog computers exist and can work with do mathematical operations on real numbers by definition, since they work with continuous values by definition.
https://www.sciencedirect.com/science/article/pii/S0885064X03000347#BIB13
https://onlinelibrary.wiley.com/doi/abs/10.1002/sapm1941201337

>>12385701
Listen, retard. Convergence is the concept of getting arbitrarily close forever. Convergence IN METRIC SPACES also involves sets of points IN METRIC SPACES.
This is a property of METRIC SPACES, you absolute idiot.
It is NOT a general property of convergence.
Fucking loud mouth undergrad brainlet copers.

>>12385289
Yeah and I an define 1/0 to be the set of rationals that whose denominator in lowest terms is divisible by at least 3 primes. That doesn't prove shit. Go back to revising analysis.

>>12385874
Convergence is as it is defined. If every open neighborhood has a point that isn't itself, then it is a limit point, and you can construct a convergent series of points whose limit is the limit point. It just so happens metric spaces use epsilon balls to make this a lot easier.
t. got an A+ in analysis 1 and 2 and having trouble coping with it

>>12386221
Neighborhood of a point*

>>12385791
i call bullshit. your analog computer has a certain degree of precision. not something exact. you get approximate math, not precise and concrete math

>>12386221
how many ways is there to approach a limit point ? can you algebra on them ?

>>12379694
What's the counter example?

Constructivists in general are just retards.

>>12386497
2z, where
z= 10△10 + 23
where △ is defined inductively as
n△1 = n
n△(m+1)=n^(n△m)

>>12386416
>How many ways is there to approach a limit point?
Infinite. If you have one sequence of points, the sequence where you only consider every nth term also converges to the same limit point.
>Can you algebra on them?
If the set has an algebraic structure. Every element of R is a limit point of R, so you can give it an algebraic structure and consider it "algebra on limit points".

>>12386408
A GPAC is a mathematical model like the Turing Machine you obtuse fuck. Of course real world applications have limitations, can you make a draw a line that is an integer number of inches long? No? Kys retard.

Why do people think there are an infinite amount of natural numbers? Surely it would be reasonable to define a maximum number, equal to something like the highest anyone ever counted to. That only seems natural.

>>12386898
Why is this more useful or more convenient?

>>12386948
It is more natural. They’re called the natural numbers, not the "useful" numbers, or the "convenient" numbers. There is nothing natural about infinity.

>>12387058
>There is nothing natural about infinity.
that's why infinity isn't a natural number

>>12379617
What's his routine, lads?

>>12384429
Based UNSW chad. Went to one of his classes once for fun (History of Mathematics), which is supposed to be a meme class for teaching majors. Was pretty comfy anyway desu.

>>12379789
You’re saying it can’t be computed, which is different. It can be done by calling it a and b and saying a+b=c and all the algebraic properties still obtain.

>hurr durr nothing's real unless you can conceive of a physical manifestation of that thing
>sqrt(2)
>nooo you can't do that you gotta only use concepts I understand!
He's just a tard that wants to pull down everything to his level of mediocrity, no wonder this is the only place that will take him seriously.

>>12386898
>MAX_INT + 1
>noooo you're not allowed to do that nobody has ever counted that high before!!!!

>>12386847
mathematics has nothing with real world applications. there are no triangles in reality. it's an ideal. it's pure and precise. get the fuck out of here with your tainted hypothetical infinite imagined models or show me a single calculation with a result. pro tip - you can't.

>>12382089
>especially the useful parts like harmonic analysis?
low iq post, lol

>>12388215
Nice argument

>>12379480
Honestly I was kind of same, but the more I learn about mathematical logic and metamathematics the more I start to see their point. His last video abut fake arithmetic basically convinced me that real number are as real as imaginary ones, which is something that was inconceivable to me a couple months ago. I want to also start reading more about this guy: https://www.jamesrmeyer.com/infinite/proof-more-reals-than-naturals.html

He also gets laughed at a lot, but I think at least on this article he brings a good point.

>>12379668
>>12379707
If a real number is defined in terms of a cauchy sequence of rational numbers, can't you just add the sequences?

>>12389177
>https://www.jamesrmeyer.com/infinite/proof-more-reals-than-naturals.html
this is an april fool joke, right?

>>12389196
Feel free to provide the reasoning as to why one infinite set can somehow have more elements than another.

>>12388210
A GPAC is a computational model of a general purpose turing complete analong computer, which can do mathematical ops on real numbers by definition. Refute me or kys nigger.

>>12389434
>which can do mathematical ops on real numbers by definition
By what definition of real numbers? Dedekind cuts? Cauchy sequences? Explain how these definitions relate to what the computer actually does. Note that if it can actually do those things, then it can solve the halting problem and essentially by extension it can solve all of maths.

>>12389321
the definition that we use agrees with the naive definition whenever it exists, i.e. when the sets are finite. that's an argument for why is the definition reasonable.

>>12389447
If you want to expand the bijection definition and create a difference between countable and uncountable, then that's fine, however, for you to claim that since no bijection is possible then one most be bigger than the other, is basically like saying that 3.5! = 3.5x2.5x1.5. You're trying to apply a property that only holds for your original domain and not for the extended one. It makes no sense that one infinite set can have more elements than another, and it should therefore not be referred to that way.

>>12379480
For many years we of the ONE TRUE FINITE AND DISCRETE UNIVERSE we oppressed by the infinity loving SODOMITES!

We suffered through school, having to endure such absurdities as infinity recurring decimals, negative numbers, pi, sqrt2 and many other such BLASPHEMIES!

We took refuge in our GOD, knowing that one day our oppression would be ended, that HE would send SALVATION in the form of a SAINT to right the wrongs of the mathematical world.

Our faith has been vindicated. For now we have SAINT WILD BURGER to cleanse this sinful Earth of the GOD CURSED SODOMITES who practice their vile and debauched mathematical fallacies!

Praise SAINT WILD BURGER! We are saved!

Under his divine leadership we shall march forth to reclaim the HOLY land of mathematics, burning books and sending all those who oppose us straight into the gaping mouth of HELL, where they shall burn for a discrete and finite amount of time.

Say it with me, Brothers and Sisters of the ONE TRUE FAITH.

DEUS VULT! DEUS VULT! DEUS VULT!

Amen.

>>12379480
I like him but I don't think it's fair to say that there must be a maximum natural number simply because we have finite computational power, that seems silly.

>>12389917
I love your posts. Wish I had saved them all for a compilation

>>12389917
amen
>shall burn for a discrete and finite amount of time
kek

>>12389635
>it should therefore not be referred to that way
So there's complete agreement about the mathematical property (nonexistence of a bijection between two infinite sets), and this "debate" is merely about what this property should be called?

>>12391514
exactly, that's why I called the article "april fool joke". it's literally just a rant about terminology, lol.

Bump

>>12379480
I would like to thank whatever meme factory shill on /sci/ led me to wildberger's lectures about 2 years or so ago
I shit you not I am using universal hyperbolic geometry as a model for 'deep learning' and doing epic, epic things at work with it these days I never thought possible, I'm also looking at his algebraic calculus course and wondering why I was never taught this. Totally non complex analysis

Reminder that all his papers are on arxiv and you can get a huge 30+ paper on his affine geometry model or algebraic calculus to write a programming library as you go through this course if you want or just to get all the theorems at once and try your own edge cases/make your own examples.

Do I agree with him that 'pure math is a fraud' kind of, but I've always worked in the applied engineering world where we just use floating point junk approximations anyway

>>12381247
Even 0 is a 'limit' of a cauchy sequence that approaches zero and goes on for infinity, so no algorithm there either it never terminates

>>12386408
Agree with this guy
Floating point is an abstraction not any kind of precise math, I have seen it fuck up so many times in practice that it's not even worth using the doctor's I work with all insist on BigInteger or BigDecimal java libraries for approximations https://www.simplexacode.ch/en/blog/2018/07/using-bigdecimal-as-an-accurate-replacement-for-floating-point-numbers/

Knuth has half a book about this in his Art of Computer Programming Series from the 1960s and it was never, ever solved BigDecimal is just a temporary hack/patch for an ongoing approximation problem with applied mathematics 'at scale'

>>12394575
That's a misuse of definitions. (0,0,0...) Is also Cauchy, fun fact...

>>12379855
Dude, if a mathematician doesn't specify which foundation of mathematics he's using, within the mathematical community, by convention, it's assumed he's using ZFC.

Your're asking this poor guy to explain to you, someone who can't even follow the links that wikipedia conviniently adds for you in its articles, what took people thousands of years to figure out properly.

>>12394697
I'm talking about 1/n limit, and yes that does appear to be cauchy if my understanding of Tao's text is correct

>>12394797
he's playing the Wildberger game where nobody can actually, properly, define what an 'infinite' set is, to Tao's credit he tries to do this in his books but even Tao seems suspicious and the exercises dry up whenever 'infinity' is involved.

Not that I know more than Tao or anybody knows more than Tao but you can kind of sense it from him that 'this is not correct'.

>>12394797
shut up bitch, it's not like you understand why nj does what he does anyways

>>12387863
I watched his lectures on Babylonian math from that. Pretty good.

>>12394806
>he's playing the Wildberger game where nobody can actually, properly, define what an 'infinite' set is
There's legitimate ambiguity here. There are demonstrably models of ZFC with different properties, e.g. whether the continuum hypothesis is true or false. And per Goedel it's not something you can get around by adding more axioms to pin down what we mean by "set" better. The natural numbers also have this problem, but at least in that case you can point to physical things like strings of strokes that seem to have the properties of natural numbers, although I'm not sure that's entirely satisfactory because at some point adding one takes you beyond numbers that can be observed physically. As I see it, this ambiguity is something we probably have to live with, and rather than trouble ourselves over it, we should find more cool ways to exploit it like the hyperreals. But Wildberger is of a different mind apparently.

>>12395038
>observed physically
Why does anyone even care about this particular quality? It seems very "control freak"ish to me, and also completely at odds with normal human experience. We can't physically observe air, but we spend basically 100% of our life immersed in it and monotonously inhaling it so as not to die.

>>12395494
3 of the 5 senses can observe air.

>>12395494
I suppose it's the desire that there ought to be something "true" about some mathematical statements.
Of course, it's not like the meaning of "true" is undebated referent either.

>>12395502
All 5 can observe it under specific conditions. As a rule, or "almost always," 0 of them can.

>>12395494
So that we know what we're actually talking about, i.e. what objects we are considering. It's not necessary for the object itself to be observed physically, just some representation of it. You want to be able to encode your mathematical objects on a computer and let it calculate stuff with them.
Right now the situation with sets is very sad. They are left completely undefined, nobody genuinely knows what is meant by a set, all they know is that they satisfy a certain collection of axioms. Actually they don't know even that, because to know that something satisfies some axioms, you have to know what that something is. You can't write down a general stuff, you can't compute with them, you can't take the union of sets, the intersection. Nobody knows what the power set operation actually does, you can prove that there can many different results each of which could follow depending on the new axioms you assume. Nobody even knows if the axioms are consistent: it could be that you could prove a contradiction with them, like 0=1. This is clearly a very very sad state, and pretty much all of it was developed as a cope by analysists who wanted to pretend what they're doing with "real" numbers is legitimate, so they obfuscated the matters by pushing the ambiguities away from "real" number and to sets. Now they claim "Here, we defined what a real number is and does!", when in reality, all they did is referred to sets which they chose as the ambiguous, undefined objects instead. This obviously doesn't fix anything.
Something being physically observed is a good standard for being rigorous. You can observe a series of strokes and so you understand what is meant by natural numbers, and you can understand what is meant by the operations on them. Nothing of the sort is possible for sets.

>>12395605
Could a "series" of strokes perhaps be a specific condition or "arrangement" of a set of strokes?
Like if the strokes line up in a specific order or progression, it's a "string" as we talked about yesterday?
So a string would analogize to a set as a square to a quadrilateral (or pick a better analogy)?

>>12396288
Maybe, I just don't see how that's useful. I really don't care very much about the way you implement the natural numbers, the important thing here is that it can be done in a rigorous and precise manner, unlike with sets.

>>12396531
Here's what I've gathered from your explanations so far:
The difference between a string and a set is that in a "string" the strokes are explicitly collected into an order, and in a "set" the strokes aren't explicitly collected into an order.
Is this an accurate summary?

If so, why do you care that the genus "Set" is unclear, if the species "Set string" is clear, and all that matters for your purposes as you've described them are "Set strings"?

>>12396531
if you can implement natural numbers, then you can implement sequences. have you ever played "who names the biggest number" as a kid? even little kids realize that no matter what number does one say, you can always add +1 to get an even bigger number. that's it. if you get this, you understand infinity and sequences. it's not necessary to bring set theory into this.

>>12379480
Anons, what are the consequences of his chromogeometry? For the uninitiated, he has taken the three general quadratic forms [math]x^2+y^2[/math], [math]x^2-y^2[/math], and [math]2xy[/math], and labeled them blue, red, and green respectively, and those forms define the "circle" in each geometry. Then given a triangle in any cartesian coordiante system, each quadratic form can used analogously to euclidean (blue) geometry to define the blue, red, and green euler lines. These turn out to be the medians of a new triangle defined by the blue, red, and green orthocenters so there is an interesting interaction between the three.

Is this anything novel, or is it simply a result of the fact that different quadratic forms over a field basically do the same thing?

he needs to make a video going over all the zfc axioms and explaining why they're bad already

>>12387863
I saw Norman Wildberger at Coffee On Campus next to the Red Centre yesterday. I told him how cool it was to meet him in person, but I didn’t want to be a typical undergrad and bother him and ask him for a consultation or anything. He said, “Oh, like you’re doing now?” I was taken aback, and all I could say was “Huh?” but he kept cutting me off and going “huh? huh? huh?” and closing his hand shut in front of my face. I queued to order, and I heard him chuckle as I walked off. When I got to the register, he cut in front and tried to order like 15 coffees without paying.

The girl at the counter was very nice about it and professional, and was like “Sir, you need to pay first.” At first he kept pretending to be tired and not hear her, but eventually he asked about the difference in size between medium and large.

When she said that the large was sqrt 2 times bigger, he stopped her and told her to "give him a real number, to prevent any numeretical infetterence,” and then turned around and winked at me. I don’t even think that’s a word. After she poured each coffee using a measuring jug, and put them in some cardboards holders, he kept interrupting her by yawning really loudly in Babylonian.

>>12397901
Not a bad idea.
>>12398748
kek

>>12398748
incredible edit

>>12397901
He would if he could. Truth is, he's a hack.

>>12401526
that's not what the word hack means. maybe you're thinking of quack or crank

>>12396698
one direction i see it going is his new def of complex numbers which throws us back to the times of Heaviside and Steinmetz . this is what i gathered from eric dollard.
http://www.gestaltreality.com/energy-synthesis/eric-dollard/symbolic-operators-steinmetz-to-pythagoras-backward-in-time/
http://www.gestaltreality.com/energy-synthesis/eric-dollard/and-in-the-beginning-versors-by-e-p-dollard-2012/
http://www.am-innovations.com/
at some point someone gonna come and use wildbergers math to introduce to humanity a new age of tech and electricity

>>12401700
nah a hack, he sux

>>12401700
fuck up retard

>>12402516
Yeah so, in a few of his videos, Wildberger reconstructs the complex numbers as a dihedral algebra over an arbitrary field, excepting the case where the field is characteristic 2. The usual case is pretty much equivalent to the complex numbers in common use, but taking the linear algebraic tack which introduces more intuitive notation imo. Maybe the interaction of the complex numbers as the three dihedral algebras will yield some interesting results.

bumpe

>>12388138
kek

>>12379480
lots of love for wildberger, but what do we do? i got the youtube backup of his channel from a month ago. we need his notes. we need a text book. we need software.
i really want a dump of all his whiteboard from all the videos. watching the videos is all good but having it in a textbook/slides, will definitely increase my learning speed. currently i have my personal notes til ~50 lecture in math foundations. there are 250+ lectures in the series and i want all their notes.
also i desperately want to read eric dollards "versor algebra" but i don't have a way to pay them, i got only some crypto. it seems like wildbergers complex numbers might dance nicely with eric dollards ideas.
also in the same ball park is pic related, want to read

cheers to all students and scholars, we are at the brink of a new renaissance

>>12406782
He has a few articles on arXive and his website has the rational trigonometry textbook he wrote.

>>12396698
all i remember is that he does geometry with some quantity S which avoids taking a square root

>>12379480
Wild nothinburger

>>12407739
Were talking about a senile mathematician, not a global "pandemic".

>>12407967
Oops, meant weird normieberger

>>12407967
>senile
He's as sharp as ever before.

Figured I'd have better chances of asking here than /mg/ or /sqt/:
Is there a geometric proof for the fact that if [math]q \in \mathbb{Q}[/math] is such that [math]q^n \in \mathbb{Z}[/math] for some positive integer n, then q is itself an integer?

>>12394569
you should totally email him, I bet he'd be excited to hear his stuff's turned out useful for practical applications. I feel like for a pure mathematician you expect that sort of thing to only come centuries after your death if at all.

>>12410317
>geometric
not from what I've seen. that's a pretty odd question to try to prove geometrically since it's stated in terms of algebra. It could probably be done with some mess of lines and squares for the n=2^k for k in N case at least since repeated squaring is ez with rulet compass constructions.

>>12408518
You're correct, but not for the reason you were hoping for.