*sobbing* i am a low iq idiot

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>why do people in this board always write these cryptic could-be-insults that only make sense in their heads
I don't know, why don't you tell us

You fell for the Gelfand meme, its a shite 90 page introduction if thats what you're needing get an AOPS textbook

kys nigger

No shame in doing some khan academy first bro. I was a high school dropout and khan literally saved my life with his site.
t. cs/math double major (in some shithole but still)

*grinning* that’s a really cool cover

go to bed tao

If you can't characterize the automorphisms of simple rings by the time you're in your forties you're ngmi

lol that's not a cryptic insult, it's a pretty straightforward joke user

Based double digits of truth.
As a engineering student, I relate to opening a book and reading what some maths-fag would spend hours writing out.

shut up jad

Modern math is bullshit and I will explain why

Modern math is full of paradoxes and contradictions. Why? Set theory and its retarded axioms. Sets can not have multiple elements, for example A = {1, 1} is not accepted, however, for some reason, they can look like this: A = {1, {1}}. Which makes no sense. You either have 1 or you don’t. So sets are not well defined in modern mathematics. Sets should never contain sets, especially not sets of themselves! This eliminates things like the power set, which allows you to take a set and multiply its size greatly by forming all permutations on that set, which is literally magic. For this reason we have absurd conclusions like Cantor’s theorem which shows that some infinities are bigger than other infinities. Yeah, that’s bullshit.

Infinite sets and power sets are both AXIOMS in modern mathematics. If you look at the axioms, some of them only exist to avoid problems caused by other axioms! And yet we still have paradoxes and contradictions, like Hilbert’s hotel. Can an infinite hotel with all its rooms full accept another guest? YES, says modern math. Just move each guest to the next room! And the problem stops there. But wait, what if the hotel manager wants to simply rearrange the guests? He should be able to do that since there is one guest for each room. So he simply tells all the original guests to go back to their original rooms, and the new guest to find the empty room. But there is no empty room! How can he gain/lose a room just because of the arrangement of guests? We can represent this problem with pure math:

A = {1, 2, 3…}
B = {0, 1, 2, …}

The mapping x~x-1 (1~0, 2~1, etc.) maps each element in A to an element in B. However the mapping x~x maps all elements but 0. According to the modern math, these sets have the same cardinality (size). Even though there is a case in which their size obviously seems different, we simply IGNORE this and focus on the case in which they’re the same! Isn’t math lovely?

Don’t even get me started on IRRATIONAL “numbers.” What is an irrational number? Can you tell me?
>a number that isn’t rational
Okay, but how do you actually CONSTRUCT an irrational number? A rational number is easy, just take two natural numbers and arrange them as a fraction. But how do we define something sqrt(2)? The Dedekind cut method just says that sqrt(2) is the set of all rational numbers to the left and right of sqrt(2). In other words, x^2 < 2, and x^2 > 2. But how is this meaningful? This doesn’t allow us to construct the number itself, in decimal form. Not only that, but a collection of irrational numbers (and rational numbers, as they are also real numbers) would just be collections of collections of rational numbers. How does that make any sense? There are probably DOZENS of way to construct the real numbers. Why? Because there is no correct way, because they’re all flawed. Most textbooks don’t even attempt to construct them! You just assume the existence of the “continuum” and that real numbers simply exist. Nevermind the fact that 0.15192629…… isn’t a legitimate representation or definition of this number (how do you define it? Try it).

So math is FULL of these counter-intuitive and useless axioms and theorems and paradoxes and contradictions, but it’s all accepted! And you’re just supposed to go along with it as if everything’s fine. Wittgenstein knew all of this was laughable, so did Kronecker and others when this stuff was being introduced. But people like to believe in things they don’t understand.

>Sets can not have multiple elements
>Sets should never contain sets
according to who?

>filtered by muh INFinity
Okay?

>just go along with it don’t worry about the paradoxes keep wasting your time over useless problems
it doesn’t have to be like this bros WHYYYYYYYYYYYY???? IM GOING INSANE AAAAAAAAAAHHHHHHHHHH

You can weed out the smart autists from the retarded ones by watching how they react when you say "number A."

>wasting time
Very subjective fren. Normoids will dismiss literature because
>Its just like, all dead Wypipo!
Do you see the similarities?

Take your meds, John

Throw that shit in the garbage and get Euler's Elements Of Algebra.

Lmfao, I will never cease to derive pleasure from you autists who can't handle infinity and sets. Never any rigor, just schizo assertions like
>{1,1} is not accepted, but {1,{1}} is, this is nonsensical
Like you're so stupid that you don't even realize that you're raising an aesthetic objection to a formal system.

>A = {1, 2, 3…}
>B = {0, 1, 2, …}

Oh no, its you again isn't it? Can you simply not accept that CARDINALITY and SIZE are different concepts, and that CARDINALITY = SIZE only for finite sets? They are different concepts, YOUR MERE PERSONAL AESTHETIC INTUITIONS ABOUT WHAT YOU FEEL ABOUT THE SETS {1, 2, 3…} AND {0, 1, 2, …} HAS UTTERLY ZERO RELEVANCE TO THE VALUE OF SET THEORY.

>a + a = 2a
wow nice "science" you got there

>two sets have the same cardinality if and only if there exists a bijection between them
>two sets have the same cardinality if and only if every injective mapping between them is a bijection
Why should we use the first definition but not the second? Please explain why, without saying “it’s just the definition!.” Both of these definitions apply to finite sets, so why do we only apply one of them for infinite sets??
>aesthetic objection to a formal system
NO. I’m simply pointing to a superior formal system. The set {1} and the set {1,2} are distinct, but the moment you collect them together in any way, you should have {1,2}. You can use a CLASS if you want to be able to group them together without combining them. But SETS do not contain duplicate elements.

Can you tell me WHY we should accept the traditional model of sets, as opposed to mine?

You outed yourself as a retard mate.

fuckin get over yourself OP
Yes.

>Why should we use the first definition but not the second? Please explain why, without saying “it’s just the definition!.” Both of these definitions apply to finite sets, so why do we only apply one of them for infinite sets??
>Can you tell me WHY we should accept the traditional model of sets, as opposed to mine?

Because that's the definition that's being used. You can use your own definitions if you want but you're no longer talking about modern math.

I think I should stop browsing Yea Forums. Full of pesud freshmen like you. Why don't you ask this at math stack exchange, let's see what responses you get from educated people and not NEETs on Yea Forums.

>Why should we use the first definition but not the second?
Because the second definition presupposes that we only work with finite sets. If you want to make the second part necessary, we can talk about only finite sets and simply use the word size, but cardinality is a broader concept than size. Size and cardinality happen to coincide when we're talking about finite sets, but reveal subtleties when talking about infinite ones. I think its completely natural to be disturbed by notions like "{1,2,3,...} and {0,1,2,...} have equal cardinality," clearly one of these sets is a subset of the other, shouldn't it be "smaller" then? And in the sense that the first set is indeed a subset of the latter, the first set is, in some sense, "smaller," but this doesn't change what the cardinality of a set is measuring. You might not like cardinality but there is no inconsistency in its definition, I think you just don't agree that cardinality is capturing a notion of "size" properly. And that's a perfectly fine opinion to have, there's tons of cool finitist, constructive stuff, etc, what I don't understand is the insistence that somehow reasoning about infinity is contradictory. Like, you agree that there exists a one-to-one and onto function from {1,2,3,...} to {0,1,2,...} right? You aren't disputing that, correct? F(n) = n-1 is clearly one-to-one and onto here. I don't think you disagree about that, it seems like you're saying the mere existence of a bijection is not encoding the size of infinite sets as reliably as finite ones.

not my problem u can't understand how useful logic is

>I’m sick of listening to Socrates, do you think the sophists would say this?
>I’m going to ask Protagoras and get some REAL answers!
Sunk-cost autists; gaming for government money. AKA academics.
None of this matters.

nice bro thanks for the good reply, not the guy youre replying to but that was rly good. makin the board better bro:)

The NPC response that I was expecting. Why should we choose the path of paradoxes when it can be avoided? Just because we can generate more and more theorems doesn’t mean the system is consistent or correct. What noticeable effect has this type of math had on society? Why are our brightest minds wasting time on these surrogate activities? Useless puzzles? If it’s not computable or constructable or applicable to the real world, then why is it so prevalent? Why is it being taught in universities? There are CONSEQUENCES to creating bad mathematical systems. It completely changes the trajectory of the species. Why the FUCK are going along with this? Bomb the universities!

You haven't shown any inconsistencies instead you've just said you don't like the definitions. If you can show an inconsistency in the accepted definitions and axioms that will be notable and people will listen. But just complaining that you don't like them will get you no where.

write a paper bro im serious, ill read it. best of luck.

>I have a diamond in my house
>John vists my house
>I realize that the diamond is missing
>Therefore John must've stolen my diamond
that's logic. let's see how useful your 2X = 350Z bullshit is

hello jew, go do your pilpul somewhere else, we're enjoying this discussion. evidently youre not. so leave

go try to program a computer bud

>The NPC response that I was expecting.
why did you ignore

>durr learn to code
typical bugman libtard response

You were the one to bring up how useful something is. Math is extremely useful for technology

logic is literally how computers work at a molecular level, you need to build those properties in when making the chips.

The PROBLEM is that you can ASSUME all type of things like the existence of an infinite set of sets of sets, but this loses sight of the PURPOSE of mathematics, which should be to solve problems in the REAL WORLD. I cannot point to a contradiction in the axiom of infinity, I simply don’t think it is useful to include it, and it is not justified because it is not intuitive. What does it even mean for an infinite set to “exist” ? How can we model this on a computer when we do actual WORK? Do you think we can take a ball and duplicate it by cutting it into pieces and rearranging it? That is what modern math says. How can anyone look at this and not realize that SOMETHING is wrong?? What is the POINT of all of this?
Yes, there exists a bijection (but even this requires assumptions). So what? Why is this useful? I don’t believe in uncountable sets because they aren’t proper sets. If I assumed the existence of infinite sets then I would believe that they all have bijections. But again, none of this is practical because infinity doesn’t exist in the real world. I can make a graph on a computer by creating bounds for my x and y values, I don’t have to ASSUME that for every x value, there is a y value.

Or, I don’t have to create bounds, but the computer will never stop graphing the function. At any point in time the mapping is still finite. Infinite sets do not “exist” in any meaningful sense.

im unironically curious about whether infinity is actually useful. I know that in some physics problems we like to set the bounds at infinity to make the problem solving easier, but im not nearly educated enough to say either way. could you explain further why infinity is useless, or someone please add something on why is is useful.

>The PROBLEM is that you can ASSUME all type of things like the existence of an infinite set of sets of sets, but this loses sight of the PURPOSE of mathematics, which should be to solve problems in the REAL WORLD
Math doesn't care about the real world. It's still funded though because of a weird propensity for pure mathematics to find application in the future. And a Yea Forums poster shouldn't be talking about applications to the real world anyway. Beam in your own eye and all that.

I know Cantor didn't just come up with cardinality out of nowhere he had been working on trigonometric series when it arose. As to whether that work on series has some application I don't know.

youre having a personal argument rather than adding any evidence for or against the proposition being argued, you must really be a jew

Please, PLEASE do not take anything that schizo says seriously. There is a vanishingly tiny minority of mathematicians that take issue with the idea of infinite sets and the like. If you'd like to learn about it read on the philosophy of mathematics, particularly about finitism, ultrafinitism, constructivism, intuitionism.

interesting thanks

(a-1)(a+1)=a2-1

What proposition? That the purpose of mathematics is to solve problems in the real world?
That is not the purpose of mathematics the proposition is wrong.

ok thanks for the recommendations. I am curious about the matter

There are so many other useful areas of research. Genetic engineering alone could be the catalyst to solving lots of our problems. If we could create super-intelligent individuals then they could tell us if all the pure math is nonsense or not much faster then just continuing to research this stuff for hundreds of years. Do we really believe that we’ll find a solution? Zeno’s paradox is apparently “solved” but it isn’t. That was 2000 years ago and we still don’t know if space is continuous or discrete. This problem will exist whether or not we continue to advance in math. We should be advancing in physics, biology, chemistry, neuroscience, etc.

i didn't know wildberger was a Yea Forums poster

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theres not one purpose to math. my question is whether infinity is useful or not, and if this guys objections about sets have merits. leaving for a dentists apt, interested to so yalls responses if its a discussion you care to have

avoiding the issue

I mean if we followed this line of thinking we could keep the math department and still get money for the sciences just by axing all of the humanities as useless. Philosophy would be the first on the chopping block since it doesn't even have the practical successes of math and suffers all the same problems of application to the real world.