/sci/ - Science & Math

>>10091594
yes

>>10091594
no

>>10091594
maybe

>>10091594
isn't it some mathfag "oh there are equivalent systems of math, so yeah we can arbitrarily change our axioms for whatever context we want, which implies we can claim whatever we want whenever we want" shit"

>>10091607
Is it true or not?

>>10091594
>Assuming something exists, despite the fact we can't ever find it.
Non-constructivists actually believe this.

>>10091594
>Axioms are true or false

>>10091685
Anon, I hate to break this to you, but many things assumed to be axioms in the past have been disproved.
How about that parallel postulate?

>>10091692
>How about that parallel postulate?
Not disproven.

I hate to break this to you anon, but you're a brainlet.

>>10091694
>not disproven
I don’t think you understand the history there. People literally believed there was no such thing as Euclidean geometry. So yeah, now you can kind of decide when to apply them, but I think your assuming math has always been done the way it has now. It hasn’t.

>>10091700
*non

>>10091700
I don't think you understand what "disproven" means.

>>10091594
This is actually a bad illustration of the axiom of choice because those choices aren't arbitrary. Also the collections are obviously finite in the picture.

>>10091594
True or false is the wrong way to look at an axiom.
Normally you look at a collection of axioms and ask if they are consistent or inconsistent.
Looking at an axiom by itself is kinda pointless.
>>10091700
Euclid had Euclidean geometry in mind when he was making his axioms. His axioms successfully described what he was trying to describe.
Nobody "disproved" the parallel postulate within Euclidean geometry.
Somebody just came along and "removed" the parallel postulate which resulted in more general geometries.

>>10091594
Why would this need to be axiomatic? Couldn't one just construct whichever function they desired? Perhaps I'm not fully understanding the criticism of the AC. Given a collection of sets, just define the function to pop out whatever element you want from a set as input.

Could someone explain a situation in which the AC is unintuitive or perhaps incorrect?

>>10091594
yes, if you choose so

>>10091813
I think your misunderstanding is that you think the axiom of choice is about constructing a _particular_ function. It's not; in the situations it appears, the axiom of choice is needed in order to claim that _any_ function exists at all.

For me, the easiest way to understand why choice is necessary was to consider the example of a single uncountable set. You don't know anything else about this set. How are you going to pick an element to give to me? There is no way for you to describe even one specific element of the set, but at the same time it feels very strange to say that you cannot just grab an element out of this giant box. This is why we need choice. (the full axiom of choice is even worse than this, since I only made a single arbitrary choice. full AoC lets you make uncountably many arbitrary choices at once.)

>>10091692
topkek
you don't have a clue what an axiom even is

>>10091607
Basically this. It's model-dependent.

>>10091835
That's actually not true. As long as you know a set X is inhabited you can fix an arbitrary element in it, then form the function {X} -> X if you want. 1st order logic lets you do this for finitely many choices, AC is only needed for infinite sets.

>>10091594
i thought it was proven that AoC was godel undecidable

>>10091692
>How about that parallel postulate?
That's not false, it's independent.

>>10092610
It's independent of Zermelo-Freankel set theory, not more or less.

>>10091914
No, you're mistaken.
Even if you're given a set of sets where you know that all sets involved are of infinite cardinality, then without the axiom of choice, it's still not a given that you construct a function.
His elaboration actually cuts quite straight to the point.

>>10091813
>Could someone explain a situation in which the AC is unintuitive or perhaps incorrect?
Yes, but you need to take a step back and reconsider what stance you want to take on provability resp. "constructibility" of mathematical objects. So here's the situation you look for:

Let's assume [math] X = {a_1, a_2, a_3, ...} [/math] is an infinite set of sets where each element, [math] a_{43} [/math] say, has seven elements, i.e. for all k we have that [math] |a_k| = 7 [math].
Okay, now I give you the following task: State a function [math] f [math] with [math] X [math] as domain, which for each [math] a_k [math], returns an element.

..
..
..
Got a function yet?
No, because I didn't tell you anything about the [math] a_k [/math] other than they they contain something.

Now if I were to say that the element of each [math] a_k [/math] have a strict order attached to them, you could actually go on. E.g. if I were to tell you all [math] a_k [/math] hold natural numbers, or if I were to tell you the [math] a_k [/math] hold tuple of fruits and no two tuples of [math] a_k [/math] have the same size. If you know anything about the [math] a_k [/math], you have a chance of going on. If I tell you all [math] a_k [/math] are orderable finite sets, then
>choose the smallest element from each [math] a_k [/math]
is one way to construct a choice function.
But in general, you can make up a context where you have nothing to work with.

The axiom of choice says "well, we're just gonna axiom you can say 'muh f' and go on with your proof."
A constructivist would say here's where you need to stop due to lack of info.

>>10092610
It's independent of Zermelo-Freankel set theory, not more or less.

>>10091914
No, you're mistaken.
Even if you're given a set of sets where you know that all sets involved are of infinite cardinality, then without the axiom of choice, it's still not a given that you construct a function.
His elaboration actually cuts quite straight to the point.

>>10091813
>Could someone explain a situation in which the AC is unintuitive or perhaps incorrect?
Yes, but you need to take a step back and reconsider what stance you want to take on provability resp. "constructibility" of mathematical objects. So here's the situation you look for:

Let's assume [math] X = {a_1, a_2, a_3, ...} [/math] is an infinite set of sets where each element, [math] a_{43} [/math] say, has seven elements, i.e. for all k we have that [math] |a_k| = 7 [/math].
Okay, now I give you the following task: State a function [math] f [/math] with [math] X [/math] as domain, which for each [math] a_k [/math], returns an element.

..
..
Got a function yet? No, because I didn't tell you anything about the [math] a_k [/math] other than they they contain something.

Now if I were to say that the element of each [math] a_k [/math] have a strict order attached to them, you could actually go on. E.g. if I were to tell you all [math] a_k [/math] hold natural numbers, or if I were to tell you the [math] a_k [/math] hold tuple of fruits and no two tuples of [math] a_k [/math] have the same size. If you know anything about the [math] a_k [/math], you have a chance of going on. If I tell you all [math] a_k [/math] are orderable finite sets, then
>choose the smallest element from each [math] a_k [/math]
is one way to construct a choice function.
But in general, you can make up a context where you have nothing to work with.

The axiom of choice says "well, we're just gonna axiom you can say 'muh f' and go on with your proof."
A constructivist would say here's where you need to stop due to lack of info.

>>10093115
ad:
* maybe now it also makes sense why the axiom of choice is equivalent to the well ordering theorem, and zorns lemma, and Hausdorff principle
>Suppose a partially ordered set P has the property that every chain in P has an upper bound in P. Then the set P contains at least one maximal element.
>In mathematics, the well-ordering theorem states that every set can be well-ordered.
>The Hausdorff maximal principle ... states that in any partially ordered set, every totally ordered subset is contained in a maximal totally ordered subset.
It's all cheat codes to make sets have some choosing structure even if you actually don't know.

* also worth pointing out that postulating some material set theory - e.g. where you force people to build everything from the empty set and then some super-sets, this gives you already a lot of necessary properties. Zermelo Freankel actually in itself wont talk about sets of fruits or humans, just about sets or sets of sets (with not infinite decending ones). This sort of boundary condition e.g. makes it possible that if I give you a finite set of things of arbitrary size, then the corresponding choice function can be shown to actually exist in this context. But it's not the case for infinite sets of sets.

Pic related is a weaker form worth considering

>>10091594
Determinism is true but thermostats can still control the temperature.

>>10091675
>Assuming the roots of a quintic exist, despite the fact that we can't ever find it.
this is you

>>10093948
constructive existence of roots =/= can write with radicals

>>10093954
yeah but you can't actually ever FIND the root so why do you know it exists?
I mean, just like how you can't actually pick something precisely from each uncountable set, how the hell are you going to pick a root of a quintic without precisely describing it to me? some bullshit, i bet. you still can't show me that the root you find evaluates to zero.
sure, you can claim that there is one there and prove that it exists, just like i can prove to you that there exists something in an uncountable set (by referring to the definition of "uncountable set"). But that doesn't mean you ACTUALLY found a root, since I never got to plug anything into the quintic and see that it evaluates to 0.

>>10093115
Based and redpilled. Thanks for the great explanation.

>>10093963
This shit needs to be moved to the philosophy department,

>>10093963
x^5 = 0
x = 0

quintic solved
get fucking dunked on nigger

>>10091813
The whole point of AC is for when you don't have a way to construct such a function.

So I am working on building my own roll stabilizers for my boat (displacement). A few naval architecture books recommend using a Naca 0010 profile for control surfaces as it offers decent lift at speeds under 18 knots without offering too much added drag. I've been looking at some of the graphs for this profile ant the stall angle is around 5 deg.

What is confusing me is that the critical angle of attack is usually around 15 deg still. Does this just mean I'm getting a bunch of layer separation/turbulence around the stall angle? And that as I move away from the stall angle things become less turbulent until I hit my critical AoA again? The foil would more than likely be operating in the 3 to 10 deg range most of the time, and I don't want it to be stalling out during that time.

I guess my question is how can the stall angle be less than the angle of attack? And past that, do any naval architects or aero engineers agree with the Naca 0010 recomendation? My vessel's working speed is 8 knots, and from the dimensions I want to use for the foil an Re of 500k-1000k.

>>10094071
You know what I'm talking about you fucking piece of garbage. Accept choice. ACCEPT IT!!!!

>>10091692
Imagine having an IQ this low.

>>10094148
>he thinks 17th century mathematicians used the same conventions he learned at community college in current year

>>10091706
draw some infinite collections. ill wait.

>>10093963
Newton Raphson method. Inb4 boo hoo it's not in the form of 4-5 elementary operations I like. We can define anything as an elementary function. Guess what, you wont ever calculate ln(3) or sin(1) exactly either, we just have methods of generating arbitrarily precise floating point numbers for this shit which is all that matters.

If it's good enough to take astronauts to space or design microprocessors with, it's good enough for your sorry ass.

>>10095925
>I can find numbers reasonably close to zero without ever finding the actual root and that's somehow the same as approximating real numbers in decimal writing
Reread the definition of limit.

>>10095925
>Guess what, you wont ever calculate ln(3) or sin(1) exactly either
You're on the right path anon, keep thinking onwards in that direction
:^)

That said, I'm fine with numbers that have a representation even if they can't in practice not computed in total decimal representaiton.

Where it becomes ugly is with numbers that are definable, and e.g. provably in some interval like [0,1], but have digits that are fundamentally uncomplutable. Specker sequence and so on.

virgin: axiom of choice

chad: transfinite induction