/sci/ - Science & Math

People are upset about unintuitive results, but they forget that nobody ever said those results were supposed to make sense if transposed into a physical context.

>>12426480
>nobody ever said
Except themselves of course, and that's why they're mad.
>b-but maths are supposed to only represent our physical world

For one, it's non-effective in the context of some moderate other set theory axioms, as evidenced by the fact that it implies Excluded Middle for all propositions.

>>12426098
Wtf is this?

our offspring will look at set theory in the same way that we looked at geocentric theory in the 16th century
it's simply one of these things where the propagation and enforcement of it is more important to the status quo, which is also invested in having it be accepted else the entire academia's credibility drops to 0 overnight, so the exploration of anything that deviates from it is forbidden and it is accepted as the ultimate unquestionable Truth of the universe

>>12426497
Consider [math]{\mathbb R}[/math] as the real line with its standard ordering, a 2-ary predicate
[math]x<y[/math].
This ordering is not what's called a total ordering, since it has no smallest element.

So let's instead let's consider a new, total orderings
[math]x <_T y[/math].
Without loss of generality, let [math]0\in{\mathbb R} [/math] be the smallest element, i.e.
[math]0 <_T y[/math]
holds true for all [math]y\in{\mathbb R}[/math].
Turns out that it's a metatheoretical result in logic that in the theory where you characterize [math]{\mathbb R}[/math], no such total ordering could ever be described.

The axiom of choice implies that for every set, there "exists" a total order on that set.
Roughly, it's an existence claim of the form [math]\forall x. !\exists y[/math]

>>12426497
Consider [math]{\mathbb N}[/math] and the order given by [math]0[/math] (or [math]1[/math] if you want) as smallest element and [math]x<y[/math] taken to be true if there exists a nonzero [math]d[/math] such that [math]x+d=y[/math].

Now for something else, consider [math]{\mathbb R}[/math] as the real line with its standard ordering, a 2-ary predicate
[math]x<y[/math].
This ordering is not what's called a total ordering, since it has no smallest element.

So let's instead consider a new, total orderings
[math]x <_T y[/math].
Without loss of generality, let [math]0\in{\mathbb R} [/math] be the smallest element, i.e.
[math]0 <_T y[/math]
holds true for all [math]y\in{\mathbb R}[/math].
Turns out that it's a metatheoretical result in logic that in the theory where you characterize [math]{\mathbb R}[/math], no such total ordering could ever be described.

The axiom of choice implies that for every set, there "exists" a total order on that set. Uhhh.

Very roughly, you can think of it as an existence claim of the form
[math]\exists F. \forall A. !\exists B[/math]
The debate is about what we want it to mean for there to "exist" a function. E.g. a function is an assignment from a value A to a result B, should the existence claim of a function require from you that given A, you can ackshually characterize B or at least how to get to it in principle?

The situation gets tricky in mathematical theories with mere extensional equality (like virtually all set theories), since comparing inputs and outputs itself becomes an intricate task.

[math]\exists![/math]

>>12426516
Sorry I don't speak autist.
What is this good for anyway? Tell me there's at least 1 (one) application and you weren't just fucking around with all that research money we gave you.

>>12426544
Engineering.

>>12426544
Math and science should not be guided by application. Stop subverting the academic institutions with American imperialist ideology.

>>12426098
Because it's false,

>>12426585
>American
>imperialist
>>12424643

>>12426516
Repeat that, but in English

>>12426585
What a fucking disappointment. If I were your father I would've hit you as hard as I can until you snap out of this faggy bs.

>>12426585
I bet they said the same thing 2000 years ago when they invented the steam engine and did absolutely nothing worthwhile with it.

>>12426497
Axiom of choice

>>12426544
Computers
Most of logic, even over uncountable sets, ends up having application to computing and proof systems. LEM and axiom of choice are important for those

>>12426544
>we gave you
Don’t kid yourself /biz/tard

>>12426499
based schizo
no reputable mathematician thinks set theory is unquestionable
so either you are schizo, retarded, memeing, or are butthurt about set theory and trying to cope

>>12426516
So y is the sum of all nonzero numbers that fulfill the x<y condition? Sorry if thats a stupid question, im more of a biochemistry guy

>>12426516
A total order says nothing about a smallest element.

>>12426098
Because low probability duh

>>12426098
Because of people like him >>12426516 who can't explain something in it's simplest forms and thus making it seem like its some confusing nonsense that's useless.

>>12427099
you're right, well-order

>>12427129
I don't know what can be made simpler without just being more vague, but I'd like to see.
You mean I shouldn't have used the word "extensional" in the last sentence? The rest seems elementary to me.

>>12427088
Not sure where you bring a sum into the picture

>>12427153
>I don't know what can be made simpler without just being more vague, but I'd like to see.
If there are multiple sets, then the axiom states that you can choose one thing from each set.

>>12427171
That's fair enough, but it's informal and doesn't make so clear why some people would reject it.
OP got the pic from the Wikipedia page, so he's probably that far.

>>12427153
Your explanation was fine, this thread is full of retards. Or maybe it’s just one samefag shitting up the place.

I think most mathematician only have an intuitive idea what a function is or should be.

Does a Scholze think of a function as a set of pairs?

>>12426098
it assumes you have an algorithm for picking things when you don't. for most applications that it gets applied to, this isn't an issue as an algorithm can easily be constructed. but in the wild world of math, it seems like a cavalier proposition.

consider computable numbers. they form an almost-nowhere-dense subset of all numbers, even though they are the only numbers we really use. in a similar way, i can imagine the axiom of choice is only valid for an almost-nowhere-dense set of mathematics. it just turns out that set is what we primarily discuss in math.

or at least that's my uneducated take on it

>>12426098
Are the balls hidden or not hidden

>>12426516
What the m&ms in the picture have to do with that?

>>12427369
m&m's?
It's just some random nerd chick

>>12427406
Not that porn actress, op's photo.

>>12427192
Not true. Functions are perfectly well defined.
A function is a relation such that each element in the domain maps to exactly one element.
A relation [math] R [/math] from [math] X [/math] to [math] Y [/math] is a subset of [math] X \cross Y [/math] and we say that [math] xRy [/math] if [math] (x,y) \in R. [/math]

>>12427414
It's choosing m&m's from a collection of baskets.
The axiom of choice says there is a function that does this choosing.

It implies well-ordering of everything by taking any set X and, roughly, keep picking choosing elements from it (in the proof the subsets of X are used) and declaring them smaller than the element you find in the next step.
Conversely, if a well-ordering exists for all sets, you can "pick the smallest" element of each basket.

Well-ordering is one simple way of the strengh of the axiom: Whatever collection B you're thrown at, you can say
>I choose its "smallest" element b
and thus you're "functionally delivered" an element b.
This thereby happens axiomatically and you're not required to know anything about B other than that it's a nonempty set.

>>12427323
>i can imagine the axiom of choice is only valid for an almost-nowhere-dense set of mathematics.
That's an interesting perspective.

>>12427414
It's choosing m&m's from a collection of baskets.
The axiom of choice says there is a function that does this choosing.

It implies well-ordering of everything by taking any set X and, roughly, keep choosing elements from it (in the proof the subsets of X are used) and declaring them smaller than the element you find in the next step.
Conversely, if a well-ordering exists for all sets, you can "pick the smallest" element of each basket.

Well-ordering is one simple way to understand of the strength of the axiom: Whatever collection B you're thrown at, you can say
>I choose its "smallest" element b
and thus you're "functionally delivered" an element b.
This thereby happens axiomatically and you're not required to know anything about B other than that it's a nonempty set.

Here's the more accurate depiction of the axiom of choice

Here's a more accurate depiction of the axiom of choice

>>12427516
>Reach into each jar, and pick out exactly one of its objects.
Nonsense!

>>12427482
How do you know its the smallest m&m

This seems like a jump

>>12427323
that's the gist of it for me too.

>>12427581
Because it is defined that way.

>>12427607
But who says wtf?

What is the mechanism that defines it as minimum? Is it just someone declaring it as the case? Wtf?!

>>12427616
He literally explained it in his post, you retard. The well-ordering theorem implies that every set can be well-ordered. The rest follows.

>>12427651
So its just that the jar before is smaller than the jar after? The m&m in jar 1 is not necessarily the smallest m&m in jar 1? Its just the smallest in the next jar?

>>12427660
No. The well-ordering theorem implies that each jar has a well-order, so each jar has a smallest element. You pick that element from the jar. Do this for all the jars.

>>12427581
>How do you know its the smallest m&m
These principles (Axiom of Choice, Well-ordering theorem, Zorns lemma) are postulating that a smallest exists and what exists can be "taken".

The non-effective nature of those axioms is why these things are not universally liked everywhere.
E.g. if you want to see Nazis metaphorically stabbing the backs of their Profs, see

https://en.wikipedia.org/wiki/Brouwer%E2%80%93Hilbert_controversy

>>12427662
Oh for fucks sake. So you know what is in the jar?! Goddammit!!!!!

>>12427699
I was under the impression that it was a random pick

>>12427699
>>12427703
>I was under the impression that it was a random pick
Me as well, isn't the axiom that the cross product of non-empty sets is non-empty?

>>12427703
Given that there certainly is no formalization of "random pick" at the low level of logic with at this point, it's certainly not a stringent reading.
I won't stop you from interpreting the logic symbols however you like.
But it feels very operational ("random picking") and so to say "a function exists that maps A's back into B's" would mean that "from a collection of A's, we can always grab into and randomly pick" or however your semantics are, seems to complicate the thing.

>>12427712
>Cartesian product
Yes, or geometrically speaking, every bundle has a section.

Note that "Cartesian product
[math] \prod_{i \in I} X_i = \left\{\left. f: I \to \bigcup_{i \in I} X_i\ \right|\ (\forall i)(f(i) \in X_i)\right\} [/math]
is a construction involving indexing sets and the second "non-empty" does not just mean "is not the set that can't have any elements" but instead "there exists a value in the image"

>>12427750
A random pick is taking 1 unit from a set of units with no bias in which unit is picked.

Apparently, you know what is inside the jar. Seems absolutely pointless. Who cares.

>>12427750
>>12427755
Please tell me

Do you know it is the smallest m&m before you grab it from the jar

At what point is the m&m known to be the smallest? The entire time? After it is removed? Fuck fuck fuck

>>12427755
Forget m&m's.

The starting position 1) is that you adopted some set theory axioms.

Your second step 2) that you have a collection X (your theory proves that X is a set, in whatever way) and you know that X is not the empty set (your theory proves that X is not the set that has no elements).
You know nothing more.

The next step 3) is to adopt the axiom of choice. It and the other axioms imply that every set of your theory is well-ordered, i.e. you can prove that a well order exists. You don't know anything about the well-order other than that it's a well-order on X.
You can now use this well-order to characterize the smallest element of X with respect to the well-order. Call this element x0.

You have assigned the set X some other set x0.

About the set X you know nothing apriori. About the set x0 you know that it's in X and that it's the smallest set with respect to some well-ordering.
You don't know more or less about the situation.

So now to your question
>At what point is the m&m known to be the smallest?
After adopting the new axiom, at the point where you consider a particular well-ordering of X, you know that the smallest element of x with respect to that well-ordering is the smallest.

>>12427763

Forget m&m's.

The starting position 1) is that you adopted some set theory axioms.

Your second step 2) that you have a collection X (your theory proves that X is a set, in whatever way) and you know that X is not the empty set (your theory proves that X is not the set that has no elements).
You know nothing more.

The next step 3) is to adopt the axiom of choice. It and the other axioms imply that every set of your theory is well-ordered, i.e. you can prove that a well order exists. You don't know anything about the well-order other than that it's a well-order on X.
You can now use this well-order to characterize the smallest element of X with respect to the well-order. Call this element x0.

You have assigned the set X some other set x0.

About the set X you know nothing apriori. About the set x0 you know that it's in X and that it's the smallest set with respect to some well-ordering.
You don't know more or less about the situation.

So now to your question
>At what point is the m&m known to be the smallest?
After adopting the new axiom, at the point where you consider a particular well-ordering of X, you know that the smallest element of x with respect to that well-ordering is the smallest.

>After it is removed?
There is no removal, that's just for you to visualize some situation. You're doing formal logic here and speak of the existence of sets and mapping. There's no "process" involved in your sense.

>>12427703
>>12427755
No! You do not know what is in the jar. You just know that the jar is not empty, and if you assume the well-ordering theorem, then you can select the "smallest" element from each jar. The statement is about existence, so the point is that you CAN do so without actually knowing what is in the jar.
>>12427763
You just know that you can order the elements in each jar such that there is a least one. You do not need to actually know what the precise elements are; just that there is a least one. Select this least element.

>>12427799
Thank you for taking the time I have a brain problem and implied definitions of things throws major curveballs in my world

I will read through this and respond with questions later just wanted to say thanks now

of X

>>12427815
How does one well-order the interval (0,1) in R, or even in Q for that matter?

>>12427815
>You do not need to actually know what the precise elements are; just that there is a least one

Let me play the autist here and make the point that you're not required to know that there exists an element in X upfront, you merely need to know that X is not the set without elements.

I.e.
[math]\neg\big( \forall x\ \neg (x \in X)\big)[/math]
vs.
[math]\exists x (x \in X)[/math]

>>12427815
Okay so you assume jar = 2 and therefore 2 = lowest

What if you make a second pull? Either jar is empty, or it has a higher or lower number? The theory only allows one pull per jar? Dumb question but can a set just be 1 unit? Wouldnt a set be a minimum of 2 units? Wait a set can be empty? Alright nevermind answered myself there.


By "well ordering" are you just implying it would be impossible to pick from the jar anything but the smallest? Like its a "smart jar"?

>>12427856
(0,1) has no smallest element

>>12427864
I know, I'm asking if and how the Axiom of Choice can apply to (0,1)?

>>12426486
>>b-but maths are supposed to only represent our physical world
KEK
mathlets btfo's

>>12427876
If you want a choice function, you need a set of sets (and I don't think you want me to view the numbers as sets).

If you want to use the implied well-ordering theorem, then, adopting the axiom of choice, I can ensure you that there's a well-ordering on (0,1).
Don't ask me to describe a well-ordering on (0,1), that's provably not possible.

>>12427893
Take Vitali's set of sets, for example. Why is that considered a valid criticism of the Axiom of Choice, if the Axiom of Choice can't be applied to open subsets of R?

>>12427893
Wow math is so bullshit, things only work under specific circumstances despite the rules implying universality

>>12427908
The requirements are fully formalized, you're not led on

>>12427906
I'm not sure what you mean by can't be applied?

Anyway, here's the equivalence
https://proofwiki.org/wiki/Well-Ordering_Theorem_is_Equivalent_to_Axiom_of_Choice

>>12427906
>Why is that considered a valid criticism
People often choose axioms to correspond to intuitions but end up not having intuitions for the consequences of the axioms - so they tweek the axioms or the logic

>>12427942
So by the Axiom of Choice, we can select a smallest element from (0,1) even if we can't know what the element is. That's the gist of it?

>>12426098
Because mathematical objects are not sentient, and as such, they cannot choose!

>>12427942
Im sorry I guess I misunderstood the purpose of math

I thought math was universal rules that describe reality not just a bunch of pretend stuff that only works within a pretend framework

>>12427750
>geometrically speaking, every bundle has a section.
Redpill me on this, I'm unfamiliar

>>12427958
Yeah I mean that's true but the fact that it's the smallest is not really a big point.
Let O be the well ordering on the reals. Pick -17 and define a well-ordering O's by letting -17 be the smallest element w.r.t. O' and otherwise use O, except you leave out wherever -17 was in the old O.

But yes, it's a strong existence claim.

An ostensibly weaker existence claim would e.g. be that if you got two sets, then the intersection is a set as well. That's an implication of the Separation axiom, but it has a feel of nothing previously unknown being added.

I've lost all interest in math now. I wanted math to describe my world, not make arbitrary fake games that tell me nothing about my world.

>>12428050
sounds like you wanted physics all along friend. Math is all about imaginary autism games, it's one level above philosophy at this point

>>12428068
>Math is all about imaginary autism games
Based math

>>12427981
You got a block E hovering over a sheet B in the sense that you're given a projection p. Since E might be fatter than B, p need not be injective.

For every b in B, you can characterize a fibre in E by [math]F_b=p^{-1}(b)[/math]. This fibre [math]F_b\subset E[/math] exactly the collection of elements [math]e\in E[/math] that p takes down into the same b. E.g. [math]p(e_1)=b[/math] and [math]p(e_2)=b[/math] means [math]e_1[/math] and [math]e_2[/math] were hovering right vertically over b. This works for any b in B.

The well-ordering theorem says every fibre [math]F_b[/math] can be ordered. Let's phrase it as you being able to identify how far away any e over b is from b. I.e. if [math]e_1<e_2[/math] w.r.t. your well-order, you might say [math]e_1[/math] is closer to b than [math]e_2[/math].

You can now map each b to it's closet e. I.e. you have a function going into the opposite direction of p. This new function maps B into "the floor" of E.
Throwing away the "closeness to B" interpretation, in any case the choice function going upwards that now exists is just a section of the E block.

This also makes it clear that there will be a lot of topoi where choice fails. You just manage to axiomize a category with map p:E->B between two objects that can't be inverted in any way. This invalidates choice in that category.

>>12428050
What games exist is also a feature of the world.
Also, the possibility for formalism doesn't necessitate that Platonism is wrong. It might well be that we can pin down a truer math. (Gödel thought this way)
Also, much of math is clearly applicable in a pragmatic sense at least.

>>12428073
If i wanted to play pretend autism games i would just play dungeons and dragons

Im so upset with this realization of the uselessness of math

>>12428082
>Im so upset with this realization of the uselessness of math
Everything is useless. Physics is a subset of mathematics dealing with physical reality

>>12428074
Yes the capacity to create and the games that exist are a feature of the world, but at this degree you are isolating so acutely to where you are fixating so deeply on .0000000001% of the elements of the world that the energy cannot be argued as being utilized optimally

>>12428086
Thanks for being a patient dude with me, I guess physics is more my thing. I cant believe it took me so long to realize math is just making games.

>>12428074

(The choice function is technically from the Cartesian product of all fibres into the respective fibres. You posulate that every fibre can be assinged one of its elements.)

>>12428103
>math is just making games.
That's but a philosphical position

Read
https://plato.stanford.edu/entries/formalism-mathematics/
Also read the Stanford Encyclopedia for physics articles desu.
Be informed - it's free

>>12428103
>I cant believe it took me so long to realize math is just making games.
That's the fun part. I do math because I have a blast doing it. I myself am a physics major, but I enjoy math more.

>>12427469
All you did was delegate the inherent ambiguity involved in the concept of a function to the notion of a "subset". You haven't solved shit.

>>12426591
its also true

>>12427192
scholze thinks about functions the same way everybody else does

>>12427978
math is both

>>12427856
The axiom of choice allows you to construct one.
AC, well-ordering theorem, and Zorn's lemma are all equivalent given the usual axioms, i.e. the axioms of Zermelo-Fraenkel set theory, but none of them can be proved or disproved (I think as far as we know) with the axioms of ZF. This means if you assume any one of them, you get the other two, but what is required is your assumption of one of them.

>>12427859
Honest question: do you have brain damage?

>>12427978
>I thought math was universal rules that describe reality
You're actually retarded if that's what you thought.

>>12428098
Jesus christ, learn sentence structure you subhuman.

>>12426516
Wow, why would you try explaining this when you don't understand what a total order is or why your example doesn't have anything weird to say about the well-ordering theorem?

>>12428518
[math] A \subseteq B \iff (\forall x\in A, (x\in B)). [/math]

>>12429712
don't bother. he's gonna ask you what a "set" is and then he'll start rambling about strokes on a blackboard.

Is wildberger unironically rihgt?

>>12427856
For Q, it's trivial: just pick the inherited lexicographic order for Z × Z over the reduced forms of fractions. More generally, if you have a countable set you can always well-order by looking at its image in [math]\mathbb{N}[/math].

>>12429711
"Total order" should be replaced with "well-order" in that post.

>>12429712
So you're definition of a subset is a certain thing that satisfies certain property. What is that thing?

>>12429778
Always has been.

>>12427978
>I thought math was universal rules that describe reality not just a bunch of pretend stuff that only works within a pretend framework
oh my

>>12426492
Excluded middle is true for all propositions. Constructivists get off the board.

>>12426098
Because it's sometimes impossible to establish any rule based on which you select the elements

>>12430382
[math]x\ \text{s.t.}\ x\in S [/math]

>>12430348
"CH or not CH" is false. Neither CH, nor its negation is true.

>>12427978
There is no such thing as "reality"

>>12430388
CH is obviously true.

What if I grab an m&m and it isnt the smallest?

>>12429812
How does that return the smallest element of Q in (0,1)?

>>12428050
logic and science are just a atheist narrative like any other and it is not proven at all it investigates anything

This is because formalized science is based on logicized maths, hyped as ‘’the language of the universe’’ by posci addicts lol

and logicized maths is based on logic

all mathematicians are logic babies addicted to ZFC and they are all platonist, ie ‘’numbers are real bro not social construct’’.

By the way truth is not found in logic. Logic is just a field by autistic pedants about well formed formulas and valuations, ie sending a formula to 1 or 0 and asking what are those valuations which are stable under inference rules. Zero truth in this, especially truth in the casual sense. Tarski truth is moronic, meaningless. Peak atheist. Just like there is no truth in science, just some stats and a stat convention for saying ‘’if p value is XXX then the result is’’true’’’’

This is why science is shit for politics and even for daily life. At best scientists can come up about some stats about some formal system. Like ‘’this material has such and wear and tear, therefore our backlog of such conditons lead to 60% of breaking in the next year’’ That’s the pinnacle of the scientific claim and all their claims remain phrased as uncertainty.

>>12430522
(0,1) has no smallest element and neither has the intersection of (0,1) with Q

>>12430522
Use AoC to obtain a well order O on (0,1) and then use O to obtain the smallest element w.r.t. that well-order.
Simple as.

>>12430386
>x sich that x is in S
That's even stronger than AoC

https://en.wikipedia.org/wiki/Epsilon_calculus

>>12430522
The set X is a member of its powerset PX.
By the axiom of choice, there's a choice function on PX (setminus emptyset) that maps every set S in PX into an element x in S. Take S=X.
So we obtain a bottom element x in X.
Just do this a transfinitely number of times.
Problem?

>>12430396
prove it

>>12430525
stale pasta

>>12430534
>>12430544
>>12430564
Thank you, I'm actually curious how the mapping works, not trolling. I know that there's no smallest element in (0,1) and I was trying to figure out how the Axiom of Choice "reaches" that element, but doesn't allow us to "see" it, so to speak.

So 0 is not a value? Why not represent it as [1] instead of [0, 1]?

0 certainly isnt higher than 1. So it must be lower. If its not the lowest value then it cannot exist.

>>12430605
You should note the distinction of the meaning "smallest" in the two cases. Under the usual order of real or rational numbers, <, there is no smallest element. We didn't need to choose such an order though, it's just the most useful one and the one that makes the most sense. Assuming that we can well-order any set, then there is an order on (0,1) which does have a "smallest" element, but this element is not the smallest in the sense of the usual order; it is the smallest only in the sense of our new order.
The example >>12429812 gave demonstrates this concept, since via his ordering on the rationals, you get 2/3<6/55, but in the usual ordering of the rationals, obviously 6/55<2/3.

>>12430625
because (0,1) means all the values between 0 and 1, not including 0 and 1.
[0,1] means all the values from 0 to 1, including 0 and 1.

If you want the smallest number in (0,1) traditionally you prove it doesnt exist by contradiction. Let x be the smallest number in (0,1). x/2<x, and x/2>0, so x is not the smallest number in (0,1), a contradiction.

>>12430585
Since the continuum can be well ordered, it is in one-to-one correspondence with some ordinal number. Therefore, there is a smallest ordinal number whose initial segment gives the cardinality of the continuum. I leave the details up to you.

>>12430664
>Claim CH is obviously true
>Can't prove it
Awww its okay bud :) you tried

>>12430672
He just did

>>12430632
So if I'm understanding that correctly, by the Axiom of Choice we can select a smallest index value in (0,1) but not a smallest number value. Did I lose anything in translation?

>>12430703
Yeah. The axiom gives more than just that, but you are pretty much spot on.

>>12430674
>He just did
>He
Nice try samefag

>>12430773
I still dont get what magic makes it the lowest numbers outside of a literal choice inside your own mind to do so

Am I constrained to the lowest number? Can I do this with middle number or highest number?

When I do 10/2 I have no choice but to have the result of 5. I have no choice in this matter.

It seems like the lowest number thing is totally up to just deciding to do it.

>>12430780
Nothing. It's exactly that; a literal choice. Objections to AC tend to occur when infinities are involved, as there is no algorithm for choosing such an element. So the axiom of choice is usually used to state the existence of a certain object even though you cannot explicitly construct it.

>>12430789
Yeah, you could just as easily choose the highest number.
If "middle number" is well defined and corresponds to exactly one number, then you could choose that one as well. The point is that such a choice exists.
Also, keep in mind the lowest number does not mean lowest in the usual sense. See >>12430703

>>12430795
Thanks, apparently I take things even more literally than mathematicians. I should just kms.

>>12430803
So why would anybody think that they dont have the ability to.... Make a choice...??

>>12430804
No. It's a very subtle point, and it's hard to grasp the subtleties when you are used to thinking in terms of finite or tangible objects. From a naive standpoint, the axiom of choice is just trivially true.

>>12430814
See >>12430795
Some people would argue that something exists only if you can construct it, or design an algorithm which would allow you to do so. Invoking AC can often lead to pathologies, such as Vitali's set or the Banach-Tarski paradox - the latter telling you that there exists a way to "cut" up a sphere in a certain way, rearrange the pieces and put them back together in order to obtain two spheres, each the same dimensions of the original sphere.
Note I write "cut" in quotations, because you're not so much cutting the sphere, mores o decomposing it it a very weird way.

>>12430773
Okay cool, thank you.

>>12430836
How would that be a paradox

Also thanks I get it now I think

>>12430842
It just contradicts basic intuition is all.

If a thing exists, it exists.
It does not matter whatsoever if I can construct it or not. My limitation is not relevant to the things existence.
I hate constructivists.

>>12430866
You ought to open your mind, or you won't make it.

INFINITE SET

>>12430904
Isnt 1 the lowest of any infinite set?

>>12430842
It's a paradox because our intuitive understanding of volumes dictates two things:

1. if you consider some rigid motion (i.e. translation or rotation) and you apply it to some set, the volume doesn't change
2. if you partition a set into several non-intersecting parts, then volume of the whole is the sum of volumes of the individual parts

Banach-Tarski construction takes a solid ball (with a definite volume), divides it into 6 or 7 parts (IIRC), rotates and translates each part, and the result is an object with larger volume. this contradicts the two properties I've listed.

resolution is that the ball is partitioned into extremely weird sets which don't have any meaningful notion of volume. therefore property 2. doesn't apply, because there's no "sum of volumes of individual parts", one side of equation is simply undefined.

>>12430396
CH is obviously false since the Proper Forcing Axiom is true.

>>12430967
Ah. How about the inverse? Anyone done this and been able to end up with a lesser volume?

>>12430981
I mean if you care only about the principle, you can start with two balls and do the exact same thing reversed to get one ball

>>12430348
So I know I am just babbling on the internet, but the whole AC controverse is hard to understand from ZF alone.

The problem is not so much AC. In reasonably strong intuitionistic systems such as Martin-Löf type theory, AC is actually derivable. Not even an axiom, just a consequence of the rules. Furthermore, MLTT satisfies a property known as canonicity, which is a way to phrase it is constructive.

The real issue is AC + excluded middle, because excluded middle changes the meaning of the existential connective of the logic. You can state AC as
[math](\forall x : A, \exists y : B\ x) \rightarrow \exists (f : \Pi x : A. B). \forall x : A. B\ (f\ x)][/math] and from there it's pretty clear that if you change the meaning of [math]\exists y : B[/math] then you're in trouble. I don't want to write a full-fledged blog post, but to insist, the real issue is AC + EM, not AC alone.

>>12431001
I meant decreasing the volume but increasing the balls

Also: fuck you retarded platonists.

>>12431007
no idea

>>12430967
It divides it into 4+1 parts.
>>12430981
Yeah, you can. The way that the decomposition and rotation / translation is done, you can invert all the steps.

>>12431001
It takes a bit more care than that. You need to check that the rotation is invertible.

>>12431007
Yes. You can do that. It's easy. Just decompose the sphere into line segments emitted from the origin, and cut them all in half, and then join half of them together in one sphere, and half together in the other sphere.

>>12431037
Sorry for being retarded but isnt that what you do to increase the volume?

how do so many moronic undergrad engineering pseuds make it onto /sci/? this isn't another anime board.

>>12431037
this doesn't work because the partition has uncountably many parts. I haven't said this, but there must be countably many parts at most, otherwise you can just partition into points and appeal to cardinality.

>>12431054
this, the point of banach tarski is not "cardinality of infinite sets is funny!"
the point is "unmeasurable sets are so nasty that they basically have many different volumes all at once, depending on how you put them together."

>>12431067
I'll never forgive vsauce for not mentioning any measure-theoretic properties but instead rambling about cardinality for 10 minutes. vast majority of people who watch the video think it's just an overcomplicated exploit of "infinity + infinity = infinity" and one can't really blame them.

>>12431046
The radius of the smaller spheres is half of the radius of the larger sphere. The volume goes with the radius cubed, so the volume of each of the smaller spheres is 1/8 of the volume of the larger spheres. The volume therefore decreased. The reason for this decrease is that we decomposed the sphere into uncountably many line segments, which all have volume zero. Whenever you do such a thing, you get nonsense like this volume change.
What we did to increase the volume was to decompose the sphere in to 4 parts, call them A,B,C, and D, (plus the center of the sphere) so that when you rotate B in a particular way, you get B U C U D, and when you rotate D in a particular way, you get A U B U D. You then combine these rotated versions of B and D with the A and C, and you see that you have obtained two copies of each of A, B, C, and D. You just need to take some extra care with the center of the sphere.

>>12431054
Oh, true. Thanks.
I'd imagine solution then might be to find a way to cut a sphere up into two smaller spheres so that the volume remains the same (this seems reasonable enough) and then apply the inverse of Banach-Tarski to get a sphere with half the volume, and then apply step 1 again.

>>12431087
To be fair, what I wrote had nothing to do with Banach-Tarski, it was just my lack of understanding in measure theory. I think most of what I've said regarding Banach-Tarski has been pretty accurate.

>>12431087
i mean it's a good intuition, and honestly it's related since it's a residue of the choice that measure theory should be beholden to countable operations. it's tough to help someone understand the result otherwise. the best thing you can do is construct vitali sets by taking quotients by rational angles or something and showing how suddenly you need infinitely many things, all with the same length, to add up to something with finite length. but then going from that to the precise statement of banach tarski is also not obvious unless you're used to measure theory.

for others in this thread, i think the best way to describe it is "measure theory makes a sacrifice in order to differentiate between the sizes of objects which have the same cardinalities, and that sacrifice includes things like the banach tarski paradox."

>>12426098
After looking at the wiki page, I don't see what the fuss is about. Is it that finite sets are fine to choose from but infinite isn't? Because if you can choose the number 7 out of {1,2,..9}, it should make sense that you can choose 7 from Z+. Is it that choosing from a set, infinite or not, is "wrong"? Because a human has to decide "stop at the 7th element"? In that case, I'd say "the 7th element" is an object in the realm of forms and a universe state where it is viewed by a consciousness is just one state, so it exists. The fact that a human has to generate action, and is pushed by determinstic forces to view it through "choice" is just how the math views math. No different even from a human perceiving numbers in the first place, or for "choosing" to construct 2 from 1+1 and 3 from 2+1....

>>12431112
I might be misinterpreting, but I don't think the construction of the Banach-Tarski paradox has much to do with measure theory. It's just a bit of group theory and that's about it. It's the resolution to the paradox that invokes measure theory. Am I wrong?
>measure theory makes a sacrifice in order to differentiate between the sizes of objects which have the same cardinalities
I'd add to that that the sacrifice is most commonly taken to be that there are certain objects (uncountably many, I think) which cannot be assigned a size.

>>12431137
Well the example you gave was of a set which had elements you can describe. What if you are unable to describe the elements of a set precisely?

>>12431138
>It's the resolution to the paradox that invokes measure theory.
Yes, obviously. The reason the Banach Tarski "paradox" is a paradox at all is because people's intuition about the volumes of objects is not correct. What the Banach Tarski paradox does is expose why volumes are nonintuitive. Who gives a shit if the mechanism is group theory. The result is a philosophical statement about measures.

>>12431138
without invoking volumes, Banach-Tarski paradox is not a paradox in the first place. just a counter-intuitive construction.

>>12431155
>Who gives a shit if the mechanism is group theory
me. :(

>>12426585
>Stop wanting things to be useful
No.

>>12431177
Then you're never going to make it in science.

>>12431154
In that case, it becomes reality branching. "A is a set with N unknown elements" then you have infinite realities with variants of sets that satisfy A, and you perform the choice on all of them, then download one into perception when more info is given.

>>12431008
What's retarded about platonism?

>The Axiom of Choice is obviously true, the well-ordering principle obviously false, and who can tell about Zorn's lemma?

>>12426585
You sound like one of those decolonize science retards

>>12431177
Math being useful is one thing, and it's nice.
Requiring from mathematicians they be useful will lead to some things never be discovered.
But it's a double edged sword either way.

>>12431464
It's a belief grounded in German logic from the 30's of the last century. There is a thing called "computer science" that happened since, and discoveries like the Curry-Howard correspondence provide serious evidence against platonism.

Let me highlight that platonism is not a scientific claim anyways, so it cannot be refuted, neither through experiments nor through pure reasoning, so it's essentially an epistemological rambling that should be discussed only after a certain number of beers.

>>12427536
>one object from every jar!
ahhh.. Ahhgh HUHhEHHGH I"m GOIN CRAZZyzyYYYYY

>>12431918
>Curry-Howard correspondence provide serious evidence against platonism
How?

>>12432008
Because different programming languages make different assumptions about their universe of discourse? Like, stuff written in C expects mutable memory, but depending how it's written it can be compatible with threads or not. Contrarily, Haskell assumes that all functions are pure. It's stupid to say "programs exist in the real world" if there are actually several incompatible notions of programs. That's about as stupid to say "CH is true!" because it depends on the objects you're manipulating.

And then you get all the emulation gizmo, like there are different kinds of hardware than can be run one atop of the other in a way reminiscent of conservative extensions.

In the end, the only thing on which languages / logical systems must agree are purely existential statements over natural numbers. Because those can be extracted to actual integers and tested, so if they disagree it means that one of your theories is inconsistent.

>>12432063
Ok and how is any of the shit you just wrote relevant to platonism?

>>12431918
Big fan of Curry-Howard, but Hegel is our last chance to meaning so we best not reject it.

>>12432089
Well he takes a pluralist stance and already said Platonism can't be decisively rejected.

>>12432063
Cont.

There are also interesting things to say about Turing-completeness, as a way to escape incompleteness theorems. People often overlook the requirement of decidability of proof checking in the statement of Gödel's incompleteness theorem. If you don't care, you can be complete, at the cost of replacing a proof by a semi-decidable computations. If you do that you essentially reflected the meta-theoretical "truth" in your theory, but instead of degenerate Tarski's models, you get a program in the end.

Programs that rely on Turing-completeness effectively perform what would amount to "experiments" about natural numbers in the real world. It might be the case that they never terminate, but if they do, you got a proof of what you were looking for. This gives even more credence to the fact that the only thing one cares about are purely existential statements about integers.

So, call me a neo-Kroneckerian if you will, but I'd gladly claim that "The universe created (existential statements on) integers, everything else is the work of your ambient programming language."

[math] |{\mathbb R}| = |\omega_2| [/math], so god will.

>>12432133
I see we're drifting more and more away from the topic of the thread.

>>12432118
And? He claimed he had serious evidence against platonism. I'm waiting to see it.

>>12432147
>I see we're drifting more and more away from the topic of the thread.
Nope, as a good mathematician I am generalizing the topic of the thread. Why only shun the axiom of choice when you can go full schizo on foundations?

>>12432172
>he accepts Powerset

>>12432133
> People often overlook the requirement of decidability of proof checking
which part necessitates it?

>>12432133
Ok and how is any of this evidence against platonism?

>>12432193
Have you ever written a program using several languages? If not, I am afraid that this argument is going to fly over your head.

>>12432187
You need to be able to enumerate valid proofs at some point, and if you don't have decidability of proof checking you can't do that.

>>12432213
I have used several different languages to write different programs. Not sure if I ever used different languages to write one program. Be kind and explain to me as if I haven't.

>>12432213
Not him, but I have.

>>12432213
>>12432133
Can you also explain why you reject Pi_1 statements about integers?

>>12432219
I'm far from an expert, but doesn't it suffice that a given theory extends a theory for which decidability holds, then? That is, if a theory is not fully, but sufficiently decidable for a Gödel sentence to be constructed?

>>12426544
why are you even here?

>>12426585
Based Platonic priest-king.

>>12432247
[math]\Pi_1[/math] statements are the prime example of the source of incompleteness of a theory. These correspond to the falsifiable statements of your theory, in a Popperian sense. You can make computational experiments to check that they hold for a finite number of instances, but never for all of them. So, in a sense, they are already too strong to form a common ground for "truth".

>>12432172
Every 'language', is just a virtual machine that you could implement as a hardware interpreter (computer). The difference in programs comes down to the difference in machine. There's no need for any PL or machine combined with its 'programs', since you can just write your logic as a boolean circuit and implement it as hardware directly.

You'll answer with a stupidly sophist statement like, "my point exactly, a boolean circuit can be mapped to the natural numbers", which is obvious because a circuit is finite. You haven't said anything profound.

Might as well say that the problem of trisecting an angle disproves Platonism because the programming language known as "compass and straightedge" can't do it. Holy shit intuitioconstructoprogrammivistoformalists are braindead. L*gic destroys brains

>>12434094
Good point. Doesn't Godel's Incompleteness Theorem imply that not everything that can exist can be logically constructed?

Guys, let this thread die.

At the very least, don't get astray with other topics lie
>>12434621
Also
>exist
>logically constructed
You'll only be a small bit smarter if you phrase your question in such loose terms.

>>12434645
>You'll only be a small bit smarter if you phrase your question in such loose terms.
Elaborate? I'm happy to learn

>>12431918
Computer science is a subset of math, not the other way around.

>>12426098

Because you can't mathematically prove or disprove it without assuming it (or assuming it's not true).

>>12434722
Maybe read the thread

It is literally picking and choosing, it is not some rule that manifests it.

>>12426098

>>12434673
>Elaborate
Gödels theorem is a mathematical theorem.
You say "not everything that can exist can be logically constructed". I don't think you'll get any undisputable such implications.

The former has mathematical implications.
The second is a common language sentence not refering to mathematical notions.