/sci/ - Science & Math

>>9715496

prove the countable AC "makes more sense". And sure there's uncountable sets. for example C. Uncountability is a fundamental concept in mathematics , and its for a reason.

Reminder that we don't know, and can't know, what the cardinality of so-called real numbers is, and saying that cardinality of R is continuum, which is defined as a cardinality of R, is just saying "the answer is whatever the answer is", so it isn't saying anything and is not proper maths.
How can you tell R is a valid mathematical object if it doesn't even have well-defined cardinality?

>>9715502
Firstly, AC leads to ridiculous results, like well-ordering theorem, Banach-Tarski paradox, existence of non-Lebesgue-measurable sets.
Secondly, there's no way to even theoretically choose an element from uncountable family of sets. For sets not greater than [math]\omega[/math] it's at least theoretically possible to finish the process in finite time, with uncountable sets it will always take infinitely long if choosing an element takes non-zero amount of time.
It's impossible to construct anything constructed by means of AC. We "know" there is well-order on reals, but it's impossible to construct it. We "know" we can cut a ball in five parts and reassemble them into two balls identical to the original one, but we can't possibly know how any of these parts looks like. AC brings nothing to mathematics, it just creates pathological counterexamples

>>9715496
>Uncountable sets are counter-intuitive
Only for brainlets.

>>9715503
>How can you tell R is a valid mathematical object if it doesn't even have well-defined cardinality?
Why is it not well-defined? What makes it less well-defined than the cardinality of the naturals?

>>9715510
>For sets not greater than ω it's at least theoretically possible to finish the process in finite time, with uncountable sets it will always take infinitely long if choosing an element takes non-zero amount of time.
Why are you invoking the meaningless notion of "time" into mathematics?

>>9715496
How are you so brainlet that you think uncountable sets are counter intuitive and dont exist

>>9715556
Are you serious? [math]\aleph_0[/math] is defined as sum of all finite ordinals, it's as well-defined a concept as it can get. [math]\mathfrak{c}[/math] is defined as [math]\mathfrak{c}[/math], which doesn't look like valid math to me.
The fact we don't and can't know for what ordinal [math]\kappa[/math] [math]\aleph_{\kappa}=\mathfrak{c}[/math] means it's not well defined

>>9715574
union of all finite cardinals*

>>9715574
> ℵ0 is defined as sum of all finite ordinals, it's as well-defined a concept as it can get.
That's not how it's defined, but if that's as well-defined as it can get then the definition of c as the cardinality of the reals is as well defined as it can get. Or we can define it as the cardinality of the powerset of the finite ordinals if that's not good enough for you.

>>9715574
>The fact we don't and can't know for what ordinal k [math]\aleph_{\kappa}=\mathfrak{c}[/math] means it's not well defined
Why?

>>9715599
>That's not how it's defined
Then how? And if you say "as the cardinality of naturals", they're equivalent.
We know what an ordinal number is, we know what a limit ordinal is, and about continuum? We know nothing, we only know that if it exists then it's strictly greater than [math]\aleph_0[/math]

>>9715496
>believes in countably infinite sets but not uncountably infinite ones
So you reject the power set axiom?

With AC I think OP might have a point (in the end it's preference), but I see no reason whatsoever that uncountability is a meme.

>>9715628
>Then how? And if you say "as the cardinality of naturals", they're equivalent.
They're only equivalent because the set of finite ordinals is countable, which is defined by the naturals.

>We know what an ordinal number is, we know what a limit ordinal is, and about continuum? We know nothing, we only know that if it exists then it's strictly greater than ℵ0.
This is like saying we know nothing about 0 because dividing by 0 is undefined. Wrong, we know plenty about c since it's the cardinality of the powerset of any countable set.

>>9715503

What are you talking about? The cardinality of the reals is Beth-1

>>9715496
infinity is a stupid jewish invention made by (((Georg Cantor))). Norman Wildberger is right,
infinity doesnt exist at all, and breaks computational precison and needs to be purged from mathematics.

>>9716228
>infinity is a stupid jewish invention
so is like half of mathematics
>infinity doesnt exist at all, and needs to be purged from mathematics
feel free to redefine calculus without it anytime

also, OP, please answer >>9715636

>>9716228
*Wild(((berger)))

>>9715548
So Gauss, Cauchy, and Kronecker were brainlets?

>>9716765
>So Gauss, Cauchy, and Kronecker were brainlets?
Yes.

>>9715510
>theoretically
>theoretically possible
>finish the process in finite time
>it will always take infinitely long
>non-zero amount of time
What do you mean?

>>9716240
>feel free to redefine calculus without it anytime

Already been done: https://en.wikipedia.org/wiki/Constructive_analysis

>>9716946
That has nothing to do with infinity. Only absolute purists like norman wilberger reject infinity.

>>9715701
>They're only equivalent because the set of finite ordinals is countable, which is defined by the naturals.
not true
>This is like saying we know nothing about 0 because dividing by 0 is undefined
not true either. Assuming AC all "reasonable" cardinalities are alephs, and because it's impossible to know which aleph is equal to continuum it's not a well-defined concept

>>9715636
>>9716240
Axiom of infinity and power set axiom are independent, so there exist models of set theory in which power set of countably infinite set is at most countably infinite (or empty)

>>9716857
It takes some time to examine the set and choose an element from it. So, say we have countable family of sets [math]A_1,A_2,...[/math] and say it takes one unit of time to choose an element from [math]A_1[/math], half an unit of time to choose an element from [math]A_2[/math] and so on, so the process will end in two units of time. Now say we have a family [math]\left\{A_i\right\}_{i\in I}, |I|>\aleph_0[/math] and it takes [math]t_i>0[/math] units of time to choose an element from [math]A_i[/math], but for any choice of strictly positive [math]t_i[/math]s [math]\sum_{i\in I}t_i=\infty[/math], therefore it's impossible to construct a choice function in finite time

>>9716968
But wildberger and such rejecting infinity, or saying there's no infinity in their systems doesn't mean it doesn't really exist, right? Just like with ZFC, there are large cardinals which we "know" exist, but their existence can't be proven within ZFC, and same goes for wildberger, you can't prove within his system that infinite sets exist, but they do exist, and are just large cardinals, right?

>>9717751
>It takes some time to examine the set
>unit of time
Sounds like a bunch of gibberish an engineer/CStard would say. Disregarded.

>>9717756
>there are large cardinals which we "know" exist
>doesn't mean it doesn't really exist, right?
What kind of ``existence" are you even referring to? Philosophy and retardation like ``platonism" belong elsewhere.
>you can't prove within his system that infinite sets exist, but they do exist
In what sense do they ``exist" if their existence is neither provable nor postulated (and even outright negated in some cases)? Does AC ``exist" in ZF since it's consistent with ZF?

>>9717738
>not true
In mathematics, a countable set is a set with the same cardinality (number of elements) as some subset of the set of natural numbers.

>Assuming AC all "reasonable" cardinalities are alephs, and because it's impossible to know which aleph is equal to continuum it's not a well-defined concept
You're just repeating the same fallacy. Any non-redundant axiom is "impossible to know" since it is independent from the other axioms, like the continuum hypothesis. So by your logic all axioms are not well defined.

>>9717738
>Assuming AC all "reasonable" cardinalities are alephs, and because it's impossible to know which aleph is equal to continuum it's not a well-defined concept
Either the system is consistent or complete, you can only choose one as your definition of well defined. I would not call a self-contradictory axiomatic system well defined, so I think the choice is clear.

>>9717823
>you can only choose one as your definition of well defined
Being well-defined has nothing to do with being consistent or complete.

>>9715510
Give me two sets whose cartesian product is empty. I'll wait.

>>9717833
Assume the existence of the empty set and take its cartesian product with itself.

>>9717836
Now make them both non-empty.

>>9717839
What axioms am I allowed to assume?

>>9717844
Vanilla ZF.

>>9717833
You don't need AC for the finite products

https://en.wikipedia.org/wiki/Kripke%E2%80%93Platek_set_theory

>not being purist

>>9717793
Construct a sequence of sets {}, {{}}, {{},{{}}},..., call them finite ordinals, take their union and here, you have an infinite ordinal without using the notion of natural numbers. And if you say construction of finite ordinals is just a construction of naturals, it's because naturals are defined to be finite ordinals.

>>9717828
Then the continuum hypothesis being impossible to know has nothing to do with being well-defined. Good job.

>>9717880
you mean (((Kripke)))–(((Platek))) set theory
>purist
that's a weird way of spelling shabbos goy

>>9717858
Let [math](\mathcal{V}_{ZF},\models_{\diamond})[/math] be the non-standard forcing model given by the scheme valued sheaf [math]\mathcal{V}_{ZF}[/math] which is obtained by localizing the canonical "vanilla ZF" sheaf at the class of non-empty proper classes. The forcing relation is defined in the standard way, but keeping track of the fact that we have adjoined uncountably many non-standard sets [math]\mathfrak{X}_\kappa[/math] for every cardinal [math]\kappa[/math] with the property that [math]\mathfrak{X}_\kappa \in_{\text{meta}} \mathfrak{X}_\kappa^\bowtie \models_{ZF} \prod_{\kappa} \mathfrak{X}_\kappa^\bowtie = \varnothing[/math]. Where [math]\bowtie[/math] is the imaginary modality operator induced by the canonical topology on [math]ZF + \mathrm{Con}(ZF) + \neg INF[/math]. It is then a routine verficiation (using the results of Cohen and Chevalley) to see that [math]\mathfrak{X}_\kappa \in_{meta} \mathfrak{X}_\kappa^\bowtie[/math] holds in every model of ZF assuming the consistency of ZF.

>>9715496
Without AC there exists a non-empty family of pairwise-disjoint inhabited sets [math]\left\{S_i|i\in I, |I|>\mathfrak{c}\right\}[/math] such that [math]\bigcup_{i\in I}S_i=\mathbb{R}[/math]

>>9717889
>impossible to know
This isn't a well-defined notion to begin with. What do you mean by "impossible to know"? Impossible to know from within ZF? How does that make something not well-defined?
>>9717912
>Without AC there exists
Any result of this form holds assuming AC too by basic set theory. See Cohen 1963.

>>9717924
What do you mean by that? Assuming AC a set of cardinality [math]\kappa[/math] cannot be partitioned into more than [math]\kappa[/math] disjoint nonempty sets

>>9717912
Proof?

>>9717939
Not him, but that doesn't imply what's in your post. It's as retarded as claiming that without AC there provably exists a surjection which has no right inverse. That would immediately contradict the fact that AC is independent of ZF.

>>9717945
>That would immediately contradict the fact that AC is independent of ZF.
How so?

>>9717958
Think.

>>9717924
I'm not the one who claims AC isn't "well defined" >>9717738

>>9717909
Could you explain the construction at a high level?

>>9718083
no

>>9715496
Atheists belong on leddit

>>9719110
atheism dismisses countably infinite sets as well

>>9715496
>Uncountable sets are counter-intuitive
>too abstract to see, construct and comprehend

>>9715496
>Saying counterintuitive=wrong
>Is intelligent
Pick only one.

>>9715496
Set theory is a complete joke anyway. Name one concrete application it has in the real world.

>>9715496
>Grug dun unnerstun so dun exast

>>9716228
>(((Georg Cantor)))
>be devout lutheran
>brainlets think you're jewish

>>9715496
They're only counter-intuitive to you because you're a finite being and can't comprehend them. As someone with uncountably large IQ, I can easily grasp such concepts on an intuitive level.

bump

>>9721710
/thread

>>9721710
Holy shit that meme is so good

>>9715496
bump