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/sci/ - Science & Math


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9332396 No.9332396 [Reply] [Original] [archived.moe]

So what about an "infinite set"? Well, to begin with you should say precisely what the term means.
Okay if you don't, at least someone should. Putting an adjective in front of a noun does not in itself make a mathematical concept.
Cantor declared that an "infinite set" is a set which is not finite. Surely that is unsatisfactory, as Cantor no doubt suspected himself. It's like declaring that an "all-seeing Leprechaun" is a Leprechaun which can see everything. Or an unstoppable mouse is a mouse which cannot be stopped. These grammatical constructions do not create concepts, except perhaps in literary or poetic sense. It is not clear that there are any sets that are not finite, just as it's not clear that there are any Leprechauns which can see everything, or that there are mice which cannot be stopped.
Certainly in science there is no reason to suppose that "infinite sets" exist. Are there and infinite number of quarks or electrons in the universe? If physicists had to hazard a guess, I am confident that the majority would say: No. But even if there were an infinite number of electrons it's unreasonable to suppose that you can get an infinite number of them all together as a single data object.

>> No.9332548

>So what about an "infinite set"? Well, to begin with you should say precisely what the term means.

Set = A container
Infinite = Larger than anything finite

>> No.9332622

>>9332548
But Wildberger is saying, does there really exist something that is larger than anything finite? (Answer = no). And if there is nothing like that, should we really be happy incorporating it in the foundations of mathematics?

>> No.9332637

>>9332622
>does there really exist something that is larger than anything finite?
>Answer = no

Okay. Prove it.

>> No.9332651

>>9332622
>But Wildberger is saying, does there really exist something that is larger than anything finite?
https://en.wikipedia.org/wiki/Axiom_of_infinity

>> No.9332662

Infinite sets are those in which an surjective function can be made between them and the set of natural numbers.

>> No.9332684

>>9332662
Don't you mean a bijective function?

>> No.9332686

>>9332684
>Don't you mean a bijective function?
Then the real numbers wouldn't be infinite (so no, just surjective)

>> No.9332687

>>9332684
No, surjective to account for larger infinities. E.g. there's no bijective function between the real numbers and the naturals, but there is a surjective one.

>> No.9332709

>>9332396
>So what about an "infinite set"?
So what about a verb in your question?

>> No.9332710

>>9332637
>Okay. Prove it.
Well, in science and application we have only dealt with finite objects. Maybe there is an infinite object, but we have not found it. Until we find an infinity that we can first study to gather intuition then we can maybe talk infinity. Euclid's axioms for geometry did not come out of nowhere, there was a lot of drawing lines in sand to get an intuition for how lines work before we even started laying out the axioms of lines. There are no axioms without intuition.

>>9332651
Yeah, you can make infinity an axiom but how do you know that is not dangerous? How do you know that simply saying "infinity is a thing" will not bring problems down the line?

>> No.9332720

>>9332710
>There are no infinities
How many natural numbers are there?

>> No.9332726

>>9332720
Ill-posed question. Does there really need to be a "number" of natural numbers? Is that a question that even makes sense? Why?

>> No.9332757

You cannot perform arithmetic on infinity.

Infinity is not a number. It is a concept. It has no more value to mathematics, specifically as a 'general term' or constant, than mathematics draws value from Superman comic books.

if you are taking a math course and the concept of infinity arises where you're expected to define the answer using infinity because it was present in the equation, you have managed to waste your tuition and should start deeply wondering about how you're gonna pay off student loans.

>> No.9332766

>>9332726
>>9332757
t. hasn't taken calculus.
Take some real math (calculus, analysis) before you start claiming bullshit.

>> No.9332795

>>9332766
I am currently a senior in undergrad. In fact, analysis is my strongest subject. That is because I am good at logic and manipulating the various definitions and theorems come naturally to me. But for this high level of understanding I paid a price: the intellect to notice the cracks of analysis. As far as I know, by the time you are doing integrals and derivatives it is all good. The problems are at the beginning, the real numbers as we know them are ill defined. They simply don't work.

Wildberger shares this sentiment. He feels that most of analysis can be saved because most of it is "true", we just have unrigorous proofs for those things. That is why I am appreciating his new course in algebraic calculus. A rationals only, well defined data structures only, no nonsense approach to analysis.

>> No.9332807

>>9332396
Do you have some big problem with the axiomatic nature of math?

>> No.9332810

>>9332710
The axioms of ZFC did not come out of nowhere, there was a lot of drawing lines in sand to get an intuition for how math works before we even started laying out the axioms of set theory. There are no axioms without intuition.

>> No.9332815

>>9332795
>The problems are at the beginning, the real numbers as we know them are ill defined
what nonsense. why don't you stop roleplaying already

>> No.9332817

>>9332795
answer this. is there a set of natural numbers?

>> No.9332826

>>9332817
According to ZFC, there is. But there is no compelling reason for why we need to have a set of natural numbers. Nowhere in nature do we even observe anything remotely similar to a set of infinite objects.

That is why, even though ZFC axiomatically states that N exists, I side with Wildberger in that N does not need to exist. We don't need a set of all natural numbers in only to use numbers. Just like you don't need the set {1,2,3} if you only use the numbers 1 and 2.

>> No.9332829

>>9332826
what kind of math are you even interested in doing then? you can't do analysis, you can't do geometry, you can't do algebra. what can you do?

>> No.9332836

>>9332829
Why not? You can do everything. Wildberger is proving that as we speak. In his series he already defined area, has computed the area of all common polygons, has shown how to do geometry in the affine plans, has done number theory (Bernoulli numbers was his last topic as of today), has done algebra and has done analysis (He tackled the area of parabolic segments and of circles, but this were introductions to the theory he wants to use, it is not yet complete).

>> No.9332840

>>9332548
>Infinite = Larger than anything finite
Really it's
>Not finite and a set
The "and a set" part is important.

>> No.9332841

>>9332836
So, how do you solve [math]\frac{dy}{dx}+y=0[/math] without transcendentals? Find roots to polynomials?

>> No.9332844

>>9332841
Well, we have to remember that most of the analysis we use today is approximately true. That means that the method or replacing e^mx and solving the characteristic polynomial is meaningful.

We just need to wait for Wildberger to reach exponentials to see how exponentials can be done rigorously. After that, roots of polynomials are no sweat. Just extend Q with the necessary roots and then you prove solutions exist in the functions of this extended Q.

>> No.9332847

>>9332396
Why do we have numbers other than 1? Wh y not just only use 1?

>> No.9332865

>>9332836
>>9332844
might as well have just said "I can do whatever wildberger does lmao"

ridiculous

>> No.9332870

As a philosopher I feel that mathematics is misusing the word/concept of infinity.

Infinity has no beginning and no end. All numbers have a beginning and an end (at least at the level we are looking at), so to say a number is infinite is like saying that the finite is infinite.

This goes for "sets" as well, a set implies a beginning and an end, a separation from other "sets", they are finite. So to call something an infinite set makes the same mistake.

>> No.9333021

>>9332844
Extending Q with the necessary roots is just the algebraic numbers. I thought the whole point of Wildberger was to stick with only rationals?
>>9332870
Not really. For example, consider the set N of natural numbers. The set is infinite (for all natural numbers m>0 the set is larger than m).

>> No.9333032

>>9333021
>I thought the whole point of Wildberger was to stick with only rationals?

Extensions do not violate Wildberger's philosophy. He has talked a lot about how the rigorous way to view the square root of 2 is as an algebraic extension of Q, not as a number. The square root of 2 cannot be assigned a number, but it can be algebraically included so we can rigorously manipulate it symbolically. The same goes for i. Watch his videos.

>> No.9333043

is everyone in this board wildpilled

>> No.9333045

>>9333021

This assumes zero is a number which it isn't. Zero is the lack of a number.

>> No.9333053

>>9333045
Zero is not the lack of a number, it's the thing you count when you count nothing. It's more abstract than counting numbers for sure but it's a number on the same level as 1/2 and -42.

>> No.9333056

>>9333053
Why the fuck is my name Cauchy lmao
I'll be Anon after this post.

>> No.9333059

>>9333053

>when you count nothing

You can't count nothing, that's impossible.

Zero is like the blank canvas for the paint to be put on.

>> No.9333064

>>9333059
>that's impossible
That's why I've said that it's an abstraction. You can't count 1/2 of a thing either but would you say it's not a number?

>> No.9333068

>>9333032
I know what field extensions are. So he just doesn't like the real numbers? I can get behind that, but I'm not quite sure how he'll be able to work exponentials in there.
>>9333045
Fine, just change m>0 to m>=1.

>> No.9333075

>>9333064

>You can't count 1/2 of a thing either but would you say it's not a number?

I'm not following.

>> No.9333085
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9333085

>>9332766
Infinity is not a number, dickshit.

>> No.9333086

>>9333068
>Fine, just change m>0 to m>=1.

How can you know something is bigger or smaller when it is infinite? These concepts cannot exist in infinity, only in finiteness.

>> No.9333093

>>9333068
He doesn't like real numbers because real numbers are the inevitable result of being lazy and accepting infinity and just saying "lol lets just say everything is a number to make our lives easier, who cares"

>> No.9333097

>>9333075
I'll try my best to explain my thought. Let's say you have half an apple on the table. Now let's say you claim you have 1/2 apple. You are right, of course, but you didn't *count* that. What you did is you counted how many objects there were (1 object: the half-apple) and you realized that this 1 object you counted was incomplete: usually an apple doesn't look like that, it's usually composed of two objects like the one you have in front of you. Therefore you can tell me you have 1/2 of an apple.

Now imagine that I place before you some alien object. Would you be able to tell me it's the half of something just by counting? No. But counting is a process that doesn't care about what you are counting. So you can't count to a half because it would mean you can always know, for any object (including the alien one no one knows about), if it's the half of something or not.

Sending from phone so I may be unclear, tell me if that's the case.

>> No.9333105

>>9333097
So it impossible to count to 1/2 in the the fundamental way it's impossible to count to zero. But 1/2 is a number. So why wouldn't 0 be a number too?

>> No.9333112

>>9333097

But if I don't know if something is a half of something, then I cannot know what it should look like whole. I can only assume it is.

>> No.9333139

>>9333112
You are right. However my point still stands: counting things is an activity that is mirrored in maths as the numbers 1,2,3,... These symbols I wrote could be anything, like shapes. That string of symbols represent the act of counting, from counting one distinct thing to the other. It doesn't matter if it's apples, half apples, boats, alien objects, etc, the first thing you count is assigned the symbol 1, the second the symbol 2, and so on. We never run out of symbols because we have the decimal system which can always generate a new unused symbol.

Now, what I described is not 'the natural numbers', they are simply a process made into symbols. If you remove yourself from the reality of counting, 1/2 becomes a thing, meaning 'the thing between 1 and 2 on the number line', but you are not counting anymore obviously.

>> No.9333150

>>9333139
What if you cut the real life object you're counting in half? Like, you have 3 apples and half of an apple you're trying to count.

One way to consider it is that now you're counting half-apples, with 1/2 being the basic unit.

>> No.9333169
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9333169

>>9333150
I will assume you're a troll or you haven't read correctly what I've written, or what I wrote is not understandable but I doubt that.
>pic related

>> No.9333201

>>9333139

Counting requires separation, a finiteness, an equal distance between each number. What does this separation? Zero does.

You could see zero as representing consciousness, with numbers (negative and positive) representing the physical (or lack of the physical) world AKA you have two apples and then eat one.

>> No.9333265

>>9332720
10^200

>> No.9333306

>>9333150
you donkey

>> No.9333638

>>9333265
Why not Graham's Number? Why not GN^GN^GN^GN^GN^GN^GN^GN...^GN where there are GN levels of exponents? If you're going to use retard math, at least explain it.

>> No.9334013

>>9332870
As a philosopher you don't know shit about math

>> No.9335042

>>9332622
> (Answer = no).
Prove it

>> No.9335089

A set whose series diverges.

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