Board 8 > ATTN B8 Math Folks

(+) Yeah! (+)

Set theory is a total fucking scam. Some infinities are larger than other infinities, but |[0,1]| > |Naturals|? Give me a fucking break.


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Pretty much every hand you see has had a dick in it at some point

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Try counting the elements of [0,1] and then maybe we'll talk.

(I don't actually know much about set theory beyond the basics...)
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pyresword posted...

Try counting the elements of [0,1] and then maybe we'll talk.

(I don't actually know much about set theory beyond the basics...)

Try counting the elements of the natural numbers and then maybe we'll talk. "Countable" my ass

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I've never studied set theory, and I would have never actually considered what you stated to be true at first reaction, but now that I think of it, it makes sense.

For decimal values, you can have anywhere from 0 to 9, across any number of digits.
For natural numbers, you can have anywhere from 0 to 9, across any number of digits.

But that's only for terminating decimals. There's still repeating decimals and irrationals.

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Yeah, putting almost zero thought into it I think azuarc's right. You could map every natural number to something like 1/n to get: 1/1, 1/2, 1/3, 1/4, 1/5, ...

But that gives you only a subset of the rationals on [0,1] to say nothing of the irrationals in that interval.

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xp1337 posted...

But right there that's only a subset of the rationals across [0,1]. Repeating decimals like 1/3 or 1/9 would be excluded

Natural numbers are "countable". However, so are rational numbers. That argument holds no water even in Cantor's twisted logic. N x N is "countable", too. As well as N x N x N ... N times. Absolute lunacy

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xp1337 posted...

Yeah, putting almost zero thought into it I think azuarc's right. You could map every natural number to something like 1/n to get: 1/1, 1/2, 1/3, 1/4, 1/5, ...

But that gives you only a subset of the rationals on [0,1] to say nothing of the irrationals in that interval.

I don't think this form of argument is fully rigorous, for the record. (At least not just mapping set A onto a subset of set B...the commentary on rationals vs irrationals might be getting at something but I'm not an expert)

For example you can similarly fail to establish a one-to-one correspondence between the naturals and the even naturals by starting with the even naturals and doing the following: 2->2; 4->4; 6->6... and you can see we've "missed" all of the odd elements of the natural numbers.

However you can establish a one-to-one correspondence between the two sets by starting with the natural numbers and doing the mapping: 1->2, 2->4, 3->6, etc, and therefore the sizes of the two sets are the "same size" infinity. (There's probably a more precise term out there)

I'm not sure what a rigorous proof of this would look like off-hand though it's probably a relatively fundamental piece of set theory so I found it's hard to find.
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NFUN posted...

Some infinities are larger than other infinities.

This statement doesn't make sense.

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pyresword posted...

"same size" infinity.

They allegedly are designated with the size "aleph-null"

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All infinite sets are infinite.

By definition, one cannot be larger than another.

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GildedFool posted...

All infinite sets are infinite.

By definition, one cannot be larger than another.

Finally, somebody with sense!

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There's a real mathematical concept they are getting at, but I've never been sure "size" is the most intuitive way to express that because of exactly what you say. In my head I think of it more in terms of how closely spaced the numbers are to each other, but that's also very imprecise terminology that I suspect would cause problems as the discussion gets advanced enough.

In this case though, the set of integers is discreetly spaced whereas the real numbers are not. For example if I start with 2, the next integer is obviously 3, but the concept of a "next" real number doesn't make sense, I think. (2.001? 2.0000001? 2.00000000000000000000001? You could go on forever)
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pyresword posted...

In this case though, the set of integers is discreetly spaced whereas the real numbers are not. For example if I start with 2, the next integer is obviously 3, but the concept of a "next" real number doesn't make sense, I think. (2.001? 2.0000001? 2.00000000000000000000001? You could go on forever)

Rational numbers are dense, ie, you can create an arbitrarily small interval and there will be a rational number within it, yet they are supposedly countable.

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pyresword posted...

In this case though, the set of integers is discreetly spaced whereas the real numbers are not. For example if I start with 2, the next integer is obviously 3, but the concept of a "next" real number doesn't make sense, I think. (2.001? 2.0000001? 2.00000000000000000000001? You could go on forever)

gamefaqs suggests i can only go on until 2.00000000000000000000000000000000000000000000000000000000000000000000000000001

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pyresword posted...

but that's also very imprecise terminology that I suspect would cause problems as the discussion gets advanced enough.

Aha! I was right!

If you open with "I'm probably wrong" then you're always right!

Edit: Responding to NFUN's point, which yes is correct
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NFUN posted...

Rational numbers are dense, ie, you can create an arbitrarily small interval and there will be a rational number within it, yet they are supposedly countable.

And actually on further reflection, I think the difference is that there are infinitely many real numbers on an arbitrarily small interval, whereas there are only a finite number of rationals on such an interval?

That's only an educated guess, but it would make sense.

Edit: no I'm dumb this doesn't make sense and is wrong
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pyresword posted...

I don't think this form of argument is fully rigorous, for the record.

i'm sure it isn't. i would be stunned if i managed to come up with something rigorous or even correct on set theory when i desperately need sleep >_>

what ever math just breaks when you fly too close to the sun

the sum of the natural numbers is -1/12? i'm out

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xp1337 posted...

the sum of the natural numbers is -1/12? i'm out

that's the riemann zeta function, not the arithmetic sum

what's cool is that if you graph the expression of the sum of natural numbers (n/2(n+1)) and find the negative area, you do get -1/12

https://www.wolframalpha.com/input/?i=integral+of+%28x%29%28x%2B1%29%2F2+from+-1+to+0

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To be clear on the distinction between the Riemann zeta function of -1 and the sum of all the natural numbers:

The Riemann zeta function is defined as zeta(s) = the analytic continuation of the sum from n=1 to infinity of 1/n^s. The analytic continuation g(z) of a function f(z) agrees with g(z) in some region of the complex plane but f(z) is analytic (a function being analytic means it has a convergent series expansion).

A much easier example is the infinite series f(z)=1+z+z^2+z^3+... . We know that this series equals 1/(1-z) for |z|<1 and that the series diverges for |z|>1. The analytic continuation for f(z) is given by g(z)=1/(1-z), which is finite everywhere except z=1. But that doesn't mean that f doesn't blow up for |z|>1!

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NFUN posted...

They allegedly are designated with the size "aleph-null"

I have a shirt with that on it

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SgtSphynx posted...

I have a shirt with that on it

I'm sorry you got conned

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I actually really like the shirt
https://cdn.hellosubscription.com/wp-content/uploads/2019/03/12075544/curiosity-box-spring-2019-vol-XI-16.jpg

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Problem is 'infinite' isn't a quantity, it's just 'unbounded', and comparing two infinities is problematic. In an attempt to do so, you're drawing boundaries, so the constructs you're looking at wind up being finite, and yeah sure you can have a finite thing that's larger than another finite thing. That's part of the nature of finite things. You can't really generalize from the finite to the infinite. Infinity isn't a real thing anyway, it's an attempt to extrapolate from the finite things we observe to something arbitrary.

Thanks Plato you fuck

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This thread makes me glad I'm not in STEM.

Honestly, I've always found Cantor's diagonal argument to be rather beautiful and profound. Not liking the hate!

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tyder21 posted...

Honestly, I've always found Cantor's diagonal argument to be rather beautiful and profound. Not liking the hate!

String theory is also beautiful and profound. Doesn't mean it isn't bunk

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GildedFool posted...

All infinite sets are infinite.

By definition, one cannot be larger than another.

X^2 is definitely larger than X, as X approaches infinity. So much so that the quotient between them is also infinity.

edit: Oh. I missed the word sets. I would still claim that there are "more" elements in the set of integers than the set of natural numbers.

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azuarc posted...

X^2 is definitely larger than X, as X approaches infinity. So much so that the quotient between them is also infinity.

edit: Oh. I missed the word sets. I would still claim that there are "more" elements in the set of integers than the set of natural numbers.

And according to the boring Cantor you'd be incorrect

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I miss the days of the Internet when people would argue about whether 0.999... = 1, those were simpler times

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Working on a bit more sleep now I'll say that while I never formally studied set theory, I have made some occasional very half-hearted attempts to self-study cardinality/countability of infinite sets and my brain basically breaks every time so I am sympathetic to NFUN here!

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I feel dumb, I have no idea what they hell you guys are talking about.

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neonreaper posted...

Pretty much every hand you see has had a dick in it at some point

Only post that makes any sense ITT

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This is the exact same argument as .999~ =/= 1 and you're a terrible person for posting it

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foolm0r0n posted...

This is the exact same argument as .999~ =/= 1 and you're a terrible person for posting it

Steiner told me I should've died on my last wrong hill so I decided to die on this one

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Cantor's principle makes perfect sense, just specific examples seem wack. If two sets are isomorphic, they are, in some important ways, the same set. F={1,2,3} is basically G={4,5,6} since you can trivially convert one to the other. They're bijective; let Fn = Gn -3 or Gn = Fn + 3. You know each and every element of G from the elements of F precisely, so they intuitively have certain principles such as size in common.

If you take the set of the even integers and halve them, you get the set of every integers, and from the set of every integer you can get the set of the evens. The sets are bijective and isomorphic, so they must have the same size in order to draw such a relation from one to another. There are as many even integers as there are integers. Look at N -> N x N, ie {1, 2,...} -> {(1,1), (1,2),...}. Let {1} -> {1,1}, {2} -> {1,2}, {3} -> {2,1}, etc, forming a triangle from the origin with each step. Given one of these cartesian coordinates, you can get an integer, and from an integer, you get a coordinate. So N x N mus have the same size as N. This can be generalized to N x N x N... as many times as you like. You can create a similar argument for rational numbers as well.

You can't do the same for integers to real numbers. If you try to do N -> [0,1], you might get {1} -> {1}, {2}->{.1}, {3}->{.01}, etc. Imprecisely speaking, you'd get stuck on the relationship Rn = 10^(Nn-1). Cantor's argument is a very elegant way to show that you can't create a bijective relationship between the natural numbers and the real numbers, and thus the sets much have different cardinalities

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