College

[Hype]

Lots of people are very clever. Especially stupid people.

That's pretty clever.

It's moderately clever, but it doesn't imply anything about stupidity. Draw the Venn diagrams.

That reminds me of something I'd like to share here.

- The universe of natural numbers is infinite.
- The universe of real numbers is infinite.

The universe of real numbers contain the universe of natural numbers. However, both universes contain the same amount of elements.

That's beautiful.

Actually that's not true. Look at Cantor's Diagonal Argument. The real numbers can't be put in one-to-one correspondence with the natural numbers, so they don't have the same number of elements.

[Hokahey]

Actually that's not true.

You had to know that was coming.

[Hype]

Actually that's not true.

You had to know that was coming.

Well, it's not... and it's coincidental that I was just talking about the Diagonal Argument the other day...

[Larry B.]

Lots of people are very clever. Especially stupid people.

That's pretty clever.

It's moderately clever, but it doesn't imply anything about stupidity. Draw the Venn diagrams.

That reminds me of something I'd like to share here.

- The universe of natural numbers is infinite.
- The universe of real numbers is infinite.

The universe of real numbers contain the universe of natural numbers. However, both universes contain the same amount of elements.

That's beautiful.

Actually that's not true. Look at Cantor's Diagonal Argument. The real numbers can't be put in one-to-one correspondence with the natural numbers, so they don't have the same number of elements.

I looked at Cantor's Diagonal Argument and I have to agree with Kronecker: "I don't know what predominates in Cantor's theory – philosophy or theology, but I am sure that there is no mathematics there."

His whole diagonal thingy only proves that there are in fact as many numbers (of any kind) as any other kind of numbers. You just have to add the following number. And they are infinite, so you can keep doing it for good.

Cantor's diagonal argument is probably the stupidest pseudo-logical argument I've seen since Thomas Aquinas' five proofs for the existence of God.

[Hype]

You and Kronecker haven't understood it.

[Larry B.]

You and Kronecker haven't understood it.

Ooooooor...

[Hype]

You and Kronecker haven't understood it.

Ooooooor...

There's no alternative here. I can show you why you haven't understood it. It's not a difficult argument, though it can be tricky to understand why it's a valid principle.

The set of natural numbers is infinite. The question, then, is whether all other infinite sets have the same order, i.e., can be put into one-to-one correspondence with the members of the set of natural numbers.

Take some smaller subset of the natural numbers, e.g., the primes (presuming that set is infinite -- it might not be). For each natural number, you can pair a prime with it, like this:

1, 2
2, 3
3, 5
4, 7
5, 11
6, 13
7, 17
8, 19
9, 23

And so on, ad infinitum. There are no extraneous members, so these sets have the same cardinality (the same number of members, i.e., the same order of infinitude). Another way of putting this is that the set of primes is countable (because the natural numbers are the counting numbers).

But if you do this with the natural numbers and the real numbers, it doesn't work because, as the diagonal argument shows, there's already at least one uncountable set, T, and this can't be put into correspondence with the Real numbers. The set, T, is uncountable because it can't be placed into correspondence with the natural numbers, and the natural numbers DEFINE countability. They are, literally, and obviously, how we count.

Luckily, even if you still don't buy this (i.e., understand it), there are other proofs of the uncountability of the Real numbers.

[Larry B.]

You and Kronecker haven't understood it.

Ooooooor...

There's no alternative here. I can show you why you haven't understood it. It's not a difficult argument, though it can be tricky to understand why it's a valid principle.

The set of natural numbers is infinite. The question, then, is whether all other infinite sets have the same order, i.e., can be put into one-to-one correspondence with the members of the set of natural numbers.

Take some smaller subset of the natural numbers, e.g., the primes (presuming that set is infinite -- it might not be). For each natural number, you can pair a prime with it, like this:

1, 2
2, 3
3, 5
4, 7
5, 11
6, 13
7, 17
8, 19
9, 23

And so on, ad infinitum. There are no extraneous members, so these sets have the same cardinality (the same number of members, i.e., the same order of infinitude). Another way of putting this is that the set of primes is countable (because the natural numbers are the counting numbers).

But if you do this with the natural numbers and the real numbers, it doesn't work because, as the diagonal argument shows, there's already at least one uncountable set, T, and this can't be put into correspondence with the Real numbers. The set, T, is uncountable because it can't be placed into correspondence with the natural numbers, and the natural numbers DEFINE countability. They are, literally, and obviously, how we count.

Luckily, even if you still don't buy this (i.e., understand it), there are other proofs of the uncountability of the Real numbers.

I do understand it, but I think the mistake is in the fact that there are conclusions being drawn solely because natural numbers and real numbers apparently must correlate in the order in which you count, which is baseless.

For instance:

1, 0.1
2, 0.2
3, 0.3
(ooooopsie!! now we're going to add a 0.25! omg, the universe is gonna implode!)
4, 0.25
5, 0.1523
etc.

i.e., all infinite number sets are the same size (duh); even infinite subsets of infinite subsets. That's the beautiful thing I was talking about.

Countability is a whole different matter. I don't think either way about countability. It doesn't take away from the paradox(?) that infinite subsets of infinite sets have the same amount of elements.

[Hype]

natural numbers and real numbers apparently must correlate in the order in which you count

That's your mistake. That's not what the argument says. It just says that they can't be put into a one-to-one correspondence, which has nothing to do with order.

[Larry B.]

natural numbers and real numbers apparently must correlate in the order in which you count

That's your mistake. That's not what the argument says. It just says that they can't be put into a one-to-one correspondence, which has nothing to do with order.

And that's what makes no sense. Of course they can be put into a one-to-one correspondence. Pick ANY number, from ANY set of numbers and it can be connected to ANY real or natural number.

Do you really buy this crap or you're just saying all this to prove that I'm wrong? Logically, Cantor's Diagonal makes no sense. At all. We have two infinite sets, how is it possible that one of the elements of one set is left without its corresponding 'pair' from the other set? It doesn't matter the value you pick in one set, the other will always contain another value to be paired with it.

[Hype]

natural numbers and real numbers apparently must correlate in the order in which you count

That's your mistake. That's not what the argument says. It just says that they can't be put into a one-to-one correspondence, which has nothing to do with order.

And that's what makes no sense. Of course they can be put into a one-to-one correspondence. Pick ANY number, from ANY set of numbers and it can be connected to ANY real or natural number.

Do you really buy this crap or you're just saying all this to prove that I'm wrong? Logically, Cantor's Diagonal makes no sense. At all. We have two infinite sets, how is it possible that one of the elements of one set is left without its corresponding 'pair' from the other set? It doesn't matter the value you pick in one set, the other will always contain another value to be paired with it.

On the face of it, you're just working with a pre-Cantor notion of infinity and insisting that there can't be orders of infinity. But that's just begging the question.
As for arguing positively, well, there I'm out of my depth because I'm not a mathematician, nor a logician. I have worked on set theory academically, but it's only in the context of the conceptual foundations of axiomatic systems, not in actually working out the details of a given axiomatic system. I understand the Diagonal Argument (and some other stuff Cantor said) in the context of the history of philosophy of mathematics, but I don't think I'm qualified to keep trying to justify it, other than to say that I see no reason not to accept it, and I see prima facie reasons to accept it. I don't think your objections to the Diagonal Argument work. If you think they do, I urge you to go talk to a mathematician somewhere and get them to help you prove it formally, so you can become famous.

[sinep]

I took a continuing education BIM class recently at City College in SF to brush up on some professional skills, and it was like $55.00.

City College has a Music School as well.
http://www.ccsf.edu/NEW/en/educational- ... music.html

Looks like $36/unit now for in-state students.
Don't know if you are looking for more of a liberal arts experience but if you are just looking to rack up some credits and learn to read music, that would be a good ticket. And for the price, you could live in a real swank penthouse anywhere in the City for less than what those private schools charge for tuition alone. And I wouldn't think that just because you have school loans that it's not real money.

building information modeling? what's your profession?

i'm in a directed studies course right now studying BIM. i need to have a model of a 3 story office building completed in REVIT by april 2nd.



haven't started yet...

I'm an architect.

No, really, I am. After 10+ years of dinking around in firms I finally decided to take the ARE, 35 hours of testing later, I just got my license.

So if anyone needs architectural services, or just some input, let me know! (boy, I'm going to have to work on my pitch if I'm going to make this sole practitioner thing work..... )


I took a revit class there, that's what people are on these days. So many revit buildings out there! The curtain wall detailing really gives them away. Do you just need floor plates and a skin or more detailed than that? Good luck!

that's awesome.

i'm pretty sure i just need floor plates and a skin. at least when i went though the 2d drawings i was provided with that seemed like all i needed to do. if there is any MPE drawings i missed that i have to include i'll probably just end up killing myself. or dropping the class. either or.

anyways, i'm just about done with a degree in construction engineering management. have any connections at any big GCs? if you hear of any cool companies looking for a good project engineer let me know. my persona on here does not accurately reflect my work ethic.

[cabangbangq]

A small update;

Today I got an acceptance letter from Sarah Lawrence College, my top choice school. The school is fantastic, it is a half hour out of NYC, has no required classes or majors; you choose the classes you want to take, have one on one meetings with professors and your don, and you generally build the curriculum to tailor to your needs. I definitely want to go to this school, sadly, tuition is $60,000. Fun times

[creep]

A small update;

Today I got an acceptance letter from Sarah Lawrence College, my top choice school. The school is fantastic, it is a half hour out of NYC, has no required classes or majors; you choose the classes you want to take, have one on one meetings with professors and your don, and you generally build the curriculum to tailor to your needs. I definitely want to go to this school, sadly, tuition is $60,000. Fun times

no college education is worth 240k

[Hype]

What the fuck. My undergrad @ the University of Toronto was $6k tuition a year and I thought that was expensive. American Private Universities are way over-priced, unless you're talking Ivy League and you're going for something that is guaranteed to pay out. (But that's the wrong reason to go to university anyway... if you want $$$ become a tradesman or something...)

[sinep]

has no required classes or majors; you choose the classes you want to take, have one on one meetings with professors and your don, and you generally build the curriculum to tailor to your needs.

is it a bachelor of arts degree in hippie studies or a bachelor of science?

[cabangbangq]

has no required classes or majors; you choose the classes you want to take, have one on one meetings with professors and your don, and you generally build the curriculum to tailor to your needs.

is it a bachelor of arts degree in hippie studies or a bachelor of science?

Found this thread while looking through my account (first time logging in about 2 months). You get a bachelor's in Liberal Arts with a concentration in ____. Right now I'm taking International Development Studies, Democracy and Diversity (much more political theory/philosophy than I anticipated) and taking classes in music theory and guitar.

I ended up taking a gap year before coming here because I didn't fill out my financial aid in time, it actually ended up being the greatest thing that could have happened to me. As for the excessive price of the school, I got 37k covered in free money from the school, only 3 grand in loans (!!!!!!) and my parents are having a better year financially so 22k isn't a horrible out-of-pocket price.

[Pure Method]

As someone who went straight from high school to college to law school, please allow me to urge you that there is no rush to get old and join the rat race. Have you considered time off to travel or save a little money? That mexico plan sounds cool to me. I have friends who took gap years before college and I think they really benefitted from it. School is awesome and enriching, so you can't make a bad decision. but living under debt, especially $250k, if what you really hope to do is be a musician, might not be appropriate. I'd also investigate as many grants and scholarships as possible.

I know this is a lot of real world shit and it sucks, but have a good long think about this. Talk to a guidance counselor, or neighbor, or family member. And ask us more questions.