Tropers that do NOT hate math

So if asked how many natural numbers there are the correct answer is aleph-null_0 and not infinity? When describing the natural number line is it correct to say it extends infinitely?

Are the hyper real numbers and surreal numbers important for this discussion?

edited 6th Aug '13 11:06:30 AM by jate88

The correct answer to "how many natural numbers are there" is ℵ_0 (read "aleph-null" or "aleph-naught"). In fact, "having cardinality ℵ_0" means the same thing as "can be put into one-to-one correspondence with the natural numbers".

Saying that the natural number line "extend infinitely" is slightly different. The "natural number line" basically means the natural numbers, considered as a subset of the real numbers. The natural numbers are unbounded as a subset of the real line — that is, they aren't contained in any interval [-x, x] — so we can say, informally, that the natural numbers "extend infinitely far out along the real line in the positive direction" (though this is somewhat unusual phrasing).

This is different in a few ways: This isn't a property of the natural numbers themselves, but rather a property of how the natural numbers are included in the real numbers. For example, we could embed the positive integers into the real numbers in a non-standard way by sending n to 1/n. This is still infinite with cardinality ℵ_0, but the whole set {1, 1/2, 1/3, 1/4, ..., 1/n, ...} is contained in the interval [0, 1], and therefore is bounded.

Also, the set of all positive real numbers can also be said to "extend infinitely along the real line" in the same way, but by Cantor's diagonal argument, the positive real numbers have strictly larger cardinality than the natural numbers. (In fact, they have cardinality 2^ℵ_0.)

So, as these two examples show, "extending infinitely along the real line" is a property that depends on the real line with the spatial structure given by its distance metric, which allows us to say things like how far apart points are and when a subset is bounded or unbounded; this measures a different thing than cardinality, though there are some loose connections between them.

The hyperreal and surreal numbers are ordered number systems that formalize the idea of "infinitesimal numbers". Some numbers in those systems are "infinite" in the sense that they're larger than any positive integer. These are different from cardinalities, but at least in the case of hyperreals, closely related to limits. The hyperreals, in fact, give an alternate formalization of the limit concepts used in calculus; as such, they are one of the basic objects of non-standard analysis.

The surreal numbers are a bit weird. The basic idea is the same as the hyperreals, but there are a lot more surreal numbers — so many more, in fact, that the surreal numbers are "too big" to form a set. (It's related to the fact that collections like "all cardinal numbers", "all sets", or "all ordinal numbers" are "too big" to themselves be sets. Collections like this are known as proper classes.) I've never worked with the surreal numbers myself, but apparently, they have some applications to combinatorial game theory and mathematical logic.

So mathematicians don't use the words infinite or infinity? The other wiki's page on Cantor's Diagonalization Argument uses the word infinite in the first couple paragraphs.

Is denying the existence of infinite sets and real numbers a valid criticism of Cantor's Diagonalization Argument?

edited 7th Aug '13 7:01:25 AM by jate88

Mathematicians use the words "infinite" and "infinity", but in a way that makes it clear from context which type of infinity is meant. (Usually, there's only one sort of infinity that makes sense in a given situation, anyway.) For that matter, mathematicians sometimes phrase things in incredibly vague-sounding ways when they're informally discussing things; once your intuition about a subject is strong enough, you can leave all sorts of things implied and still get the meaning across.

The philosophical stance of denying the existence of infinite objects is known as finitism, a philosophical position held by only a small minority of mathematicians. Cantor's diagonal argument is valid given the framework of standard mathematics, but finitists choose to work in a framework that doesn't allow for infinite sets at all — even the natural numbers — so I don't think that theorem can be formulated at all in a finitist framework, because it doesn't allow for a set of natural numbers or a set of real numbers. (I think you can see why not many mathematicians pay attention to finitism.)

So there are different types of infinity? Not just sizes. Why don't they just stop using the word infinity and invent new words for the characteristics or behaviors they're describing?

NJ Wildberger doesn't seem like a finitist. The last few videos in his lecture on mathematical foundations talk about a type of infinity.

• K-12 Education
• Infinite Sets
• Square Root Two
• Real Numbers

edited 7th Aug '13 10:07:05 AM by jate88

That's what I explained here — the words "infinity" and "infinite" are used for a bunch of different mathematical concepts. There are more precise words (such as cardinality, order type, unbounded, etc.) that are used in mathematical contexts for when people need (or want) to be more precise.

However, the vagueness of the word "infinite" isn't a problem as long as it's clear (to everyone present) which meaning is intended. Like I said, mathematicians can get away with a lot of vagueness and still understand each other; usually, with a solid enough mathematical understanding, it's clear which interpretation makes sense.

Filter, Ultrafilter, Ultrapower construction, partial ordering, inclusion. What are all these things?

edited 9th Aug '13 6:53:30 AM by jate88

I LOVE MATH.

Things that Wikipedia has articles on. Any more specific questions?

edited 9th Aug '13 8:35:36 PM by Enthryn

I find Wikipedia tends to be a bit too technical and jargon-filled with its explanations. Not that it's necessarily a bad thing, but it can be a bit daunting for the layman who's just looking for an introduction to the topic.

But of course, there's always Google.

I know Wikipedia has articles on those. When I go to a page to find out about something I keep having to go to another page linked in the article to figure out what something else is. To be fair tvtropes does the same thing if memory serves correctly but wiki walks aren't as fun on the other wiki.

Maybe a better question would be "What all would an applied mathematician, scientist, or engineer need to know about infinitesimals in order to use them accurately?"

edited 10th Aug '13 9:50:18 AM by jate88

Hey, OP here.

Is there a site that focuses on .9_ = 1? If not, I'm interested in creating one.

The Wikipedia article is pretty thorough. What's the need?

edited 26th Aug '13 6:44:53 PM by Enthryn

I just felt like making a website with all the explanations, related concepts and Q&A. Plus I just decided to start practicing web design, and I could learn as I make a site.

If you just feel like making the site anyway, why worry whether someone else has already done it? That won't make it any less valuable as web design practice.

I know, but it doesn't hurt to see, so I don't make a duplicate.

From the Random Thoughts thread:

Is this even a thing that can be done?

Explained.

Hey, in my high-school calculus class we were doing integration by parts and we had ∫(x^2)(ln|x+1|)dx and when we were going through it in class we just wound up doing the integrating by parts twice and got two u*v's that canceled and our final integral was what we started with.

Anybody have some help on this? Sorry if the wording of this is not all that great.

edited 15th Apr '14 7:13:23 PM by RaichuKFM

How does that happen...?

= (x^3)(ln|x+1|)/3 - ∫((x^3)/(3x+3))dx

= (x^3)(ln|x+1|)/3 - ((x^4)/(12x+12)) - ∫((x^4)/(12(x+1)^2))dx

...or did you do it the other way around? I don't remember what the integral of ln|x+1| would be offhand, but you would be repeatedly taking the derivative of x^2, which becomes a constant in two steps.

edited 16th Apr '14 7:19:29 AM by Noaqiyeum

A google search for hyper real numbers has me wondering if all subsets of the natural numbers are finite.

Counterexample: the natural numbers are a subset of the natural numbers, and are not finite.

Are there any other examples?

The set of all natural numbers greater than n.

The set of all natural numbers divisible by n.

The set of all prime numbers.

The set of all numbers in the fibonacci sequence.

The vast majority of subsets of the natural numbers are infinite. More precisely, since the set of natural numbers is countably infinite, the set of subsets of the natural numbers is uncountably infinite (by Cantor's theorem), but the set of finite subsets of the natural numbers is only countably infinite (because each finite subset can be represented as a finite string in a finite alphabet, and there are only countably many such strings).