The set of "ordinary" real numbers are just as large as the set of complex numbers.

Hmmm, now here's a question. If I have an infinite set and you have an infinite set, but my set also includes all of yours, and also has members that yours doesn't, can it be classed as bigger?

edited to add: Essentially Real numbers are a subset of Imaginary Numbers, and so the set of Imaginary numbers must somehow be bigger, even though they are both infinite. A weird idea, but.......

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A Belgian friend I met in Japan who is also a working mathematician pointed me to the book Goodbye, Descartes by Keith Devlin, professor of mathematics, Dean of the School of Science at St. Mary's College, and senior researcher at Stanford University's Center for the Study of Language and Communication. One of those "few other ideas" is likely what Devlin has been working on, that is, Situation Theory. I expect this is one of those "higher levels of math" that is nevertheless considerably fundamental. A quote or two...
And here's one that might be a little controversial....

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Math is a symbolic representation of what happens in our real world. the highest level of mathematics that we have attempts to use pre-existing mathematical models to explain how things can exist beyond our current experience. Have you ever heard of a zero dimensional object? They exist, but they cannot be measured by the same scalers that we use as standards. So we represent them with a new symbol indicating that "this object must be measured differently". This is the highest form of mathematical expression, and it is not relegated to one brance of science. Where ever there is an unknowable but definite quantity, quality, or action, how we measure it is math at its best.

Actually, I think it goes beyond that. As Murray Gell-Mann puts it...

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Yes, sometimes. Cantor proved this. Basically, each set has something called cardinality, and two sets with the same cardinality can be mapped to each other, element for element. For instance, there is no way to take the integers and reals and pair them up, element by element. There is, however, a way to pair up the integers and rationals (fractions). Therefore, integers and rationals are the same size, but the set of reals is bigger than both.

When dealing with infinite numbers, things can get confusing. For example, there are exactly the same number of even integers are there are integers. If the sets were finite, you'd get twice as many integers as even ones, but because they are infinite, there are exactly as many.

As another example, suppose you take the set containing all the integers. I take all the elements of your set and put them in mine, and then sprinkle a couple of fractions in there, and maybe a car or a boat. My set is still the same size as yours.

There's a neat and related thing called the continuum hypothesis. I think that it would be very interesting to do math assuming that it isn't true, but I haven't seen much in that vein. Most people like it a lot.

There's also something called the Lebesgue measure that can be used to measure the "size" of infinite sets. Some sets, however, are non-measurable, and you get cool things like the Banach-Tarski paradox.

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snarkophilus has a good answer, I just want to add that it depends on how you define "bigger." If you use a strict superset/subset relation, e.g., every integer is also a rational, but some rationals are not integers, then by this relation you could claim that the set of rationals is "bigger" than the set of integers. But the answer given by snarkophilus (and some other earlier comments) uses a definition by which two sets are considered to be of equal size if there is a one-to-one mapping between them. A perhaps counter-intuitive result is that an infinite set can often be mapped one-to-one to a proper subset of itself. An even simpler example than the real vs. complex or integers vs. rational would be the set of integers versus the set of even integers. Clearly the set of even integers is a proper subset of the set of integers, so by the definition towards which you seem inclined, the set of integers would indeed be "bigger." But there exists a one-to-one mapping, map each integer to its double, and this is a one-to-one mapping between the set of integers and the set of even integers. For each integer, there is a corresponding even integer, and for each even integer, there is a corresponding integer. So using cardinality to determine which set is "bigger," we would conclude that these two sets are of the same size, even though one clearly includes everything the other includes and more.

I´d say that both real and imaginary numbers are subsets of complex numbers. Indeed, a real number can be regarded as a complex number with an imaginary part equal to zero (r + i0).

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In one set you have two degrees of freedom, in the other you only have one. Shouldn't that count for one set being "larger" in the same sense that a plane contains infinite lines (even though they both contain infinite points)?

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In my humble opinion, I would suggest that the highest level of math would be one with both explanatory and preditictive power for how the entire universe works. Of course this is cliche and is the goal of the GUT's and we do not know how simple or complex it might be. Still, if we ever could achieve this goal I think it would qualify. At this point in our search we are not even positive about how many dimensions that we exist in and could we be inside some other "space" that we are not aware of.

I agree that it has to be "turtles all the way down". But, which way is "down"?

Numbers are represented by individual points so that's all that needs to be compared. Two infinite sets are considered to be the same size if their members can be put into one to one correspondence with each other. This can be done with the reals and the complex numbers.

"This statement is false" is not really a paradox. Some statements have no truth value. "This statement is true" is another such statement. This should not be confused with statements of unknown truth value. "I will win the next PowerBall lottery" is either true or false but we don't know which. The truth or lack thereof of "This statement is false" is not unknown. It is neither true nor false.

This is what I'm having difficulty seeing though. You can't put points on a line into one-to-one correspondance with all points on a plane, can you?

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Here is someone's attempt to do just that, with some comments on some issues that arise. Of course, since a lot of people at this forum believe 4.999~ and 5 are different numbers, they shouldn't have any problem with the proposed method.

That's not right. He's just mapping the numbers a+ai to the set of real numbers, not the entire plane.
Ah - nevermind. I think I see what he's trying to do. I still don't like trying to put a two dimensional object in only one dimension though.

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Yup... in fact, it can be done with the reals and any finite dimensional real space. That means that there must be a line of infinite length which, when drawn inside a square, fills the square completely. It also means that a similar line exists for a cube, and for a hypercube, and so on.

You don't have to like it. You just have to believe it.

There is a suggestion that all of the information in a black hole is stored at its surface. That would be a reduction from a three dimensional space to a two-dimensional space, in the real world!

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I thought that was where the whole concept of fractal dimension came into play. You can have a fractal line that sort of fills a plane. As the fractal dimension proceeds from 1 to 2, the crazy-curve-shape fills the plane with greater and greater density. Here's the thing though - you need more than one finite number to describe your position along a crazy-curve-shape of dimension > 1, because the curve is infinitely convoluted. Just one real number won't do, unless you plan returning infinity for any deviation whose linear distance from the base-point is finite.

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Some others have answered your question of my statement, but I thought another example might be helpful.

The set {1,2} is smaller than {1,2,3}, because it is contained in it, right? But what about {1,2} and {2,3}? Neither is contained in the other, but they both have the same number of elements. Similarly, it is obvious that the odd positive integers have the same number of elements as the even positive integers, right? That's because for every odd number, there is an even number (the number just 1 more than it).

But, as montebianco mentions, we can say the same thing for the integers and the even integers. Although the even integers are completely contained in the integers, there is still one unique even integer for every single integer, and vice versa. That's why we say the sets are the same size: the size of the integers is the same as the size of the even integers.