Jarlaxle

Is it generally accepted that infinity in actuality is impossible, or is this still highly debatable?

Jarlaxle

ohwilleke

Infinity is concept which describes a host of common place and ubiquitous aspects of our reality. For example, there are an infinite number of points between your fingers. There are an infinite number of possible light frequencies. There are an infinite number of real numbers, an infinite number of imaginary numbers, an infinite number of integers, and so on.

Jarlaxle

So infinity is conceptual only, such that there can not exist an actual infinite number of events? (i.e. there can be an infinite number of possibilities, but not all possibilities can occur)

Automaton

Singularities may represent an actual function of infinitude in reality. Although I disagree with singularities for the fact that I don't think something with no spatial dimensions can exist. Some Big Bang cosmologies suppose that our universe started off with infinite heat, but since heat is a form of kinetic energy, wouldn't the particles had to have oscillated over an infinite distance, which is ludicrous?

It's a tricky issue.

Jarlaxle

Can one infinite be larger than another?

I understand that the number of integers is infinite, and likewise the number of positive integers is infinite. But are they exactly the same size? Mathematics suggest that they are the same size since there is a 1 to 1 relationship between them.

However, the number of real numbers is larger because there is not a 1 to 1 relationship. Hence the "number" of real numbers is deemed to be uncountable (bigger than infinite).

From what I currently understand there is finite, infinite, and uncountable. Are those the only 3 groups of quantities?

Jarlaxle

scumble

Hi there.

I reckon you have to be careful about concepts of infinity. What you're talking about is mathematical concepts of infinity.
I'm not sure if a human can say if infinity has any relevance in the real world, to be honest. Mathematics has certain limits. Some techniques have been developed to deal with infinite numbers, but it's only any use if it helps model a real process (from a physicist's point of view at least).

Blake

Yes, one infinity can be larger than another. You are correct that the number of integers is the same as the number of natural numbers, and that the number of real numbers is an infinity that is larger than the number of integers. However, it isn't an uncountable infinity. The formal name for the integer-infinity is "aleph-null," the one for the real-number-infinity is "aleph-one," and there are an infinite number of infinities progressively larger than each of these.

The mathematician who is best known for work with infinity is Cantor (first name Georg, I think).

Blake

CardinalMan

You are correct in saying the cardinality (size of) the natural numbers equals the cardinality of the integers, which is usually denoted by aleph_0. The real numbers do indeed have a larger cardinality, and hence are called uncountable. Any infinite set of cardinality larger that aleph_0 is called uncountable (hence the set of reals is uncountable) because there is no way to "list" them by "counting" them off one-by-one.

Mathematicians define aleph_1 to be the next larger cardinality after aleph_0. The question of whether or not the real numbers have cardinality aleph_1 (meaning that there is no size of infinity strictly between the natural numbers and the real numbers) is a very famous problem in mathematics known as the Continuum Hypothesis (because the collection of real numbers is often referred to as the Continuum).

Now the history of the Continuum Hypothesis is quite interesting. Georg Cantor spent most of his life trying to solve it, but failed. Some mathematicians began to wonder whether or not we could really solve the Continuum Hypothesis. Kurt Godel (I believe in the 1940's) gave an ingenious argument that proved that, from the basic axioms of set theory, one could not prove that the Continuum Hypothesis is false. This still left open whether it could be proved true. However, in the sixties, Paul Cohen gave an even more ingenious argument proving that, from the basic axioms of set theory, one could not prove that the Continuum Hypothesis is true. Hence the Continuum Hypothesis is independent of the other axioms of set theory, the framework that modern-day mathematicians work in.

The question of how to deal with this independence leads to a variety of different philosophies of mathematics. A growing number of set-theorists now believe that the Continuum Hypothesis is false, and that the cardinality of the real numbers is aleph_2, the infinity just after aleph_1. Of course, by the comments above, they can not prove this statement, but they have come out with plausibility arguments that use mathematical analogies. It is still far from obvious how this well be settled in the long run.

I've written a brief whimsical description of the <a href="http://www.math.uiuc.edu/~mileti/infinite.html" target="_blank">The Different Sizes of Infinity</a>. The ending cuts off quickly, but I plan to add more soon (once this semester ends).

The infinities keep right on going. We have aleph_0, aleph_1, aleph_2, etc. but once we finish here we have a sort of "limit" cardinality called aleph_omega just above these. We then continue with aleph_(omega+1), aleph_(omega+2), etc. In fact, there are more sizes of infinity than there are objects in any given infinite set.

CardinalMan

owleye

Jarlaxle....

I think the question you are asking is whether an infinite number of things actually exist in the world. For some reason, most of your respondents want to focus on modern mathematics, thereby either wanting to impress us with their knowledge, or to equate the domain of a particular axiomatic system with the domain of reality.

It was Aristotle, of course, who held that infinities were never actualized. They existed only potentially, without the possibility of being completed. Indeed, with Aristotle's logic, it is not possible to construct entities having an infinite number of elements. Plato, on the other hand, being more mathematically inclined, was untroubled by infinities. Indeed, rationalist followers of Plato (Augustine, Galileo, Leibniz) had no difficulty with positing infinities). Empiricists, following Aristotle, however, could not conceive how this was possible.

Interestingly, Kant, rather falling in line with Aristotle, found a middle course, despite that he had only subject-predicate logic to work with. Mathematical infinities can be applied to the world on the presupposition of the independent existence of space and time (which was largely defined in accordance with Euclid's geometry and Newton's laws of motion). Objects exist in space, and inherit its properties, one of which is that space is infinitely divisible (thus objects existing in space are infinitely divisible). But to admit this, Kant had to require that such infinities deal only with phenomena, or things as they appear to us. Space and time (and their mathematical properties) are merely forms of perception, and do not actually exist in themselves, as objects are thought to do. As such, for things as they are in themselves, mathematics has nothing to say. It is only logic that would apply to them, and this restricts it to finite entities (though the number of elements comprising its finititude is not limited -- any finite number can be dealt with).

However, since the late 19th century, with the advent of quantificational logic, developed by Frege, and particularly when the era of modern physics dawned, mathematics is not generally thought to be constitutive, but instead serves as a model embedded in a given theory of the world. As such, it is no longer necessary to require that mathematical entities have objective reality in order to establish a successful theory. The mathematical points of space and time, even though we can construct a geometry having an infinity of them, do not necessarily exist, objectively.

On the other hand, despite that there are attempts currently underway to model space in accordance with its having discrete, rather than continuous, properties, I am pessimistic that they will succeed (though I suppose a new property that shares aspects of both continuity and discreteness is possible). In particular, I have heard no solution to the problem Hermann Weyl raised with respect to the extreme difficulty of accomplishing this feat (it refers to the problem of tiling a space having more than one dimension).

owleye

CardinalMan

I am doing neither. Jarlaxle has three posts posted on this thread, one of which said:

This is a mathematical question about the nature of infinity and deserves a mathematical answer. I was replying to this post, and Bill the Cat's response, because there were some misconceptions present (the real numbers are uncountable but do not necessarily have size aleph_1). If you want to reply to other more philosophical questions about infinity, please do so. That was not my intention.

On a slightly related note, it is my opinion that in order to have a reasonable philosphical discussion we first need to know the facts underlying the discussion. I find it silly to philosophize about the nature of matter and energy without thorough knowledge of basic physics. Similarly, I find it silly to speculate philosophically about the nature of infinity without a firm understanding of the mathematical facts about infinity.

CardinalMan

Edited for spelling

[ April 23, 2002: Message edited by: CardinalMan ]</p>