Infinity Paradox

[dangerdan 0]

I've been reading and thinking a lot about the nature of infinity and particularly how paradoxical it is.Example: there are infinite numbers, however within this set of numbers there are also infinite square numbers. The same principle applies to odd, even and prime numbers. An infinite set within an infinite set. Example: infinity plus one equals infinityExample: there are infinite numbers. there are infinite numbers between 0 and 1.Example ( a worrying one!): if you have infinite income, no matter what income tax is levied at, you pay infinite income tax.The best way to visually think about infinity is to consider any graph which features an asymptote. An asymptote is when a graph approach's a value without ever actually reaching. The graph continues infinitely, but there are also infinite possibilities without the graph reaching the point it is asymptotic to.

[Nabb 0]

I think the key point which you are missing is that infinity is concept and not a real number, which is why the standard axioms do not hold.

Edited June 15, 2009 by Nabb (see edit history)

[dangerdan 0]

I'm not missing that point of course I understand that infinity is only a concept. It is this conceptual analysis of infinity that allows you to arrive at the paradoxes I pointed out.

[HDuffRules 0]

Georg Cantor in the late 1800s is responsible for the manner in which infinity is treated in modern logic/mathematics applications. Cantor mathematically demonstrated that infinity comes in different sizes.

Let us consider all the counting numbers (1, 2, 3, 4, 5... etc)
Let us also consider all the even numbers (2, 4, 6, 8... etc)

These two forms of infinity are actually the same size. There are arguments to be made that either set is twice as large as the other, but this is not the case. For every number we have in the second set, there is a corresponding number in the first set (and vice versa). This gives us a 1:1 correspondence ratio, which then leads to the conclusion that that both sets contain the same amount of numbers, and thus represent the same infinity.

However, if we consider the infinity of all fractions (yes, all of them), we will find that this infinity is larger than the two infinities we have previously considered.

Let us consider the counting numbers again, and then try to conceive how we will depict the fractions. We can effectively partition the counting numbers up so that they form intervals (from 0-1, 1-2, 2-3, etc.), and some simple logic will tell us that there are an infinite number of fractions that fits in every single interval. Extrapolating, since we have an infinite number of intervals and an infinite number of fractions in each interval, we conclude that the infinity of fractions is larger than the infinity of counting numbers (Incidentally, it is bigger by one order of magnitude. Whatever that actually means, I have no real idea.)

A more rigorous, mathematical proof is provided here:
http://www.bellevuecollege.edu/math/

Edit: I noticed that I got a little off-topic and didn't totally address your concerns.
The reason that the different sizes of infinity are relevant is that in your scenario, the income infinity is larger than the tax infinity. Assuming that you're not getting taxed at 100% (at which point I would consider moving to a different country) and that the tax rate is constant (I don't want to go into progressive/regressive tax schemes right now, although it shouldn't affect the results at all), you still have some money to spend, and thus the tax infinity MUST be smaller than the income infinity. If this explanation is insufficient, consider a real-world (sort of) example, which I've modified slightly.

Assume that everyone is always employed (great world) and that everyone is taxed at 25% (even great worlds have taxes). Finally, assume that everyone lives forever (OMG AWESOME).

What happens in this scenario? You can see that your income is infinite. Your taxes are also infinite. However, you still get 75% of your income to spend, demonstrating that the income infinity is larger than the tax infinity.

Edited June 15, 2009 by HDuffRules (see edit history)

[truefusion 3]

I've been reading and thinking a lot about the nature of infinity and particularly how paradoxical it is.

You use math in all of your examples. Technically, the phrase "goes on forever" can be substituted for that which is declared infinite. Yet, is "infinity" merely a concept or is it possible to prove? You can't show that your math equations are logical. And while it may not be necessarily safe to assume that the universe is infinitely big, people would have a hard time picturing running into a "wall" at the end of the universe. Can something go on forever, or must all things come to an end? If there is no such thing as something that is infinite, then wouldn't it follow that proving that something cannot go on forever is easy? Or would finite, then, be hard to prove? You may be able to show in theory that infinity is hard to impossible to prove, but doing so does not necessarily mean that you have shown that there can only be finite.

Example ( a worrying one!): if you have infinite income, no matter what income tax is levied at, you pay infinite income tax.

This example allows for no form of income tax to be levied at a person.

[Nabb 0]

However, if we consider the infinity of all fractions (yes, all of them), we will find that this infinity is larger than the two infinities we have previously considered.

This is incorrect, these infinite sets have exactly the same cardinality (see for example the pairing function which maps NxN onto N). The link you provided demonstrates that the real numbers (which we can create a one-to-one mapping against the power set of integers) has a higher cardinality from that of the integers.

The reason that the different sizes of infinity are relevant is that in your scenario, the income infinity is larger than the tax infinity.

Again, incorrect. These infinities are of the same cardinality. The correct approach to determine the remaining income would be to take the limit of x-(x/t) as x approaches infinity (where t is the tax rate, between 0 and 1). This diverges to infinity as x approaches infinity, and hence one would still have infinite income.

[HDuffRules 0]

This is incorrect,

Again, incorrect.

Okay fine, make me feel stupid