Odd.

In the world of mathematics, the idea of infinity is both fascinating and complex. The ability to count infinity is a topic that generates much interest and discussion. Let's explore this idea by starting with a base number, such as 1, and then counting only the odd numbers. Then we double each number to observe an interesting phenomenon: we can theoretically halve the 'time' it takes to count to infinity.

Imagine you start counting at 1. Then you move on to 3, 5, 7, and so on. These are all odd numbers. Counting only odd numbers is equivalent to selecting half of all positive integers. If you double each of these numbers—so 1 becomes 2, 3 becomes 6, 5 becomes 10, and so on—you are effectively counting every second integer.

By counting only the odd numbers and then doubling them, you skip a step (the even numbers that you do not count individually but generate by doubling). At first glance, this seems like a more efficient way to go through the numbers 'faster', especially if your goal is infinity. However, in the mathematical sense of infinity, this method does not actually save time. Infinity remains an unreachable limit, regardless of the method of counting.

The paradox of this approach lies in the concept of infinite sets. Mathematicians such as Georg Cantor have shown that there are different 'magnitudes' or cardinalities of infinity. For example, the set of all integers and the set of all odd numbers are both infinite, but interestingly enough, they are in bijective correspondence with each other—any odd number can be associated with an integer. This means that even though we have a method to halve the 'time', both sets are the same size in terms of infinity.

The idea of "reaching" infinity faster by taking every second step is more of a thought exercise than a practical application. It underlines the complexity and intriguing nature of infinity in mathematics. It shows us that even with clever tricks, some concepts, like infinity, remain fundamentally elusive.