Measurements of infinity are impossible, or at least impossible according to the usual notions of size. If you cut infinity in half, each half is still infinite. In an imaginary scenario known as Hilberts grand hotel, if a weary traveler arrives at a fully occupied hotel of infinite size, no problem. You simply move the guest in room 1 into room 2, the guest in room 2 into room 3, and so on ad infinitum. In the process, youve accommodated all the previous guests and freed up room 1 for the new arrival. Theres always room at the infinity hotel.

We can play games with infinity, but we cannot visualize it. By contrast, we can visualize flying horses. Weve seen horses, and weve seen birds, so we can mentally implant wings on a horse and send it aloft. Not so with infinity. Its unvisualizability is part of its mystique.

One of the first recorded conceptions of infinity seems to have occurred around 600 B.C., when the Greek philosopher Anaximander used the word apeiron, meaning unbounded, or limitless. For Anaximander, the Earth and the heavens and all material things were caused by the infinite, although infinity itself was not a material substance. About the same time, the Chinese employed the word wuji, meaning boundless, and wuqiong, meaning endless, and believed that the infinite was very close to nothingness. In Chinese thought, being and nonbeing, like yin and yang, are in harmony with each otherthus the kinship of infinity and nothingness. A few centuries later, Aristotle argued that infinity does not actually exist, though he conceded something he called potential infinity. The whole numbers are an example. For any number, you can always create a bigger number by adding 1 to it. This process can continue as long as your stamina holds out, but you can never get to infinity.

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Indeed, one of the many intriguing properties of infinity is that you cant get there from here. Infinity is not simply more and more of the finite. It seems to be of a completely different nature, although pieces of it may appear finite, such as large numbers or large volumes of space. Infinity is a thing unto itself. Everything we see and experience has limits, boundaries, tangibilities. Not so with infinity. For similar reasons, St. Augustine, Baruch Spinoza, and other theological thinkers have associated infinity with God: the unlimited power of God, the unlimited knowledge of God, the unboundedness of God. God is everywhere, and in all things, inasmuch as He is boundless and infinite, said Thomas Aquinas. Beyond the religious sphere of the immaterial world, physicists believe that there might be infinite things in the material world as well. But this belief can never be proved. You cant get there from here. Most of us have our first glimmerings of infinity as children, when we look up at the night sky for the first time. Or when we go to sea, out of sight of land, and gaze upon the ocean extending on and on until it meets the horizon. But these are only glimmerings, like counting to a few thousand in Aristotles potential infinity. Were overwhelmed. But we havent even come close.

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