There are infinite numbers between 0 and 1. There's .1 and .12 and .112 and an infinite collection of others. Of course, there is a bigger infinite set of numbers between 0 and 2, or between 0 and a million. Some infinities are bigger than other infinities. A writer we used to like taught us that. There are days, many of them, when I resent the size of my unbounded set. I want more numbers than I'm likely to get, and God, I want more numbers for Augustus Waters than he got. But, Gus, my love, I cannot tell you how thankful I am for our little infinity. I wouldn't trade it for the world. You gave me a forever within the numbered days, and I'm grateful.

We know that there is an infinite, and we know not its nature. As we know it to be false that numbers are finite, it is therefore true that there is a numerical infinity. But we know not of what kind; it is untrue that it is even, untrue that it is odd; for the addition of a unit does not change its nature; yet it is a number, and every number is odd or even (this certainly holds of every finite number). Thus we may quite well know that there is a God without knowing what He is.

The notion of infinity had long since been introduced into mathematics, but this infinity was what philosophers call a becoming. Mathematical infinity was only a quantity susceptible of growing beyond all limit; it was a variable quantity of which it could not be said that it had passed, but only that it would pass, all limits.

The notion of infinity had long since been introduced into mathematics, but this infinity was what philosophers call a becoming; Mathematical infinity was only a quantity susceptible of growing beyond all limit; it was a variable quantity of which it could not be said that it had passed, but only that it would pass, all limits.

[B]ecause that which is finite is always bounded with reference to something... it is necessary that there should be no end... [N]umber also appears to be infinite, and mathematical magnitudes, and that which is beyond the heavens. And since that which is beyond is infinite, body also appears to be infinite, and it would seem that there are infinite worlds; for why is there rather void here than there? ...If also there is a vacuum, and an infinite place, it is necessary that there should be an infinite body: for in things which have a perpetual subsistence, capacity differs nothing from being. The speculation of the infinite is, however, attended with doubt: for many impossibilities happen both to those who do not admit that it has a subsistence, and to those who do. ...It is ...especially the province of a natural philosopher to consider if there be a sensible infinite magnitude.

As for the mathematical infinite, to the extent that it has found a justified application in science and contributed to its usefulness, it seems to me that it has hitherto appeared principally in the role of a variable quantity, which either grows beyond all bounds or diminishes to any desired minuteness, but always remains finite. I call this the improper infinite [das Uneigentlich-unendliche].

Nature constantly begins the same things over again, years, days, hours, spaces too. And numbers run end to end, one after another. This makes something in a way infinite and eternal. It is not that any of this is really infinite and eternal, but these finite entities multiply infinitely. Thus only number, which multiplies them, seems to me to be infinite.

The transfinite numbers are in a certain sense themselves new irrationalities and in fact in my opinion the best method of defining the finite irrational numbers is wholly dissimilar to, and I might even say in principle the same as, my method described above of introducing transfinite numbers. One can say unconditionally: the transfinite numbers stand or fall with the finite irrational numbers; they are like each other in their innermost being; for the former like the latter are definite delimited forms or modifications of the actual infinite.

The Infinite has to be a relative concept. Go any distance: an infinite space means that there is more to be explored.

I don't know how many people know this, but often it's mindblowing when you learn, that some infinities are bigger than others. [...] The number of counting numbers...so 1, 2, 3, up to infinity...the numbers you would use to count things, that's infinite. The number of irrational numbers...so the numbers you cannot represent as a fraction, okay, there are more of those than there are counting numbers, by far. So these are orders of infinity. Then there are more transcendental numbers than there are irrational numbers. So that's a number you'll never find as a solution to an algebraic equation. So pi is a transcendental number. e is a transcendental number. These are magic numbers that show up in mathematics. And it turns out there's an even bigger infinity of those than there is of these other two classes of numbers. And they use the Hebrew letter aleph in ranking. So it's aleph-1, aleph-2, aleph-3, aleph-4. I think there are five levels of infinity.

Infinity...is used in physics simply as a shorthand for "a very big number.”

There is finitude — unless this were true, infinity would have no meaning. The contrast of finitude and infinity arises from the fundamental metaphysical truth that every entity involves an indefinite array of perspectives, each perspective expressing a finite characteristic of that entity. But any one finite perspective does not enable an entity to shake off its essential connection with totality. The infinite background always remains as the unanalysed reason why that finite perspective of that entity has the special character that it does have. Any analysis

There cannot be a single, simple body which is infinite, either, as some hold, one distinct from the elements, which they then derive from it, nor without this qualification. For there are some who make this (i.e. a body distinct from the elements) the infinite, and not air or water, in order that the other things may not be destroyed by their infinity. They are in opposition one to another — air is cold, water moist, and fire hot—and therefore, if any one of them were infinite, the rest would have ceased to be by this time. Accordingly they say that what is infinite is something other than the elements, and from it the elements arise.

A great deal of misunderstanding is avoided if it be remembered that the terms infinity, infinite, zero, infinitesimal must be interpreted in connexion with their context, and admit a variety of meanings according to the way in which they are defined.