The concept of infinity and what it is has baffled philosophers for more than two thousand years.

Here are some definitions of infinity:

1. Unlimited extent of time, space, or quantity; eternity; boundlessness; immensity.

2. Endless or indefinite number.

3. A quantity greater than any assignable quantity of the same kind.

Intuitively, we think of infinity as something or entity so humongous (or so minute) that its size or numerical value cannot be determined at all.

Imagine the number of viruses (the smallest known biological entities) on the earth, the number of quantum “particles” in the universe, the amount of matter and energy (including dark) or the size of the expanding universe itself. In other words, infinity lurks everywhere if one looks carefully.

Mathematicians, physicists and cosmologists have developed techniques to deal with extremely large or small quantities. However, these cannot determine what is infinite because if so, then the final value determined is not infinite. This is the conundrum or paradox of infinity.

In the 19th Century, Georg Cantor, using set theory (which he created), proposed a revolutionary approach to define mathematical infinity.

From the wiki on infinity

A set of elements can be defined as infinite if the set has a seemingly paradoxical quality: a subset of elements in an infinite set can be matched up, one-to-one, to all of the elements in a set.

Consider the following:

1,2,3,4,5….......(the set of natural numbers) (NN)

1.3.5.7,9….......(the set of odd numbers) (ON)

There is a one-to-one correspondence for every number in both sets (if the number of numbers in both sets are assumed to be inexhaustible), therefore NN=ON and both sets are infinite.

Mathematically brilliant, except for the oddness of the outcome which arises from the definition of an infinite set whereby a subset (ON) is equal to the whole set (NN).

The part is equal to the whole contradicts Euclid’s Principle: “The whole is greater than the part”.

The other objection to Cantor’s approach is the reliance on induction. Unlike a finite set, an infinite set has undefined number of elements and it cannot be assumed there will be one-to-one correspondence for every element of both sets based on the experience of finite counting.

The next objection comes from finitism

In the philosophy of mathematics, finitism is an extreme form of constructivism, according to which a mathematical object does not exist unless it can be constructed from natural numbers in a finite number of steps. In her book Philosophy of Set Theory, Mary Tiles characterized those who allow countably infinite objects as classical finitists, and those who deny even countably infinite objects as strict finitists.

Another approach to inquire into the nature of infinity is to define vagueness as the essence of infinity. In the quantum realm, quantum objects are vague objects and empty space is not empty, it has “vacuum energy” and out of this infinite sea of vague reality, matter and energy emerges. Cosmologists theorize that after the big bang, space expanded exponentially (inflation) faster than the speed of light and is still expanding in the observable universe, but the actual universe is much larger than the observable universe.

William Blake’s sublime opening stanza in “Auguries of Innocence” aptly describes finding infinity:

To see a world in a grain of sand,

And a heaven in a wild flower,

Hold infinity in the palm of your hand,

And eternity in an hour.