The definitive answer involves definitions. In order to answer the question “does infinity exist” we must be clear about what we mean by the terms “infinity” and “exist”. To this end, we will examine the definitions returned by Google’s own dictionary.
The first definition of infinity: Infinity has two definitions, the first of which is “the state or quality of being infinite”. This leads us to look up ‘infinite’ which also has two definitions.
The first definition of infinite: The first definition is “limitless or endless in space, extent, or size; impossible to measure or calculate”. To understand the first part of this definition we can apply it to the sequence of natural numbers (1,2,3,4,…). Here we can appreciate that however high we go, we can always add 1 to get the next number, and in this sense the sequence is endless.
Unfortunately we cannot demonstrate this property of endlessness in the real world. Any counting process, either on a computer or in a human brain or implemented some other way, will inevitably hit a physical limit where it cannot count any higher. Taking this on-board, we could claim that endlessness means there is no defined end point within the implementation constraints.
The second part of this definition talks about being ‘impossible to measure or calculate’. But if we understand what an endless sequence is then we already appreciate that trying to measure its length does not make any sense. And so on one hand this second part of the definition could be dismissed as just being superfluous, but on the other hand it could have been purposely written to imply that an infinite object can exist but cannot be measured. If so, this infers a non-finite object can exist without providing any explanation how.
The second definition of infinite: The second definition of infinite is “another term for non-finite”. For the first definition we visualised the endlessness of natural numbers. From this, we know we can never have all natural numbers because we can always add 1 to the largest (within implementation constraints). This second definition directly contradicts the first as its hidden implication is that we can gather all natural numbers together in an infinite object. And we are still left with no clue of how to construct something that is non-finite.
The second definition of infinity: Returning to the definitions for infinity, the second definition claims to be the one used in mathematics. It says “a number greater than any assignable quantity or countable number (symbol ∞)”. Most mathematicians claim infinity is not a number and yet this mathematical definition claims it is. To understand the rest of this definition we first have to understand “assignable quantity” and “countable number”.
As there does not appear to be a mathematical definition for “assignable quantity” I suspect that word ‘assignable’ has been used for the purpose of misdirection. Assignable or not, it is far from clear how numbers can reach a limit or how this thing called infinity can be greater than this limit.
Similarly there is no definition for a “countable number” but Google offers up the definition of a “countable set” in its place. This definition assumes the set of all natural number exists, and so even if we could use this “countable set” we would have a definition that already assumes an infinite object exists.
Can we use these definitions?
If there were just one definition for infinite, and it was ‘endless aside from implementation constraints’ then we could say with some confidence that the sequence of natural numbers was infinite. But there would be no need to use the word ‘infinite’ at all because ‘limitless’ or ‘endless’ would covey the concept with greater clarity. And as we have already seen, the concept has several definitions that make little sense. So if we describe something as infinite then we infer these multiple meanings, which can be described as unclear at best.
We could go on to discuss the meaning of ‘exist’ but, as we have failed to get a satisfactory definition for infinity, there is little point.
And so what is the definitive answer?
Many people insist infinity must exist. They claim it is obvious because they believe there exists ‘an infinite number of’ numbers. But the problem remains that the definitions of the core concepts do not stand up to reasonable scrutiny.
And so the definitive answer to “does infinity exist” is that the question cannot be answered as ‘infinity‘ does not have any clear meaning.
See ‘Investigation of infinity in mathematics‘ for more about the definition of infinity.