This question determines the boundary of science and art, mathematics and philosophy, Ego and Super Ego. For example, the mathematicians have varying beliefs because it is not something which has been defined to a point that can be agreed upon, whereas the reals, HYPER-REAL's, integers and what have you, have been. This is curious for a number of reasons, and helps lead us to the result.
First, let us acknowledge that there are 3 basic axioms of group thought: 1) We agree on an axiom 2) We don't agree on an axiom 3) Some other position, which may be considered the set of all sets that are not Axiom 1 or 2.
If we agree that the symbol ? (called 'infinity') represents the set of all integers (or reals or whatever number system you want to use), and has no cardinality, that could be a definition. That it represents something UNREAL, one could contend that the reals are a subset of the 'unreals', since numbers are not real in the sense that they have any real-world matter (or energy), and are simply the representation of an idea. Clearly, we begin to move away from computational mathematics and into the realm of philosophy (perhaps physics, religion or psychology, etc.) because the goal of computations using unreal numbers (or specifically ?) is to be able to consistently apply axioms to get repeatable results. Thus, we could give ? similar properties as 0, 1, or some other symbol, or we could give it some OTHER property.
We have now created another divergence in which we get a trinity of yes/no/maybe, similar to our 3 postulates above. By induction, we can see that reasoning through a definition of ? creates a 'schism triad' which will continue diverging fractal-like as we approach infinity. How might this be represented symbolically?
First, a few definitions: GUN - Greatest Unbounded Number ? - schism triad (a triangle, which does not represent delta, it represents the 3 possible paths available to our decision tree) r - the resolution of the current level (think digits of precision)
Using some familiar symbols:
? = (GUN/?)*e^r
in other words, infinity represents the largest numeric value divided by either 1,2 or 3, times exp to the power of the specified resolution (precision). So when we consider the first schism triad axiom, which category do you believe is correct?
Let's start with case 1, that you agree there are 3 axioms. Then, you want to look at ? to 10 digits of precision. Simply multiply the largest number you want by e^10, and use this value in your calculations.
Let us consider case 2, that you do NOT agree there are 3 axioms. Then, use the largest number in your calculator, divide by 2 and multiply the result by e^n (where n = the resolution/ precision...), and use this value for ? in your computations.
For case 3, you get the idea.
What happens is that when certain boundaries are approached, you move from one level of resolution to the next order of magnitude (hence the use of e). Since you have a limit of 3 choices of 'belief' in the definition, the ? is really a recursive 3, which divides a torus such that the result is a phi spiral of a tri-state that continually 'folds back on itself.' The closed loop of infinity coincidentally implies its hyperbolic torus - spiral nature. I would prefer the 3-petal lemniscate to ?, which would allow me to remove the cumbersome '/?)*e^n' and thus express the inherent 'three-ness' of infinity, which allows the unfolding of the planar space into 3-space and higher dimensions. Of course, if in your belief system you have 4 axioms, you could use a 4-petal lemniscate, but then you'd be wrong, because 4 axioms are implied by the TAO dualism state.
Getting back to my main point, in order to even attempt a Serious definition of ?, one must add some kind of axiom to any number system, that gives it the life of randomness, or initial conditions, or curved space, or manifold topology, or whatever unfolds in the future which thought leaders will agree upon. I hope in my example above, it has given rise to thinking about the problem of ? in a whole new way (and if not, sue me). Since none of the PhD's can agree on a definition of ?, nor can they use it in a consistent, logical fashion that makes sense to everyone, we should make defining infinity to be one of those mathematical puzzles which, when truly 'solved', that person wins a million dollars and a Field's Medal..