It cannot be rightly said that infinity has a nature unless the nature of infinity includes everything and more. Infinity is so wide open that it must include all that is conceivable as well as all that is inconceivable. In other words the nature, of infinity cannot be known conceptually in any way.
It cannot be known if there exists anything infinite or not. Science doesn’t postulate infinite states because the very most infinite conception, the universe, is thought to have an edge where expanding light from the Big Bang is moving away from that singular event. Don’t ask what that frontier edge is moving into or whether it will cease to expand and fizzle out or collapse due to gravity. It’s a complete waste of time; 25 billion years and ticking.
Yet without infinity the finite cannot exist. If we look past Descartes to the real evidence of I Amness, it is the propensity to ask questions at all not the thoughts that form questions. There is an underlying consciousness that is necessary for anything to exist conceptually or otherwise. The very nature of consciousness, being not a thing or nothingness has to be infinite by definition. It is immeasurable. It has no beginning or end. It includes the unconscious which is totally unknowable because nothing can occur in deep sleep..
The fact of infinity is a more dominant trait in a person than anything that can be known about them because it can never be absent by definition. It can’t be limited by thought. It has to contain all time and space within itself. It has to be the ground of every event that has ever taken place in the universe. To be infinite is neither to be big or small, on or off, existent or nonexistent or any other description that the knowable thoughts of anyone, including God, can reach. Infinity must be now exactly as it has been forever in the past and forever in the future.
Have an infinitely good time forever!