I keep hearing people say that if time was infinite then it must have taken an infinite amount of time for us to reach the present, which is impossible, therefore time is finite. What do you think of this? Is it a logically sound argument?

Hmmm, interesting point. personally I think that time defies classification, and will remain to do so for as long as it continues...
(However long that may be. )

some things are best left unknown

Couldn't agree more, its good to have something intangable around.

Infinity doesn't exist. It is only a concept invented my math geeks to act as a default bucket for numerical series. Nothing is infinite. Every sub-atomic particle in the universe is a positive integer - not infinity.

There are two ways of looking at "infinity".

One way is like you suggest - that if something is infinite then the chances of anything randomly happening are infinitely small.

The other is if you begin with a non-random event within infinity then everything related to that event will behave as if the infinity that surrounds it doesn't exist.

If time is a consequence of space and space goes through cyclical 'reboots' (Big Bang - Big Crunch/Rip - Big Bang) then time is bounded by these events. It is finite, but eternal.

I'd say that it is not true. That is the same arguement that if you always cross half the distance to the other side of the room, that you will never get to the other side. Time is moving at a steady rate, just as a person would move at a steady rate across the room.

All measurements, such as length, width, height, weight, mass and time can be infinitely divided. Even after you reach the quantum scale you can still measure off halves, quarters and so on. And supposedly the universe is without end, so going big you should be able to double all physical distances infinitely.

ehm..yes...but what is more infinite, infinite or infinite²?