I'm astounded at how many deep thinkers we have on Mayhem .....

Amongst other people, Galileo addressed the issue of adding and subtracting from infinity in his

*Two New Sciences* (1638), and 'concluded that the ideas of less, equal, and greater apply to finite sets, but not to infinite sets.' That last phrase is directly lifted from a wiki entry - see

http://en.wikipedia.org/wiki/Galileo's_paradox for the reasoning behind that statement. So Galileo's answer would be that infinity-1 breaks the rules and does not really exist.

After Cantor, an alternative approach, and the one usually used by current mathematicians, is the use of transfinite arithmetic, as evidenced by Hilbert in his famous 'Hotel' illustration, where he uses Cantor's pairing argument to show that 'infinity+1=infinity' (at aleph-null cardinality). Similarly, infinity-1 would also equal infinity. And, before you ask, yes, there is an aleph-infinity cardinality. It's called aleph-aleph null ....

Have a look here:

http://io9.com/5809689/a-brief-introduction-to-infinity Nothing deep with this - it's just maths, which is a simple process of following the rules. Unlike some the problems in life that I do have difficulty with, such as why I cannot get the transom on my boats to stay at 90 deg to the keel, and why my wife complains if I use her shoes to hold a sheet of balsa down when gluing....