suppose A and B are two people,they can live forever,will never die.if omega exists really,one day,the have an omega-metre long run race on a straight road more than omega metres long.BOTH A and B run a step per second,but A runs 2m per step,B runs 1m per step.clearly B runs slower than A,so the distance between A and B will be further and further.but in ordinal arithmetic we know that 2*omega=omega,so A and B will finish the race in the same time,omega seconds.the question is,how can B finally catch up A?
related en.wikipedia.org/wiki/Zeno%27s_paradoxes#Proposed_solutions– DruxMar 21, 2014 at 13:04
You cannot "finish" an infinite long race ... After a finite amount of time (let N seconds) A will be far ahead to B but he still have to run for an infinite amount of time. And you cannot either stay at the finishing line waiting for the winner : for also if you started your travel well in advance with respect to the start of the race, you will never reach the end...– Mauro ALLEGRANZAMar 21, 2014 at 13:13
Here you are slightly abusing logic. 2 omega is not 1 omega by "ABSOLUTE" value they are same by the ORDER of magnitude, which means it is same Type of infinity. So by order of magnitude yes you are correct they will be at the same time. But by "ABSOLUTE" value of time B will be slower. It is a question of a debate/thinking/research how to operate with infinity. Clearly what is possible is to compare ORDER of infinities magnitude but not EXACT absolute value, which based on this example ALSO may exist.