So I just want to make sure I have this straight. You are telling me that on (0,1) you have aleph-nought rationals and that on (0.5,0.75) you also have aleph-nought rationals.
Am I correct in thinking then that the intersection of the rationals contained on the intervals (0,1) and (0.5,0.75), when regarded as being seperately "infinitely divided" into rationals, would have a smaller cardinality than aleph-nought, but still be infinite? This seems like a contradiction, so it confuses me. Am I making some logic mistake somewhere, or is there some concept that covers this which I am clearly unaware of? Thanks a bunch for answering.
I looked up cardinal number and I get aleph-nought, but what specifically they mean by aleph-one, aleph-two, etc. confuses me.