MC, That's right. To understand this, you need to find something on cardinal numbers and their arithmetic. The original author was Georg Cantor, but Wikipedia should have more recent references. Cantor basically said that sets have the same cardinality if they can be put into a 1-1 correspondence. For example, we can pair 1 with 2, 2 with 4, 3 with 6, etc. Therefore the positive integers and the even positive integers have the same cardinality, even though one properly contains the other. To understand how this works, you'll just have to get a good source and "get dirty" with the details. Best wishes. Ben
> Date: Tue, 30 Apr 2013 20:12:05 -0400 > From: firstname.lastname@example.org > To: email@example.com > Subject: Re: RE: How does infinitesimal exist? > > 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.