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Consecutive Reals 
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#37
Jan1514, 09:42 AM

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#38
Jan1514, 10:01 AM

P: 1,622

In any case, there are lots of reasons to use infinite sets. The first is utility and the widespread use infinite sets have found describing the natural world. The second is aesthetics in that doing math without infinite sets, while possible, gets quite ugly. The third is that even talking about problems in elementary arithmetic gets quite difficult if you discard the notion of infinite sets. For example, one common proof technique works by establishing that a proposition is true for the base case and then showing that if it holds for some arbitrary case, then it must also hold for the next one. Unfortunately without infinite sets you cannot quantify the statement over the set of all natural numbers so this natural proof technique gets broken. One way to fix this involves turning ordinary theorems in arithmetic into metatheorems, much in the same way proper classes are dealt with in ZFC, but this is less than satisfying in my opinion. 


#39
Jan1514, 10:10 AM

P: 82




#40
Jan1514, 10:21 AM

P: 1,622




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