Dedekind's Theory of Irrationals I have gone through one video lecture about Dedekind's theory of Irrational numbers. I just want to make sure that what I have understood about it is right! Please let me know if I need to know something about it. We divide the whole set of numbers into two classes which is known as a cut/section. Let us say L and R the lower and upper classes respectively. Properties: 1. Each element in the upper class is greater than the elements in the lower class. 2. Each element should belong to either of the classes. 3. If the cut is a rational number, then L had the greatest number and R will not have the least number or 4. If the cut is a rational number, then L doesn't have a greatest number and R has the least number. 5. If the cut is an irrational number then both the classes will not have the greatest & least members respectively. Since the aggregate of the classes are larger than the rational numbers themselves, the additional members are called as irrational numbers. Please go through it and let me know any more points ðŸ˜Š 
Hi For 2 you should note any irrational number cut does not belong to L or R 3 and 4 can be combined by stating L contains all real numbers less than the Dedekind Cut , R contains all real numbers greater than or equal to the Dedekind Cut. If a cut is a rational number then it belongs ro R. :) 
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Thanks for that ðŸ˜Š 
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Note my edited post , I forgot to include the word 'cut' between the words 'number' and 'does' :) 
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@#$%!!! I fixed it , thank you 
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The point is to construct the reals out of the rationals. At the start of the construction you don't have any irrational numbers. 
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At first, there are only Rational numbers and we divide it into two classes, later when we found that there are gaps in between Rational numbers, we have named the additional numbers as Irrationals and together the Reals. 
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Thank you. It's nice to get challenges from you :D That way I can learn more things :) 
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