this isn't homework this is from a book im reading for fun(adsbygoogle = window.adsbygoogle || []).push({});

L is the set of all negative rationals and zero and all numbers whose square is less than 2. R is the set of all numbers whose squares are greater than 2. this defines[itex]\sqrt{2}[/itex]

so thats cool i understand that, the irrational number is the interstitial region between the two sets.

what i don't understand is how applying the dedekind to the reals you get reals. maybe by analogy in that if you apply it to the rationals and the cut is inside one of the sets then the cut defines a rational number?

x is real and has a square less than 2. this defines anLclass with no largest member and anRclass with smallest member [itex]\sqrt{2}[/itex]

when we cut the rationals we say that if L is everything with a square less than 2 then that set has no greatest number and conversely or w/e R is the set of all numbers with squares greater than 2 but this set has no bottom.

what is different in the definition of the cut in the reals that makes it include the square root of 2 in R?

edit

ahh i understand wow i'm dumb it makes sense that if we are now including irrationals in field(right word?) then sqrt(2) is included in one of the real sets. and since its square 2 it should be in the set that does not contain the numbers whose squares are less than 2

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# Dedekind Cut

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