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August 20th, 2017, 10:58 PM   #1
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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 😊
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August 21st, 2017, 12:14 AM   #2
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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.


Last edited by agentredlum; August 21st, 2017 at 12:25 AM.
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August 21st, 2017, 12:16 AM   #3
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Quote:
Originally Posted by agentredlum View Post
Hi

For 2 you should note any irrational number 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 less than or equal to the Dedekind Cut. If a cut is a rational number then it belongs ro R.

Sorry! I forget to include that point

Thanks for that 😊
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August 21st, 2017, 12:21 AM   #4
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Quote:
Originally Posted by agentredlum View Post
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 less than or equal to the Dedekind Cut. If a cut is a rational number then it belongs ro R.

I think here you have mentioned less than or equal to instead of great than or equal to!?
Thanks from agentredlum
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August 21st, 2017, 12:21 AM   #5
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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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August 21st, 2017, 12:23 AM   #6
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Originally Posted by Lalitha183 View Post
I think here you have mentioned less than or equal to instead of great than or equal to!?
Oh shoot , you're right!!!

@#$%!!!

I fixed it , thank you

Last edited by agentredlum; August 21st, 2017 at 12:28 AM.
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August 21st, 2017, 10:40 AM   #7
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Quote:
Originally Posted by Lalitha183 View Post
We divide the whole set of numbers into two classes ...
Which set of numbers?
Quote:
Originally Posted by Lalitha183 View Post
5. If the cut is an irrational number ...
If you already have the irrational numbers, then what is the point of the exercise?

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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August 21st, 2017, 07:09 PM   #8
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Quote:
Originally Posted by Maschke View Post
Which set of numbers?


If you already have the irrational numbers, then what is the point of the exercise?

The point is to construct the reals out of the rationals. At the start of the construction you don't have any irrational numbers.
Sorry about including all the points together!

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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August 21st, 2017, 07:36 PM   #9
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Originally Posted by Lalitha183 View Post
Sorry about including all the points together!

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.
Yes correct. Just wanted to make sure that was clear. If someone ever challenges us to prove that the set of reals exists, we can whip out this argument. "If you believe in the rationals we can construct the reals." Otherwise nobody ever cares about it. Nobody actually believes that a real number is a pair of sets of rationals. Now we're into philosophy! There are many constructions of the reals. The real numbers are the abstraction represented by all of our models. They aren't the models themselves. In which case ... what are they exactly? Don't worry this won't be in your math curriculum
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August 21st, 2017, 07:41 PM   #10
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Thank you.
It's nice to get challenges from you That way I can learn more things
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