Newbie here. I am a bit confused here - If Dedekind cuts are subsets of rational numbers, how could real numbers be constructed by those Cuts, as they include rational numbers only? Maybe the "cut point" is also considered part of the Cut, which might be an irrational number?
To my understanding, a Dedekind cut just says, "here you have a number line of some random points (easiest is propably 0 to 1) and we now cut that in half. Take that new point (1/2) and consider one of the original points (I´ll take 0). Now cut that in half again and you get 1/4. do that againa nd again infinitely many times and you get a number arbitratrily close to 0. Now, we didn´t actually always have to use the left side (imagine you have a number line), but could have gone to the right side at any point. Thus we now have gotten every possible rational number there is between one and zero" Now we can do the same thing between any other two points. As long as you took rational numbers, this actually plots the set of all rational numbers. But what if you took squareroot of 2 at the start insted of one (so now you imagine the number line from 0 to sqrt 2 (squareroot 2)) which is obviously irrational. If we do the same procedure we´ll get all the irrartional numbers between 0 and sqrt 2. Both sets together are the real numbers for this interval, and all possible intervasls thus construct the real numbers. That´s it to the Dedekind cut. Now any individual "slice" we took from that is obviously a subset of the real numbers because we took only the rational and irrational, thus the real numbers into consideration. Howevere all these sclices together make up the real numbers. Thus all possible slices united are in fact the real numbers. This property is actually pretty common and not at all limited to the Dedkind cut. Another example would be as follows: Imagine you took a set of natural numbers up to 10. Now you say the set of any individual number (so 1,2,3,...10) is each a subset of 10 which is true since all of these sets are in their entirety in the set of the natural numbers up to 10. If you unite them all you get the set of all natural numbers up to 10 again.
Great video! What I don’t understand is how dedekind cuts create the real numbers if they are subsets. If you define a set a a collection of subsets, and those subsets only contain rationals, how to the reals arise? Are the reals a set of all supremums of the dedekind cuts? I was also wondering how we can create a dedekind cut for the cube root of two before having defined what a cube root means in the reals.
@obikenobi3629 Great question. In every construction of the real numbers, the definition is a bit surprising. In the Cauchy sequence definition, real numbers are defined as Cauchy sequence of rationals modulo Cauchy sequences converging to zero. This sequence and all of its representatives can be thought of as a subset of Q as well!
Yes. Let's take for example pi. You can define a subset of the rational number such as every element of the set is less than any partial sum of a series that converges to pi. Then, you have defined a Dedekind cut that represents pi.
OK...Uncle Doug here...explain to me how you can write so straight across the screen??? I mean as a retired engineer, I like straight lines. However, for the life of me, I can't even sign my name anymore without trailing off the page…
@princez2835 Every irrational number can be realized as a Dedekind cut! I will add more example videos on this, but most of the famous irrational numbers can be represented by a convergent infinite series (convergent needs to be properly defined, but the Least Upper Bound Principle will help us), and this representation will help us build the cut.
@@mikethemathematician thanks! but i have another question. if real numbers are subsets of the rational numbers, than does it mean that all the real numbers are inside the rational numbers? It doesn't make sense since rational numbers do not contain irrational numbers.
@princez2835 Good question! If we look at the set of rational numbers {p: p^3 < 2} (this is a Dedekind cut), then we can identify this set with its least upper bound which happens to be 2^{1/3}, which is an irrational number! Likewise, for any rational number r, we can identify the Dedekind cut {p: p< r} with its least upper bound of r. There is certainly a lot to unpack here, but you got it @princez2835!
Newbie here. I am a bit confused here - If Dedekind cuts are subsets of rational numbers, how could real numbers be constructed by those Cuts, as they include rational numbers only? Maybe the "cut point" is also considered part of the Cut, which might be an irrational number?
To my understanding, a Dedekind cut just says, "here you have a number line of some random points (easiest is propably 0 to 1) and we now cut that in half. Take that new point (1/2) and consider one of the original points (I´ll take 0). Now cut that in half again and you get 1/4. do that againa nd again infinitely many times and you get a number arbitratrily close to 0. Now, we didn´t actually always have to use the left side (imagine you have a number line), but could have gone to the right side at any point. Thus we now have gotten every possible rational number there is between one and zero"
Now we can do the same thing between any other two points. As long as you took rational numbers, this actually plots the set of all rational numbers.
But what if you took squareroot of 2 at the start insted of one (so now you imagine the number line from 0 to sqrt 2 (squareroot 2)) which is obviously irrational. If we do the same procedure we´ll get all the irrartional numbers between 0 and sqrt 2.
Both sets together are the real numbers for this interval, and all possible intervasls thus construct the real numbers.
That´s it to the Dedekind cut.
Now any individual "slice" we took from that is obviously a subset of the real numbers because we took only the rational and irrational, thus the real numbers into consideration. Howevere all these sclices together make up the real numbers. Thus all possible slices united are in fact the real numbers.
This property is actually pretty common and not at all limited to the Dedkind cut.
Another example would be as follows:
Imagine you took a set of natural numbers up to 10. Now you say the set of any individual number (so 1,2,3,...10) is each a subset of 10 which is true since all of these sets are in their entirety in the set of the natural numbers up to 10. If you unite them all you get the set of all natural numbers up to 10 again.
I think Terence Tao , constructed Real Number set perfectly in his book , I really love that , maybe it's purely algebraic but it's SO strong !
@AmirErfanian Yes Terry is one of a kind!
Could you make a video with the construction of the dedekind cut correspondent to pi?
Sure thing!
Great video!
What I don’t understand is how dedekind cuts create the real numbers if they are subsets. If you define a set a a collection of subsets, and those subsets only contain rationals, how to the reals arise? Are the reals a set of all supremums of the dedekind cuts?
I was also wondering how we can create a dedekind cut for the cube root of two before having defined what a cube root means in the reals.
@obikenobi3629 Great question. In every construction of the real numbers, the definition is a bit surprising. In the Cauchy sequence definition, real numbers are defined as Cauchy sequence of rationals modulo Cauchy sequences converging to zero. This sequence and all of its representatives can be thought of as a subset of Q as well!
is it possible to construct a unique dedekind cut for an uncomputable number?
Yes. Let's take for example pi. You can define a subset of the rational number such as every element of the set is less than any partial sum of a series that converges to pi. Then, you have defined a Dedekind cut that represents pi.
Great stuff mate appreciate the effort!
Thanks @LeonTagleLB
OK...Uncle Doug here...explain to me how you can write so straight across the screen??? I mean as a retired engineer, I like straight lines. However, for the life of me, I can't even sign my name anymore without trailing off the page…
Thank goodness you aren't designing conveyer belts that trail off! Did you know that real numbers are really defined as Dedekind cuts?
No, but when I order a Easter ham I specify a Dedekind cut in stead of spiral cut. @@mikethemathematician
excellent
I’m glad you liked the video!
Great video
Thank you!
where are my irrational numbers?
@princez2835 Every irrational number can be realized as a Dedekind cut! I will add more example videos on this, but most of the famous irrational numbers can be represented by a convergent infinite series (convergent needs to be properly defined, but the Least Upper Bound Principle will help us), and this representation will help us build the cut.
@@mikethemathematician thanks! but i have another question. if real numbers are subsets of the rational numbers, than does it mean that all the real numbers are inside the rational numbers? It doesn't make sense since rational numbers do not contain irrational numbers.
and are there any examples of the subset of Q is not bounded above? I think real numbers are not bounded above@@mikethemathematician
@princez2835 Great question! The integers are a subset of Q that is not bounded above.
@princez2835 Good question! If we look at the set of rational numbers {p: p^3 < 2} (this is a Dedekind cut), then we can identify this set with its least upper bound which happens to be 2^{1/3}, which is an irrational number!
Likewise, for any rational number r, we can identify the Dedekind cut {p: p< r} with its least upper bound of r.
There is certainly a lot to unpack here, but you got it @princez2835!