I don't understand Dedekind cuts

[Warp]

I have now tried for an hour to make ChatGPT make me understand how this works, without success, so I humbly ask for a clearer explanation.

The succinct way to express what I don't understand is this:

A "Dedekind cut" simply splits the set of numbers into two, without dropping any values and retaining ordering, so that every value on the left of the cut is less than every value on the right of the cut. Let's apply this to the set of rational numbers:

1. Every rational number splits the set of rationals into two subsets, and this split is unique.
2. Every irrational number splits the set of rationals into two subsets, and this split is unique.

(In this context "unique" means that two different numbers used to do the split never produce the same subsets of the rationals. Every unique number produces a unique split, no duplicates.)

This makes it sound like the set of rationals and the set of irrationals are of the same size. Yet, somehow, there are (uncountably many) "more" of the second type of cuts than the first type. How can a countably infinite set have uncountably many unique splits of this kind? It defies intuition and logic.

 

[Hill]

This makes it sound like the set of rationals and the set of irrationals are of the same size.

I don't see, why ^^^^.
You could add, e.g.,
3. Number 5 splits the set of rationals into two subsets, and this split is unique.
Does it make it sound like the set of rationals and the set 'number 5' are of the same size?

 

[Dale]

In this context "unique" means that two different numbers used to do the split never produce the same subsets of the rationals. Every unique number produces a unique split, no duplicates.

You may be misunderstanding unique in this context. It doesn't mean that you can not produce the same subsets in different ways. What it means is that if two different ways of producing a split lead to the same split, then those two ways are a representation of the same number, the smallest number in the upper cut.

So, for an almost trivial example for the lower cuts $$A=\{a \in \mathbb{Q}:2 a<1\}$$ and $$A'=\{a \in \mathbb{Q}:6 a<3\}$$ Since $A=A'$ the numbers defined are the same number. The cut for $A$ is the rational number $1/2$ and the cut for $A'$ is the rational number $3/6$. The fact that $A'$ is a duplicate of $A$ means that $1/2$ and $3/6$ are the same unique number.

This makes it sound like the set of rationals and the set of irrationals are of the same size.

You compare the sizes of sets by finding a one-to-one mapping between the sets. If there is such a mapping, then the sets are the same size. In the Dedekind cuts, the number identified by the cut is the smallest element of the upper cut. So, consider the lower cuts $$A=\{a \in \mathbb{Q}: a^2<2 \cup a<0\}$$ and $$A'=\{a \in \mathbb{R}: a^2<2 \cup a<0\}$$There is a least element in the upper cut for $A'$, but the upper cut for $A$ has no least element. So this cut does not place the reals and the rationals in a one-to-one mapping. There exist elements in the reals that do not map to elements in the rationals through cuts.

This mapping maps every number in the rationals to a number in the reals, but not every number in the reals to a number in the rationals. So it is not one-to-one, and there are more elements in the reals by this mapping. By itself this is not a conclusive proof that the cardinality of the reals is larger than the cardinality of the rationals since you have to prove the non-existence of a different one-to-one mapping. But certainly this mapping is not one-to-one so it does give credence to the idea that the reals may be bigger than the rationals.

 

[Warp]

There is a least element in the upper cut for A′, but the upper cut for A has no least element. So this cut does not place the reals and the rationals in a one-to-one mapping. There exist elements in the reals that do not map to elements in the rationals through cuts.

This mapping maps every number in the rationals to a number in the reals, but not every number in the reals to a number in the rationals. So it is not one-to-one, and there are more elements in the reals by this mapping.

I was not talking about the set of real numbers but about the set of irrational numbers, both in the description of the problem and my sentence "this makes it sound like the set of rationals and the set of irrationals are of the same size."

If the argument is "since there is no 'smallest rational number that's greater than the irrational number used for the split', then the set of irrationals is larger", the same argument could well be used in the reverse:

"Since there is no 'smallest irrational number that's greater than the rational number used for the split', then the set if rational numbers is larger."

As mentioned at the end of my original post, my confusion can be summarized as:

"How can a countably infinite set have an uncountable amount of unique Dedekind cuts?"

It sounds like cutting the set of rationals into more parts than there are rationals, which completely defies intuition.

 

[Warp]

Talking a bit more with ChatGPT about this subject, I think I got a sort of understanding breakthrough, after understanding two key facts:

1. Using two Dedekind cuts on the set of rational numbers is, essentially, selecting a subset of the rational numbers (between the two cuts).
2. There are uncountably many subsets of the rational numbers.

(Fact 2 might not be immediately obvious, but I'm assuming it's for a similar reason why the set of all possible infinite strings is uncountable.)

In other words, we are not really mapping unique cuts to unique rational numbers. Instead, we are essentially counting how many possible subsets of the rationals there are.

 

[berkeman]

Talking a bit more with ChatGPT about this subject,

Please remember that AI is not allowed as a reference in the technical PF forums. You haven't posted your AI discussion here in this thread, so you are okay for now, but I just wanted to remind you in case you were tempted.

 

[Dale]

I was not talking about the set of real numbers but about the set of irrational numbers

Oops, that is true.

I don’t think that Dedekind cuts can constitute any sort of mapping between the rationals and the irrationals. A Dedekind cut at an irrational does not map to any rational numbers.

If the argument is "since there is no 'smallest rational number that's greater than the irrational number used for the split', then the set of irrationals is larger"

That is not a valid argument about the size of the two sets. The only way to make that kind of argument is to make a mapping between rationals and irrationals. The Dedekind cut approach does not do that.

It sounds like cutting the set of rationals into more parts than there are rationals

Yes. Every rational corresponds to a Dedekind cut of the rationals. But not all Dedekind cuts of the rationals correspond to a rational.

A Dedekind cut of the rationals with a smallest element of the upper cut corresponds to that rational. But there are Dedekind cuts of the rationals with no smallest element of the upper cut. So indeed there are more cuts than there are rationals: every rational can be mapped to a cut, but not every cut can be mapped to a rational.