Doubts on the Vitali Set - Marco's Story

In summary, the Vitali set is constructed by defining equivalence classes based on the difference of two numbers being rational. These classes are then used to form a set, which is shown to be a disjoint union of sets (after a rational translation). However, one cannot create a set containing all irrationals, leading to a contradiction in the interval [0,1] being the disjoint union of a zero measure set and a non-measurable set. This is due to the incorrect assumption that there is a set containing all irrational numbers.
  • #1
muzialis
166
1
Hello all,

I have a doubt on the Vitali set.
In its construction we define equivalence classes, two numbers belonging to one if their difference is rational.
Then we "pick" a member per class, forming a set. It is then shown that the interval [0,1] is a disjoint union of such set (after a rational translation), hence after some reasonsing this set is non-measurable.
It seems to me that one such set is formed by a rational number plus all irrationals.
Now, the interval [0,1] is also the disjoint union of the rationals and the irrationals.
Hence, as the rationals are a set of zero lebesgue meausure, and as the irrationals are not measurable (differing by only one member from the the set built during the Vitali construction), one could conclude the interval [0,1] is the disjoiunt union of a zero measure set and a non - measurable set, which seems a contradiction.
Can anybody point out my mistake as I can not?

Thanks

Marco
 
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  • #2
muzialis said:
Hello all,

I have a doubt on the Vitali set.
In its construction we define equivalence classes, two numbers belonging to one if their difference is rational.
Then we "pick" a member per class, forming a set. It is then shown that the interval [0,1] is a disjoint union of such set (after a rational translation), hence after some reasonsing this set is non-measurable.
It seems to me that one such set is formed by a rational number plus all irrationals.
No, there is no such set containing all irrationals. For example, [itex]e- \pi[/itex] is not a rational number so [itex]e[/itex] and [itex]\pi[/itex] cannot be in the same equivalence class.

Now, the interval [0,1] is also the disjoint union of the rationals and the irrationals.
Hence, as the rationals are a set of zero lebesgue meausure, and as the irrationals are not measurable (differing by only one member from the the set built during the Vitali construction), one could conclude the interval [0,1] is the disjoiunt union of a zero measure set and a non - measurable set, which seems a contradiction.
Can anybody point out my mistake as I can not?

Thanks

Marco
Your error, as I said, is in thinking that there was such a set containing all irrational numbers.
The set of all irrational numbers, between 0 and 1, is not, of course, "non-measurable"- it has measure 1.
 
  • #3
muzialis said:
Hello all,


It seems to me that one such set is formed by a rational number plus all irrationals.

This is where you went wrong.
 
  • #4
Correct indeed, many thanks
 

1. What is the Vitali Set and why is it important in mathematics?

The Vitali Set is a mathematical construct that was first introduced by the Italian mathematician Giuseppe Vitali in the early 20th century. It is a non-measurable set, meaning that it cannot be assigned a precise numerical value. This set is important because it challenged the foundations of mathematics and sparked debates about the nature of infinity and the concept of measure.

2. What is Marco's story and how does it relate to the Vitali Set?

Marco's story is a thought experiment that was created by mathematician and philosopher Bertrand Russell to illustrate the paradoxes that arise from the existence of non-measurable sets like the Vitali Set. In the story, Marco is a traveler who encounters a genie that grants him the ability to perfectly replicate a given figure using a fixed set of points. This leads to the paradox of the Vitali Set, where Marco is able to create an unmeasurable set using a finite number of points.

3. Is the Vitali Set a real, tangible set or just a theoretical construct?

The Vitali Set is a theoretical construct, meaning that it does not exist in the physical world. It is a mathematical concept that was created to challenge our understanding of infinity and measure. However, the implications and paradoxes that arise from the Vitali Set have real-world applications and implications in mathematics and philosophy.

4. How do mathematicians reconcile the existence of non-measurable sets like the Vitali Set?

There is still ongoing debate and discussion among mathematicians about the existence and implications of non-measurable sets like the Vitali Set. Some mathematicians argue that these sets are purely theoretical and do not have any real-world applications, while others believe that they have important implications for the foundations of mathematics. Ultimately, the concept of non-measurable sets challenges our understanding of infinity and the concept of measure, and it continues to be a topic of interest and discussion in the mathematical community.

5. Can the Vitali Set be proven to exist or not exist?

No, the existence of the Vitali Set cannot be proven either way. It is an undecidable statement, meaning that it cannot be proven true or false using the standard axioms of set theory. This is because the existence of the Vitali Set leads to logical contradictions and paradoxes, and there is no consensus among mathematicians on how to resolve these issues. As a result, the existence of the Vitali Set remains an open question and a source of debate in mathematics.

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