Set Theory to Show Transcendental Numbers Exists

In summary: This is what caused many people to criticize Cantor's work, because they thought that there were different sizes of infinity."In summary, Cantor showed that most numbers are transcendental by proving that the set of real numbers is much larger than the set of algebraic numbers. He also showed that most numbers can be considered transcendental, which caused many people to criticize his work.
  • #1
ashnicholls
50
0
Does anyone know how Cantor showed the existence of Transcendental numbers. How can he say that most numbers are transcendental?

Is that why everyone critised it?

Cheers Ash
 
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  • #2
The algebraic numbers are a countable set.
 
  • #3
Yes sorry I should have said I undestand that, but how does he go on to say most numbers are transcendental, only a few are surely??
 
  • #4
Does he mean though, if a number is not countable, its non algebraic so its transcendental?

Is that the gist of it?

Cheers
 
  • #5
I believe the existence of transcendental numbers was established before Cantor was born.

As already stated, the algebraic numbers are countable.
Since the real numbers are uncountable, what does that tell you about the set of transcendental numbers?
 
  • #6
ashnicholls said:
Does he mean though, if a number is not countable, its non algebraic so its transcendental?

Is that the gist of it?

Cheers

Countable pertains to sets, not real numbers. The SET of algebraic numbers is countable. The set of real numbers is uncountable. Thus the set of algebraic numbers is 'almost none' of the totality of real numbers.
 
  • #7
yes sorry I know he didnt prove there existence, he just made a method to show other number were transcendental.

eerrrrrrrrrrrrrrrrrrr of course, all real are uncountable, so most are transcendental, obvious, thanks.

Any more info you can give?

How do all of you know all of this?

Cheers Ash
 
  • #8
sorry that has gone straight over my head, group/set theory is not a strength of mine.

Cheers
 
  • #9
ashnicholls said:
sorry that has gone straight over my head, group/set theory is not a strength of mine.

Cheers

Then what do you think countable/uncountable means if you don't know your set theory. Please, let us know what you think 'countable' means.
 
  • #10
No I understand countable, ie The sets has a 1-1 correspondence with N.

But don't really get what you mean:

Countable pertains to sets, not real numbers. The SET of algebraic numbers is countable. The set of real numbers is uncountable. Thus the set of algebraic numbers is 'almost none' of the totality of real numbers.

Can you explain this any simpler?
 
  • #11
And I have found some where in my notes that:

The of Cardinality of A, the set of algebraic numbers, is aleph null, which is correct, A is countable, algebraic numbers are not transendental but it then says on the same line, so there are many transcendentals, how can you come to that conclusion by sayin algebraic numbers are not transendental?
 
  • #12
Right, I thought I had it but I have lost lost it now.

Am i right in thinking that he said if a set is countable, then it does not contain transendental numbers. So as the set of real numbers is not countable it contains transendental numbers,

Is that the logic behind it??

If so how did he work that out, how can he say that a number is transendental or not on whether you can count the set that it is in is countable or not?

Or am I completely of line with what's is going on?

Cheers Ash
 
  • #13
The set of real numbers is not countable. The set of real numbers is the union of the algebraic numbers and the transcendentals. The algebraic numbers are a countable set, thus the transcendental numbers must be an uncountable set (or the reals would be the union of two countable sets, hence countable which would be a contradiction).

When we say 'thus almost all of the real numbers are transcendental' we are saying that countable sets are, in some sense, a lot smaller than uncountable sets. In fact, if you were to pick a number uniformly at random from the interval [0,1] then it would be transcendental with probability 1.
 
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  • #14
ok thanks for your help, I will run with that thought now.

Hopefully that is all I need.
 
  • #15
Oh there is one more thing why did people critise this method?
 
  • #16
Because they thought it nonsense to have 'different sizes of infinity', perhaps.

It is possible to edit your posts, rather than having consecutive 1 line posts.
 
  • #17
Yes ok sorry, I was just being lazy.

What is the proof that you can't count the set of real numbers?
 
  • #18
Cantor's diagonal argument, amongst others.
 
  • #20
Cheers,

Is what I have written here correct?

"With these proofs he could then go onto show that any real number that is not in the set of algebraic numbers is a transendental number. And he also showed that because the set of real numbers is far larger than the set of integers, and because the set of algebraic numbers has the same cardinality as the set of integers because it is countable, this implies that the set of real numbers is far greater than the set of integers. This also implied that the number of transendental numbers is larger than the set of intergers and that most numbers can be considered as transendental numbers."
 
  • #21
"With these proofs"? Which proofs?

Use clear and exact language. "the number of transendental numbers is larger than the set of intergers" is neither clear nor exact. You should say "there are more transcendental numbers than integers" or "the set of transcendentals is larger than the set of integers."
 
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  • #22
The proofs were stated above, I didnt want to list the whole thing, i just wanted to check that what i had written made sence.

ok cheers. but the rest makes sence?
 
  • #23
"And he also showed that because the set of real numbers is far larger than the set of integers, and because the set of algebraic numbers has the same cardinality as the set of integers because it is countable, this implies that the set of real numbers is far greater than the set of integers." is far too long a sentence.

Try: "In addition, he showed that algebraic numbers are countable (their set have the same cardinality with the set of integers). Since the real numbers are uncountably many, there must be uncountably many non-algebraic numbers. In fact, the set of non-algebraic numbers must have the same cardinality with the set of real numbers. These non-algebraic numbers are called transcendental numbers."
 
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1. What is set theory and how does it relate to transcendental numbers?

Set theory is a branch of mathematics that deals with the study of sets, which are collections of objects, such as numbers. Transcendental numbers are a type of real number that cannot be expressed as the roots of a polynomial equation with integer coefficients. Set theory is used to prove the existence of transcendental numbers by constructing a set of numbers that contains both algebraic and transcendental numbers.

2. How do you prove the existence of transcendental numbers using set theory?

The proof of the existence of transcendental numbers using set theory is based on the idea that there are more real numbers than there are algebraic numbers. This can be shown by constructing a set of numbers that contains all the algebraic numbers and then showing that there are still real numbers left over. These remaining numbers are then proven to be transcendental.

3. Can you give an example of a transcendental number?

Yes, one example of a transcendental number is pi (π), which is the ratio of a circle's circumference to its diameter. Pi cannot be expressed as the root of a polynomial equation with integer coefficients and therefore, is a transcendental number.

4. How does the existence of transcendental numbers impact mathematics?

The existence of transcendental numbers has a significant impact on mathematics as it demonstrates the incompleteness of the real numbers. It also has implications for other branches of mathematics, such as calculus, where transcendental numbers are used extensively.

5. Is the existence of transcendental numbers widely accepted by the scientific community?

Yes, the existence of transcendental numbers is widely accepted by the scientific community as it has been proven using rigorous mathematical techniques. The proof has been verified and accepted by mathematicians and scientists around the world.

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