Set Theory to Show Transcendental Numbers Exists

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Discussion Overview

The discussion revolves around Cantor's work related to the existence of transcendental numbers and the implications of countability in set theory. Participants explore how Cantor's findings suggest that most real numbers are transcendental, the nature of algebraic versus transcendental numbers, and the criticisms surrounding Cantor's methods.

Discussion Character

  • Exploratory
  • Technical explanation
  • Debate/contested
  • Mathematical reasoning

Main Points Raised

  • Some participants inquire about how Cantor demonstrated the existence of transcendental numbers and the claim that most numbers are transcendental.
  • It is noted that algebraic numbers form a countable set, while the set of real numbers is uncountable, leading to implications about the nature of transcendental numbers.
  • One participant suggests that if a number is not countable, it is non-algebraic and therefore transcendental, questioning if this is the correct interpretation.
  • Another participant points out that the existence of transcendental numbers was established prior to Cantor's work, prompting a discussion on the implications of countability.
  • There is a mention of Cantor's diagonal argument as a proof that the set of real numbers cannot be counted.
  • Some participants express confusion regarding the concepts of countability and uncountability, seeking simpler explanations.
  • One participant attempts to summarize the implications of Cantor's findings, but others challenge the clarity and accuracy of their statements.

Areas of Agreement / Disagreement

Participants generally agree on the countability of algebraic numbers and the uncountability of real numbers, but there is no consensus on the implications of these concepts regarding transcendental numbers. The discussion contains multiple interpretations and some confusion about the underlying principles.

Contextual Notes

Some participants express uncertainty about set theory terminology and concepts, indicating a need for clearer explanations. There are unresolved questions about the nature of transcendental numbers and the reasoning behind Cantor's conclusions.

ashnicholls
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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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The algebraic numbers are a countable set.
 
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??
 
Does he mean though, if a number is not countable, its non algebraic so its transcendental?

Is that the gist of it?

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

Cheers
 
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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