Continuity and uncountability

[Strange]

21 minutes ago, pengkuan said:

I think that the set of all even natural numbers corresponds to the natural number ...10101010.

That is not a number because it has an infinite number of digits

[uncool]

As strange has said, that is not a natural number. If you think otherwise, then please write that many '1's in a row. For example:

I can write 2 1s: 11

I can write 15 1s: 111111111111111

[pengkuan]

5 hours ago, Strange said:

That is not a number because it has an infinite number of digits

This means that the set of natural numbers {1,2,3…} contains only numbers with finite number of digits. Suppose the number of members of N is n.

Then, the set of even numbers corresponds to 10…101010, with n digits.

[wtf]

7 minutes ago, pengkuan said:

This means that the set of natural numbers {1,2,3…} contains only numbers with finite number of digits. Suppose the number of members of N is n.

Then, the set of even numbers corresponds to 10…101010, with n digits.

When you evaluate that string using standard positional notation, how do you do it?

 

[Strange]

4 hours ago, pengkuan said:

This means that the set of natural numbers {1,2,3…} contains only numbers with finite number of digits.

So there is a largest natural number?

[studiot]

Accomodating such questions is one of the reasons that the distinction between countable and uncountable infinities was introduced.

You need to learn the distinction.

[uncool]

11 hours ago, pengkuan said:

This means that the set of natural numbers {1,2,3…} contains only numbers with finite number of digits. Suppose the number of members of N is n.

Then, the set of even numbers corresponds to 10…101010, with n digits.

N has infinitely many members, each of which has finitely many digits. So there is no natural number "n" that denotes the number of elements of N.

[Strange]

11 hours ago, pengkuan said:

This means that the set of natural numbers {1,2,3…} contains only numbers with finite number of digits. Suppose the number of members of N is n.

Then, the set of even numbers corresponds to 10…101010, with n digits.

A number with n digits must be larger than N. So your number 10…101010 representing the set of even numbers is not in the set of natural numbers and so needs to be added. Repeat ad infinitum to prove that the set of natural numbers is infinite and the number N does not exist.

 

[pengkuan]

21 hours ago, wtf said:

When you evaluate that string using standard positional notation, how do you do it?

 

I cannot.

17 hours ago, Strange said:

So there is a largest natural number?

No, there is not. I said "This means that the set of natural numbers {1,2,3…} contains only numbers with finite number of digits" to answer your reply below.

On 17/10/2018 at 10:43 PM, Strange said:

That is not a number because it has an infinite number of digits

According to you, ...10101010 is not a natural number. So, ℕ does not contain numbers with infinite digits. In this case, only numbers with finite digits exist in ℕ. This is why I said "Suppose the number of members of N is n". If this means that there is a largest natural number, then this is derived from your  reasoning.

 

16 hours ago, studiot said:

Accomodating such questions is one of the reasons that the distinction between countable and uncountable infinities was introduced.

You need to learn the distinction.

There can exist other way to accommodate sets with different size. I have proposed related set in my article. For example, 

{1,2,3...n} is related with {1,2,3...2^n}. When n becomes infinity, both sets are ℕ, but their sizes are different, while being both countable.

11 hours ago, uncool said:

N has infinitely many members, each of which has finitely many digits. So there is no natural number "n" that denotes the number of elements of N.

"each of which has finitely many digits" means that they are all finite. n is meant to represent any finite number.

11 hours ago, Strange said:

A number with n digits must be larger than N. So your number 10…101010 representing the set of even numbers is not in the set of natural numbers and so needs to be added. Repeat ad infinitum to prove that the set of natural numbers is infinite and the number N does not exist.

 

Sorry, the letter N in my post that you reply meant ℕ. So, there is confusion.

[uncool]

6 hours ago, pengkuan said:

"each of which has finitely many digits" means that they are all finite. n is meant to represent any finite number.

Any finite number? So the string "101010101010" by itself represents the set of all even numbers?

Can you type out the specific string you think represents the set of all even numbers? No ellipsis - the exact string.

 

[studiot]

8 hours ago, pengkuan said:

There can exist other way to accommodate sets with different size. I have proposed related set in my article. For example, 

{1,2,3...n} is related with {1,2,3...2^n}. When n becomes infinity, both sets are ℕ, but their sizes are different, while being both countable.

All non finite countable sets have the same 'size'

Perhaps you are mixing up the set and the members themselves?

What is the sequence rule that produces this set please?

{1,2,3...2^n}.

[pengkuan]

On 18/10/2018 at 4:29 AM, wtf said:

When you evaluate that string using standard positional notation, how do you do it?

 

I cannot

15 hours ago, uncool said:

Any finite number? So the string "101010101010" by itself represents the set of all even numbers?

Can you type out the specific string you think represents the set of all even numbers? No ellipsis - the exact string.

 

I cannot

14 hours ago, studiot said:

All non finite countable sets have the same 'size'

Perhaps you are mixing up the set and the members themselves?

What is the sequence rule that produces this set please?

{1,2,3...2^n}.

Sorry, This set is in fact that: {1,2,3...m-2,m-1,m} with m=2^n.

 

[uncool]

On 10/19/2018 at 5:42 PM, pengkuan said:

I cannot

Precisely - which means you have no natural number corresponding to the set of even numbers, which means you have no bijection.

[pengkuan]

4 hours ago, uncool said:

Precisely - which means you have no natural number corresponding to the set of even numbers, which means you have no bijection.

But does the set of even numbers exist? It is assumed to exist by an axiom which allows infinite set to exist. However, infinitely big natural number is not allowed to exist. If the set of even numbers exist, what is the number of its members? Aleph0 is not a natural number. We do not have bijection not because there is not, but because there is no axiom for infinitely big natural number to exist. This is an incoherence of set theory because a thing exists, the number of the members of infinite set, but this number does not exist in the same theory that creates this thing .This is the cause that no natural number corresponds to the set of even numbers. 

 

[studiot]

11 minutes ago, pengkuan said:

But does the set of even numbers exist?

Yes.

 

You are asking a mathematical question of existence, not a physical one.

The mathematical statement there exists a set G such that...........

means that set G and its conditions do not conflict with each other or the rules employed.

So the statement there exists a number C, such that C = 6 + 6

is consistent with both number theory and standard arithmetic rules.

It does not mean that I can go down to the supermarket and buy a dozen 'number Cs'.

 

Compare this with the mathematical statement

There exists a positive number D such that D = 6 - 12

D does not exist since there is a contradiction of conditions, although there is not contradiction of the rules of arithmetic.

In other words I can point to a contradiction (or rule) that forbids  the existence of D.

 

 

Edited October 22, 2018 by studiot

[Strange]

21 minutes ago, pengkuan said:

But does the set of even numbers exist?

Of course. 

22 minutes ago, pengkuan said:

However, infinitely big natural number is not allowed to exist. 

Irrelevant. 

22 minutes ago, pengkuan said:

If the set of even numbers exist, what is the number of its members?

The cardinality of the set is not a (natural) number. 

24 minutes ago, pengkuan said:

This is an incoherence of set theory because a thing exists, the number of the members of infinite set, but this number does not exist in the same theory that creates this thing .

There is no rule that says the cardinality of a set has to be a member of the set. 

 

[pengkuan]

On 23/10/2018 at 1:57 AM, Strange said:

Of course. 

Irrelevant. 

The cardinality of the set is not a (natural) number. 

There is no rule that says the cardinality of a set has to be a member of the set. 

 

Thanks.

On 23/10/2018 at 1:42 AM, studiot said:

Yes.

 

You are asking a mathematical question of existence, not a physical one.

The mathematical statement there exists a set G such that...........

means that set G and its conditions do not conflict with each other or the rules employed.

So the statement there exists a number C, such that C = 6 + 6

is consistent with both number theory and standard arithmetic rules.

It does not mean that I can go down to the supermarket and buy a dozen 'number Cs'.

 

Compare this with the mathematical statement

There exists a positive number D such that D = 6 - 12

D does not exist since there is a contradiction of conditions, although there is not contradiction of the rules of arithmetic.

In other words I can point to a contradiction (or rule) that forbids  the existence of D.

 

 

If there is not rule for an object, what will the rule that allows it to exist? An axiom? For example, the set of natural numbers is infinite. Nobody can write it down. So, if it is accepted to exist, it is because of an axiom, isn't it?

 

[studiot]

25 minutes ago, pengkuan said:

If there is not rule for an object, what will the rule that allows it to exist? An axiom? For example, the set of natural numbers is infinite. Nobody can write it down. So, if it is accepted to exist, it is because of an axiom, isn't it?

Look again at my explanation of

"There exists."

 

It is not necessary to have a rule which states explicitly there exists and humperdink, if the existance of a humperdink does not conflict with any establish rules or axioms.

You can define a humperdink, along with its properties, to be wanything you want, so long as there is no conflict.

 

You can then correctly assert "there exists a humperdink"

 

Your problem comes when someone comes along and points out that there is, in fact, a conflict as I did with my simple example of a non negative D.

 

My compliments to Strange for some incisive thinking. +1

 

[Strange]

17 minutes ago, studiot said:

My compliments to Strange for some incisive thinking

You say that as if it were unusual!  

[taeto]

1 hour ago, studiot said:

It is not necessary to have a rule which states explicitly there exists and humperdink, if the existance of a humperdink does not conflict with any establish rules or axioms.

You can define a humperdink, along with its properties, to be anything you want, so long as there is no conflict.

You can then correctly assert "there exists a humperdink"

And interestingly, this is actually not completely correct.

There are some cases, even in ordinary arithmetic, when it can happen that there is no conflict at all with rules or axioms to prevent the existence of the humperdink.

Examples arise when a humperdink means a solution to certain "diophantine equations"; they are built from addition and multiplication of integer numbers and some variables that can be given integer number values. A humperdink then means an assignment of values to the variables for which the equation is satisfied.

An example of a diophantine equation would be something like \(x^3 + 2x^2y^2 - 4y^3 +5 =0.\) For this particular one it is probably either easy to solve it, or to prove that no solution exists, when integer numbers are substituted in for \(x\) and \(y.\) But it is not too hard to imagine that it might take some amount of work to decide either way.

Some diophantine equations have been constructed for which it is impossible to prove that they have solutions, and impossible as well to prove that they have no solutions.

Therefore in such a case, you cannot simply assert that one exists, nor assert that none exists. Whenever you make such an assertion, you have to prove it. It is not enough that it does not conflict with rules or axioms. Of course Gödel was the pioneer of such discoveries.

Edited October 24, 2018 by taeto

[uncool]

There actually is an axiom which is essentially that the natural numbers exist (more precisely, that we can construct something which follows the Peano axioms):

There is a set S such that the empty set is an element of S, and such that if x is an element of S, then {x} is an element of S.

This is called the axiom of infinity, and the natural numbers can be constructed from it.

(There are other constructions, like "if x is an element of S, then the union of x and {x} is an element of S"; all that matters is that the operation chosen is injective and doesn't send anything to the empty set)

Edited October 24, 2018 by uncool

[studiot]

9 minutes ago, uncool said:

There actually is an axiom which is essentially that the natural numbers exist (more precisely, that we can construct something which follows the Peano axioms):

There is a set S such that the empty set is an element of S, and such that if x is an element of S, then {x} is an element of S.

This is called the axiom of infinity, and the natural numbers can be constructed from it.

(There are other constructions, like "if x is an element of S, then the union of x and {x} is an element of S"; all that matters is that the operation chosen is injective and doesn't send anything to the empty set)

If instead of asserting an axiom, you started

 

Assume a set S exists such that.....etc

or

Let there be a set S such that.....

or

What would such a statement be in conflict with?

19 minutes ago, taeto said:

And interestingly, this is actually not completely correct.

There are some cases, even in ordinary arithmetic, when it can happen that there is no conflict at all with rules or axioms to prevent the existence of the humperdink.

Examples arise when a humperdink means a solution to certain "diophantine equations"; they are built from addition and multiplication of integer numbers and some variables that can be given integer number values. A humperdink then means an assignment of values to the variables for which the equation is satisfied.

An example of a diophantine equation would be something like x3+2x2y2−4y3+5=0. For this particular one it is probably either easy to solve it, or to prove that no solution exists, when integer numbers are substituted in for x and y.  But it is not too hard to imagine that it might take some amount of work to decide either way.

Some diophantine equations have been constructed for which it is impossible to prove that they have solutions, and impossible as well to prove that they have no solutions.

Therefore in such a case, you cannot simply assert that one exists, nor assert that none exists. Whenever you make such an assertion, you have to prove it. It is not enough that it does not conflict with rules or axioms. Of course Gödel was the pioneer of such discoveries.

A similar question to taeto

[uncool]

What do you mean by "be in conflict with"?

If you mean that such an assumption should lead to a contradiction, then I think your concept of an axiom is faulty. 

The closest thing to sensible I can see is if you are asking "What models of the other axioms are excluded by this assumption?" To which the answer is models where all sets are finite. In some sense, to get infinity, you have to start with an infinite set in the first place. 

Edited October 24, 2018 by uncool

[studiot]

24 minutes ago, uncool said:

What do you mean by "be in conflict with"?

If you mean that such an assumption should lead to a contradiction, then I think your concept of an axiom is faulty. 

The closest thing to sensible I can see is if you are asking "What models of the other axioms are excluded by this assumption?" To which the answer is models where all sets are finite. In some sense, to get infinity, you have to start with an infinite set in the first place. 

Inconsistent with.

 

Do all the other axioms you are referring to, taken together, preclude the existance of an infinite set?

 

I disagree with your use of the word model.

Models follow they don't precede.

Edited October 24, 2018 by studiot

[taeto]

43 minutes ago, studiot said:

If instead of asserting an axiom, you started

Assume a set S exists such that.....etc

What would such a statement be in conflict with?

Not really with anything.

If you say "Assume a set S exists such that 2+2=5", then you reach a contradiction. In which case you just resolve that no such set exists, and you deduce that 2+2 is not 5.  

If you say "Assume a set S exists such that P. Then Q", where P is an undecidable proposition, and you prove your statement, then you have proved an implication: if S exists, then Q is true. This happens very often in algebra, which has lots of undecidable statements, such as P="G is an abelian group" and the likes. Then by showing "P => Q: x and y are in G implies x+y=y+x" you have shown that Q is true given a particular circumstance P.