Continuity and uncountability

[uncool]

1 hour ago, pengkuan said:

A set that has an end.

That's not a precise definition. However, even using that imprecise definition, it is clear that the set of even numbers is not finite. 

[taeto]

3 hours ago, pengkuan said:

A set that has an end.

Can you make a finite set out of the minutes on a clock, since it does not seem to have an end, and it just moves around in cycles?

3 hours ago, pengkuan said:

The disagreement between you two shows that infinity is really confusing.

I am on the safe side here, because I am just the one saying that there is no such thing as a particular concept of "infinity" in mathematics. No way i can get confused about something that is not there. 

But I see where strange is coming from as well, because apart from mathematics itself, it is not uncommon to refer to "infinity" as that aspect of math that deals with some kind of theory of infinite structures. It is difficult to set precise borders between theories that do and theories that don't. If you are only interested in arithmetic with natural numbers, then every mathematical object that you will ever consider will be a finite object, namely a natural number. And yet, the thing which is "arithmetic with natural numbers" is the prime example of something in real life which consists of infinitely many other things, namely all the infinitely many natural numbers. That makes it natural for observers of mathematical practice to suggest that "infinity" is an actual thing in mathematics. 

3 hours ago, pengkuan said:

A set that has an end.

Just like the set \(\{ \ldots , -4,-3,-2,-1,0 \}\), which has end \(0\)?

Edited November 17, 2018 by taeto

[pengkuan]

8 hours ago, Strange said:

Maybe we mean different things by "[well] defined"

Like what we have to give a definition to what we are discussing in order to understand each other.

5 hours ago, taeto said:

there is no such thing as a particular concept of "infinity" in mathematics.

Then, how can we discuss about infinite set?

5 hours ago, taeto said:

namely all the infinitely many natural numbers. 

Can we say that "the infinitely many natural numbers" are all finite?

6 hours ago, uncool said:

8 hours ago, pengkuan said:

A set that has an end.

That's not a precise definition.

Can you give a precise definition in order to speak the same thing in our discussion?

5 hours ago, taeto said:

Can you make a finite set out of the minutes on a clock, since it does not seem to have an end, and it just moves around in cycles?

The context is about set of natural numbers. So, a set of different natural numbers that has an end.

[uncool]

I can give several definitions that are equivalent under the usual axioms of ZFC. The one that seems useful here is: A set A is finite if there exists a natural number n such that A is in bijection with the set {0, 1, 2, ..., n-1}.

Edited November 18, 2018 by uncool

[taeto]

13 hours ago, pengkuan said:

Then, how can we discuss about infinite set?

Because some sets are finite, and some are not finite. In a context of using mathematical language, I would be justified to say that "infinity" is just a symbol \(\infty\), which is a kind of letter. And letters are neither sets nor are they infinite. It is used to denote limits and improper integrals, among other things. In the particular contexts when \(\infty\) is used as the name of an actual set, it is  not required to be an infinite set.

13 hours ago, pengkuan said:

Can we say that "the infinitely many natural numbers" are all finite?

It is grammatically correct, if it means that each of the natural numbers is finite. The properties of each natural number do not depend on how many there are of them.

13 hours ago, pengkuan said:

The context is about set of natural numbers. So, a set of different natural numbers that has an end.

I think that was not the context in which the question was asked. I believe the question concerned sets in general.

Edited November 18, 2018 by taeto

[pengkuan]

On 2018/11/18 at 5:21 AM, uncool said:

A set A is finite if there exists a natural number n such that A is in bijection with the set {0, 1, 2, ..., n-1}.

We agree on the definition of finite set. But what is the definition of infinite set? Like the set of all even numbers?

[wtf]

Just now, pengkuan said:

We agree on the definition of finite set. But what is the definition of infinite set? Like the set of all even numbers?

A set that's not finite. 

[pengkuan]

On 2018/11/18 at 2:10 PM, taeto said:

Because some sets are finite, and some are not finite. In a context of using mathematical language, I would be justified to say that "infinity" is just a symbol ∞ , which is a kind of letter. And letters are neither sets nor are they infinite. It is used to denote limits and improper integrals, among other things. In the particular contexts when ∞ is used as the name of an actual set, it is  not required to be an infinite set.

Can we have a definition of infinite set? Without a proper definition, work on infinite set does not have sense.

5 minutes ago, wtf said:

A set that's not finite. 

Se cannot work with this "definition".

[taeto]

21 minutes ago, pengkuan said:

Can we have a definition of infinite set? Without a proper definition, work on infinite set does not have sense.

We certainly can. Such as: "A set \(X\) is infinite if there exists a proper subset \(Y\) of \(X\)  and an injective function \(f : X \to Y\)." Which means that if \(a\) is an element of \(X\), then \(f(a)\) is an element of \(Y\), and if \(a\) and \(b\) are two different elements of \(X\), then \(f(a)\) and \(f(b)\) are two different elements of \(Y\). And there is at least one \(y \in Y\) such that \(y\) is not an element of \(X\). As an example, the set \(X = \mathbb{Z}\) of all integers is infinite by this definition, because the set \(Y=2\mathbb{Z}\) of all even numbers is a proper subset of \(X\), and \(f(n)=2n\) defines an injective function from \(X\) to \(Y\).

 

Edited November 19, 2018 by taeto

[pengkuan]

7 minutes ago, taeto said:

We certainly can. Such as: "A set X is infinite if there exists a proper subset Y of X   and an injective function f:X→Y ." 

Why X is infinite?

If X={1,2} and Y ={1}, would X and Y fit this definition?

[taeto]

Just now, pengkuan said:

Why X is infinite?

If X={1,2} and Y ={1}, would X and Y fit this definition?

No, because any function from \(X=\{1,2\}\) to \(Y=\{1\}\) maps both \(1\) and \(2\) to \(1\), so is not an injection. The condition is that if \(a\) and \(b\) are different, then \(f(a)\) and \(f(b)\) are also different.

 

27 minutes ago, taeto said:

 

You already got a perfectly precise from wtf anyway, as I just noticed. And your reply was that you "cannot work with it". You are obviously not serious, so I am out of here.

[studiot]

1 hour ago, taeto said:

And there is at least one y∈Y such that y is not an element of X

?

[wtf]

3 hours ago, pengkuan said:

Can we have a definition of infinite set? Without a proper definition, work on infinite set does not have sense.

Se cannot work with this "definition".

The definitions given were: 

* A set is finite if it can be bijected to a natural number 0, 1, 2, 3, ...

* A set is infinite if it's not finite. 

That is a perfectly clear definition that's workable in practice. 

It's true that it was given in two different posts. @uncool mentioned the def of finite and then I added that a set is infinite if it's not finite.

You can't pretend this wasn't clearly stated.

By the way I'll just clarify a technicality. @uncool said that a set is finite if it can be bijected to the set {0,1,2,...,n-1} for some natural number n.

I  said that a set is finite if it can be bijected to some number n. In my definition I'm using the fact that in set theory, we actually define the number n as the set {0,1,2,...,n-1}. So the number 1 is modeled as {0} and the number 5 is modeled as {0,1,2,3,4}. You'll note that in every case the cardinality of n is indeed n. That's a nice feature of this particular way of defining the natural numbers. It's not the only way, there are others. But this is the standard if no other encoding is mentioned. So my formulation and @uncool's are the same because in standard math, the number n IS the set {0,1,2,3,4,...,n-1}.

 

Edited November 20, 2018 by wtf

[uncool]

3 hours ago, pengkuan said:

We agree on the definition of finite set. But what is the definition of infinite set? Like the set of all even numbers?

Infinite means not finite. A set A is infinite if it is not finite, that is, if for any n, there is no bijection f: A -> {0, 1, ..., n-1}. 

It's not hard to prove that the set of even numbers is infinite. 

[studiot]

9 hours ago, uncool said:

Infinite means not finite. A set A is infinite if it is not finite, that is, if for any n, there is no bijection f: A -> {0, 1, ..., n-1}. 

It's not hard to prove that the set of even numbers is infinite. 

To add background to the excellent definitions from wtf and uncool:

Cantor chose the word 'transfinite' instead of infinite since he developed several different infinities.

The point is you divide the universal set into only two parts and define one part as XXX.

Everything else is then in then in the other part not or non XXX.

The trick is to chose the right part for the definition.

 

@taeto, did you get your X and Y mixed up?

[taeto]

13 hours ago, studiot said:

?

Yes, thank you. Should not try this late evening after an afternoon of lectures.

There is an \(x \in X\) which is not an element of \(Y.\) Which is the meaning of \(Y\) is a proper subset of \(X\). 

[wtf]

6 hours ago, studiot said:

 

The point is you divide the universal set into only two parts and define one part as XXX.

 

Jeez Louise @studiot, there is no universal set. That's Russell's paradox. 

Edited November 20, 2018 by wtf

[pengkuan]

19 hours ago, pengkuan said:

19 hours ago, wtf said:

A set that's not finite. 

Se cannot work with this "definition".

Yes, your definition is correct. But is it used to prove that a set is countable?

For example the rationals are countable. This is proved by counting along the diagonals of the plane N*N. 

The set {1/1,2/1,1/2...} is bijected to {1,2,3...n,...}, thus is countable. Is the definition used here?

Edited November 20, 2018 by pengkuan

[studiot]

2 minutes ago, wtf said:

Jeez Louise @studiot, there is no universal set. That's Russell's paradox. 

Forgive me I should have said "the appropriate universal set", since there term as a relative one.

In this case the universal set is the set of all entities possessing the properties of finiteness and nonfiniteness.

This has nothing to do with Russell.

[wtf]

7 minutes ago, studiot said:

 

In this case the universal set is the set of all entities possessing the properties of finiteness and nonfiniteness.

 

Nonsense. Garbage. Wrong. @Studiot why are you doubling down on this error? 

What is an "entity?" Do you see that you are using unrestricted comprehension, so that Russel's paradox has EVERYTHING to do with this? My God man this is GARBAGE. 

There is no set of all finite sets. There is no set of all infinite sets. There is no set of all finite or infinite sets. And for exactly the same reason. You can't form sets via unrestricted comprehension. There is no technical term in math called an "entity." You are just making this up.

 

 

Edited November 20, 2018 by wtf

[studiot]

5 minutes ago, pengkuan said:

Yes, your definition is correct. But is it used to prove that a set is countable?

This is a good answer

Quote

In general, the terms “countable” and “denumerable” are synonyms. Some mathematicians use “denumerable” to refer more specifically to an infinite countable set, such as the natural numbers. This is done because in many contexts, finite sets aren't interesting, and mathematicians are lazy folks who don't want to have to say “infinite” when that's already understood.

https://www.quora.com/What-is-the-difference-between-a-countable-and-a-denumerable-set

3 minutes ago, wtf said:

Nonsense. Garbage. Wrong. @Studiot why are you doubling down on this error? 

What is an "entity?" Do you see that you are using unrestricted comprehension, so that Russel's paradox has EVERYTHING to do with this? My God man this is GARBAGE

Yes that was a bad description.

I tend to think in pictures, not algebra.

I have no trouble with universal sets in pictures.

So go take a cold shower before you overheat.

[wtf]

51 minutes ago, studiot said:

 

So go take a cold shower before you overheat.

If you're wrong on the facts, admit you're wrong on the facts. Tossing insults doesn't reflect well on your character. 

Perhaps OP "thinks in pictures and not in actual mathematical facts." Why do you get a pass on a silly statement like that?

I've seen you rip through engineering math problems with skill and precision on two forums. I'm curious as to why you don't introspect and say to yourself, "I'm great at engineering math. Maybe I should review Russell's paradox."

I know for a fact that when you're working a differential equation you don't indulge yourself in flights of fancy. I'm sure you must agree with this point. You know what you're doing and you solve such problems with authority. If you are a little weak on set theory, you could study up. Not toss insults and claim that flights of imagination are more important than getting the math right. You would never make that argument when you do engineering math. Agreed?

Edited November 20, 2018 by wtf

[studiot]

But I'm not wrong, though we do seem to have a difference of opinion.

Differences of opinion occur in mathematics as well as other disciplines, as noted in part of my recent post.

Quote

Quote

In general, the terms “countable” and “denumerable” are synonyms. Some mathematicians use “denumerable” to refer more specifically to an infinite countable set, such as the natural numbers. This is done because in many contexts, finite sets aren't interesting, and mathematicians are lazy folks who don't want to have to say “infinite” when that's already understood.

https://www.quora.com/What-is-the-difference-between-a-countable-and-a-denumerable-set

Whilst I have already agreed that what I said was ill thought out and therefore amounts to balderdash, that was no reason to pour out vitriol.

You could have discussed more like I did with taeto's slip.

 

However since the OP is indifferent to my help I see no purpose being served in my further presence in this thread.

If you wish to discuss Russell and the set of all infinite sets of numbers properly, then this should be done in another thread, should you wish to start one.

 

[uncool]

2 hours ago, pengkuan said:

Yes, your definition is correct. But is it used to prove that a set is countable?

For example the rationals are countable. This is proved by counting along the diagonals of the plane N*N. 

The set {1/1,2/1,1/2...} is bijected to {1,2,3...n,...}, thus is countable. Is the definition used here?

No, that definition of finitude is not the definition used to prove that a set is countable, because countability and infinitude are different concepts.

However, I want to draw your attention to something. Note that every fraction appears after a finite number of steps. That is, if you gave me a fraction, I could tell you exactly at what step you reach it - and I could represent the number by writing it as "1 + 1 + 1 + 1" and eventually stop. For example, 2/4 is reached in the 12th step, that is, the 1+1+1+1+1+1+1+1+1+1+1+1th step. 

Edited November 20, 2018 by uncool

[studiot]

11 minutes ago, uncool said:

No, that definition of finitude is not the definition used to prove that a set is countable, because countability and infinitude are different concepts.

+1