Is this correct(countable vs uncountable)

[Azael]

I am trying to explain to someone the difference between countable and uncountable. I am not exactly 100% sure myself and a strictly mathematical definition doesn't help me.

So now when I was thinking about it I came up with this explanation and I am wonder if it is accurate?

a way to look at the difference between a countable and uncountable set would be to imagine you are number 4.

If you are contained in a countable set you can look to your right and find for instance 4.1 and you can look to your left and find for instance 3.9. There are discrete values around you because 4.05 or 3.99 are not allowed. That means you can pinpoint the exact number that is closest to you and can count your way to any allowed number. You can jump to 4,1 and then 4,2 ect and count each step until you reach any desired number.

If you are contained in a uncountable set however you can look to your right and find no nearest number. If I say 4.1 is the nearest its not true because 4.01 is even closer and then we have 4.001 and so on to infinity, you have no closest neightboor, just a bunch of numbers that get infinitly closer without end. No discrete values are around you. So you can not jump count your way to 4.1 since there are just no defined places to jump to. Its impossible to count your way to any number and that makes it uncountable.

 

[TD]

The rationals (Q) are countable, but if you're number 5/3, can you tell me which ones are on your left and right?
If you say "x is on my right", then I say: what about (x+5/3)/2? You see the problem? The idea wasn't all bad though...

 

[Azael]

Yeah I se now :( Do you know anyway that the difference countable vs uncountable can be explained simply with words?

 

[matt grime]

Discrete sets can be uncountalbe, and uncountable sets can be not discrete. "Next to" or 'infinitely close' have no meaning in arbitrary sets, and countability or otherwise is purely a statement about sets, not about things like 4.005.

A set is countable if it can be put in bijection with a subset of the natural numbers. I.e. if there is a way to label every element of the set with numbers 1,2,3,4,... or some finite subset of 1,2,3,4...

The natural numbers are tautologically countable, so are the integers: the integers can be listed thus;

0,1,-1,2,-2,3,-3,...

so they are clearly countable.

 

[CRGreathouse]

Yeah I se now :( Do you know anyway that the difference countable vs uncountable can be explained simply with words?

Your description is a good (layman's) one for dense vs. not dense, I think.

For countable vs. uncountable, I'd say that countable sets can be put on a list, while uncountable ones are "too big" for any list.

 

[Azael]

I guess that's why I never got into maths and choose physics
I can work with the definitions, manipulate the expressions and get the desired answeres but I never have a grasp for what I am doing and I can't express it in words so that creates a barrier and distance in my mind and a feeling that I never understand it

 

[TD]

Unless you feel comfortable enough with understanding the definition, without having the need to put them in 'other words'?

 

[Azael]

Discrete sets can be uncountalbe, and uncountable sets can be not discrete. "Next to" or 'infinitely close' have no meaning in arbitrary sets, and countability or otherwise is purely a statement about sets, not about things like 4.005.

A set is countable if it can be put in bijection with a subset of the natural numbers. I.e. if there is a way to label every element of the set with numbers 1,2,3,4,... or some finite subset of 1,2,3,4...

The natural numbers are tautologically countable, so are the integers: the integers can be listed thus;

0,1,-1,2,-2,3,-3,...

so they are clearly countable.

That makes a lot of sense and is the clearest explanation I have seen so far. Thanks

 

[Azael]

Unless you feel comfortable enough with understanding the definition, without having the need to put them in 'other words'?

Never been able to. I guess that is something that comes naturaly when working with definitions for a long time??
I almost got that relation with stochastic calculus in the short class in that subject that I took though, so there would be hope incase I take anymore classes. A good review of basic calculus would be needed though since the classes I took focues on problem solving and not so much on theory.

 

[Alkatran]

A countable set is one where you can number all of the elements and have no numbers left over. For example, the set of all positive even numbers can be numbered like so:
1 -> 2
2 -> 4
3 -> 6
etc...

An uncountable set is one where you can't number all of the elements. Another way of saying it is that you can't list an uncountable set, even with an infinitely long list. The real numbers are uncountable because, given a list of real numbers, you can construct a real number not on the list.

 

[matt grime]

A countable set is one where you can number all of the elements and have no numbers left over.

you really should specify that by 'number' you actually mean natural number (the positive integers). Plus you are adopting the convention that finite sets are not countable, my convention was that finite sets are countable, but that is just a convention - I see both regularly.

 

[matt grime]

That makes a lot of sense and is the clearest explanation I have seen so far. Thanks

What other definition can you have had?

 

[Azael]

What other definition can you have had?

This one for instance wasnt so clear

http://en.wikipedia.org/wiki/Uncountable

The first part is but the rest isn't clear to me.