Countability: Subjective or Objective?

[Swapnil]

"Are there more real numbers between 0 and 1 or between 0 and 2?"

If you ask this question to a present day mathematician, he/she would answer that they have the same amount of numbers. Why? Because for every x in the set of numbers between 0 and 2 (call this set A), there is a corresponding number x/2 in the set of numbers between 0 and 1 (call this set B). Thus both set A and B have the same number of elements.

But this type of reasoning seems very subjective to me. If instead of mapping from x -> x/2, you map from x -> x/3, then you conclude that there are more elements in set B! Furthermore, if you map from x -> x, then you conclude that set A is bigger! Thus, by changing your mapping you can just about say any thing: |A| > |B|, |A| < |B|, or |A| = |B| !

I don't get it. I thought math is suppose to be objective, not subjective?

 

[CRGreathouse]

It's absolutely objective. I don't know how your x -> x/3 mapping proves anything, but here's a simple way to deal with equal cardinalities. If you have sets X and Y, and X maps (1-1 and onto) to a subset of Y and Y maps (1-1 and onto) onto a subset of X, then |X| = |Y|.

 

[matt grime]

Step 1: you use bijections to count cardinality. That is correct.

Step 2: you don't use cardinality to talk about cardinality. That is incorrect.

It isn't maths that is subjective here, but your arbitrary decision to use two different notions of 'size'.

 

[Hurkyl]

In particular

But this type of reasoning seems very subjective to me. If instead of mapping from x -> x/2, you map from x -> x/3, then you conclude that there are more elements in set B! Furthermore, if you map from x -> x, then you conclude that set A is bigger!

By the very definitions of the ordering, these tell you that $| A | \leq | B |$ and $| A | \geq | B |$ respectively.

It is a mistake to think that they imply |A| < |B| or |A| > |B|.

Your intuition about finite sets has misled you. You should generally avoid using it when dealing with infinite sets. (Ideally, you work with the definitions, and in the process you build up an intuition for infinite sets)

 

[Swapnil]

So why should I have those $$\geq , \leq$$ signs as oppose to $$<,>$$signs.

 

[CRGreathouse]

So why should I have those $$\geq , \leq$$ signs as oppose to $$<,>$$signs.

Do you know what those mean for cardinalities?

 

[matt grime]

We don't have to answer that question (well, we already have). Instead you have to answer the question: why do you think you should use >,<?

If X is a proper subset of Y, that does not imply |X|<|Y|. If you think it does then you are using the wrong notion of cardinality. Cardinality is based upon bijections, not containment (though containments do give useful implications about cardinality, just not the one you think). Your intuition is evidently based upon finite sets.

 

[Hurkyl]

So why should I have those $$\geq , \leq$$ signs as oppose to $$<,>$$signs.

Because that's how those symbols are defined.

The definition of $|A| \leq |B|$ is that there exists an injective map A -> B.

The definition of |A| < |B| further requires that there does not exist a bijection A -> B.

(Note that |A| < |B| is not always the same as $\neg(|B| \leq |A|)$ unless you assume the axiom of choice)

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

 

[CRGreathouse]

Cardinality is based upon bijections, not containment (though containments do give useful implications about cardinality, just not the one you think).

I agree, though I must admit I use injections via the Cantor-Bernstein theorem more often than actual bijections.