Is this a stupid question? Meaningless?

[daon]

Hello all! I have a perplexing thought for everyone to chomp on (at least I think so)... I was tutoring a student on infinite series when they incorrectly made an answer: "The sum of all the reals." That got me thinking.

If we donote the sum of a set A (given such a structure exists as in the case of a field) by:

\(\displaystyle \phi \(\mathbb{A}\) = \L \sum _{a \in \mathbb{A}} a\).

Just what is...

\(\displaystyle \phi \(\mathbb{R}\) = \L \sum _{a \in \mathbb{R}} a\).

Is it zero? Some order of infinity? Is it meaningless? What about:

\(\displaystyle \phi \(\mathbb{N}\)\), \(\displaystyle \phi \mathbb{\(\(0,1\)\)}\), \(\displaystyle \phi \(\mathbb{Q}\)\), \(\displaystyle \phi \(\mathbb{R}_+\)\)?
(\(\displaystyle \mathbb{R}_+\) meaning positive reals).

Would \(\displaystyle \phi \(\mathbb{R}_{+}\) = \phi \(\(0,1\)\)\)? This comming from the fact that there exists a bijection between the sets, although I'm not sure if that is related at all.

Related, what about \(\displaystyle \L \prod _{a \in \mathbb{R}_{\ge1}} a\)?

This whole idea may be pointless. Thought it might make for an interesting discussion...

 

[tkhunny]

Circular or self-destructive.

It isn't likely to make sense that the sum of all the Reals is a Real Number, otherwise you have missed one. If this is so, addition isn't closed over the Reals. That's not a good result.

 

[daon]

Circular or self-destructive.

It isn't likely to make sense that the sum of all the Reals is a Real Number, otherwise you have missed one. If this is so, addition isn't closed over the Reals. That's not a good result.

I didn't mean to imply the sum would be a real number, but rather some magnitude of infinity, maybe zero, or some undefined element that might belong to another mathematical structure.

But what you said would make sense, if and only if the sum were zero. If the sum of all reals were zero, then you haven't missed one (any further additions of phi(R) would only be adding zero). From a set theoretic point of view, for every real number r, there exists a -r so that they'd cancel.

Also, if the product of all real numbers (minus zero) is 1, we'd have a similar situation. I was really looking to explore "how big" a sum of phi(R+) was in relation to "how big" phi((a,b)) (0<a<b) were.

 

[tkhunny]

Points to ponder. Surely there is an esoteric paper sitting about, somewhere.