Notes:Infinity notation

Overview

I think I have made a mistake with the notation:

• Suppose we write [ilmath]\bigcup_{n=1}^\infty[/ilmath], if we have [ilmath]\bigcup_{n=1}^\infty A_n[/ilmath] for a sequence [ilmath] ({ A_n })_{ n = 1 }^{ \infty } [/ilmath] all is well, from the expression we can tell it means the union of all terms in the sequence.

However take:

• [math]\bigcup_{n=1}^\infty X^n[/math], where [ilmath]X^n[/ilmath] is to be interpreted as all [ilmath]n[/ilmath]-tuples of elements of [ilmath]X[/ilmath]
  • Does this mean all finite tuples, or does it include [ilmath]X^\mathbb{N} [/ilmath]?

Typically when we write [ilmath]\bigcup_a^b[/ilmath] we mean starting at [ilmath]a[/ilmath] and proceeding towards [ilmath]b[/ilmath] in the obvious way, and including [ilmath]b[/ilmath], for example:

• [ilmath]\bigcup_{i=1}^5 A_i[/ilmath] is [ilmath]A_1\cup A_2\cup A_3\cup A_4\cup A_5[/ilmath], so when we encounter an [ilmath]\infty[/ilmath] (which in this case... if anything means [ilmath]\aleph_0[/ilmath]) we should attempt to include it!

Possible solution

The solution currently being considered is:

• [ilmath]\bigcup_{n\in\mathbb{N} } A_n[/ilmath], this has the advantage of:
  • [ilmath]\left[x\in\bigcup_{n\in\mathbb{N} }A_n\right]\iff\left[\exists n\in\mathbb{N}(x\in A_n)\right][/ilmath] (by definition of union), this is exactly what we mean when we write this.

Counterpoints

1. What about [ilmath]\sum^\infty_{n=1}a_n[/ilmath]? Should we write [ilmath]\sum_{n\in\mathbb{N} }a_n[/ilmath] instead? This also has [ilmath]\sum_{i=1}^5 a_i[/ilmath] being the sum from [ilmath]a_1[/ilmath] to [ilmath]a_5[/ilmath] inclusive.
   • This is sidestepped by saying:
     • [math]\sum^\infty_{n=1}a_n[/math] is an expression/notation/syntatic sugar for writing [math]\lim_{n\rightarrow\infty}\left(\sum_{k=1}^na_k\right)[/math]
   • Of course also we cannot sum infinite terms, nor is there an [ilmath]a_\infty[/ilmath] term in a sequence. We can only sum finitely many times (in a ring, or group)

This page is some notes on a solution to this problem, and to mention "irregularities" that may result.

Practical problems

1. A lot of pages use [ilmath]\bigcup_{n=1}^\infty[/ilmath]