Notes:Infinity notation

Overview

I think I have made a mistake with the notation:

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

However take:

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

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

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

Possible solution

The solution currently being considered is:

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

Counterpoints

1. What about $\sum^\infty_{n=1}a_n$? Should we write $\sum_{n\in\mathbb{N} }a_n$ instead? This also has $\sum_{i=1}^5 a_i$ being the sum from $a_1$ to $a_5$ inclusive.
   • This is sidestepped by saying:
     • $$\sum^\infty_{n=1}a_n$$ is an expression/notation/syntatic sugar for writing $$\lim_{n\rightarrow\infty}\left(\sum_{k=1}^na_k\right)$$
   • Of course also we cannot sum infinite terms, nor is there an $a_\infty$ 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 $\bigcup_{n=1}^\infty$