Infinity in logic implicit (compact) notation.

Question

This question is about notation.

Is this true?

$$\bigvee_{i=1}^{\infty}~p_{i} \Longleftrightarrow \bigvee_{i=1}^{}~p_{i}$$

I mean, will it represent the same if I write $\infty$ or left it blank on top of $\bigwedge$ and $\bigvee$ ?

Answer 1

I think the notation "$\bigvee_{i=1}p_i$" is noticeably less clear than the other. Another option would be "$\bigvee_{i\ge 1}p_i$" if you really want to avoid upper bounds for some reason.

Ultimately, I don't see why you wouldn't put in upper bounds: it makes things more clear and doesn't add any real length or complexity notation-wise.

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For what it's worth, I personally prefer $$\bigvee_{i\in\mathbb{N}}p_i$$ since there are times index sets other than $\mathbb{N}$ are used (say, $\mathbb{R}$). But this is definitely an ideosyncracy of mine: "$\bigvee_{i=1}^\infty p_i$" is fully unambiguous in my opinion (if $i$ were intended to range over reals, I think "$\bigvee_{i\in [1,\infty)}p_i$" would be used instead).