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Is |{10, (8+2)}| = 2? (cardinality of the set)

I'm not sure about this because 8+2 = 10. I know the cardinality of the set {10, 10} = 1, but since 8+2 is a different representation of 10, it could be different? Any one can enlighten me on this?

Thanks!

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$8+2$ is a different representation of $10$, but it's still $10$. –  EuYu Oct 7 '12 at 2:29
    
No. The cardinality is 1. –  Shahab Oct 7 '12 at 3:05
    
If you reduce it to $\{10,(10)\}$, can you get rid of the brackets? i.e. is the second element just a number, or something else like a 1-tuple? –  Robert Mastragostino Oct 7 '12 at 3:32
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2 Answers

up vote 2 down vote accepted

Unless (like Asaf seems to be assuming) we're working at a much higher level of sophistication than it seems here, the elements of a set are simply numbers -- not expressions, representations or splotches of ink or chalk.

So when you write $\{10,(8+2)\}$ what you mean is that the number denoted by "$10$" is in the set, and the number denoted by "$(8+2)$" is in the set, and nothing else is. These two expression happen to denote the same number (which can also be described as "ten" or "one less than eleven" or 0x0A or "$1+1+1+1+1+1+1+1+1+1$"), so that number is the only element of the set in question, so its cardinality is $1$. Period.

It is somewhat common to understand sets intuitively as "lists of things where the order doesn't matter (and neither does repetitions)". This understanding can be misleading unless one is extremely careful about what a repetition is (not to speak of what a "list" means if it has infinitely many elements). What is really going on is:

A set of something that you can ask "is this one of your elements?" of for every "this" in the universe. The set consists of its yes/no answers to all of these questions, neither more nor less.

So when you write $\{10,(8+2)\}$ you're speaking of a set that answers "yes, $X$ is one of my elements" if and only if $X=10$ or $X=(8+2)$. But no matter what $X$ is, "$X=10$" and "$X=(8+2)$" are either both true or false, so that is the same as answering "yes, $X$ is one of my elements" if and only if $X=10$. And therefore $\{10,(8+2)\}$ is (a name for) the same set that $\{10\}$ is a name for. A set consists only of its answers, so when the answers are the same we're looking at the same set.

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You are a bit right, and a bit wrong.

The set contains two distinct terms. However from the axioms of PA/rings/fields (or whatever axioms you work with) we can prove the terms are equal, and the set is equal to the singleton.

So as terms, the set has two elements. But if one works within a theory (as one usually does) then the set has provably but one element.

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