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June 6th, 2019, 11:00 PM   #1
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Null set paradox

Paradox: if any member, or lack thereof, in the null (i.e., empty) set is countable, it requires the concept of zero, so it is not truly empty.
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June 6th, 2019, 11:20 PM   #2
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June 6th, 2019, 11:46 PM   #3
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Empty set:
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The empty set is the unique set having no elements
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June 7th, 2019, 04:58 AM   #4
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Quote:
Originally Posted by Loren View Post
Paradox: if any member, or lack thereof, in the null (i.e., empty) set is countable, it requires the concept of zero, so it is not truly empty.
You are mixing several different mathematical notions and pretending they are somehow related. To see why this is absurd, consider the related "theorem" you could prove using the exact same argument.

Theorem: All sets containing a 1984 Honda Accord, must contain at least 1983 other elements. The proof is obvious. Since any such set requires the concept of 1984, it can't truly have fewer than 1984 elements.
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June 7th, 2019, 05:05 AM   #5
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Originally Posted by Loren View Post
Paradox: if any member, or lack thereof, in the null (i.e., empty) set is countable, it requires the concept of zero, so it is not truly empty.
Think of 0 as the additive identity. That means it is something, not nothing.

-Dan
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June 7th, 2019, 11:01 AM   #6
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I wish you all success in a field I know much less about.

Quote:
The empty set is the unique set having no elements
I was trying to grasp (the count of) how many elements the null (empty) set has, and whether the null set could exist without a concept of zero.

For instance, could any set exist without the concept of numbers?

No wonder why zero was so late on the number scene!
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June 7th, 2019, 11:27 AM   #7
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whether the null set could exist without a concept of zero.
Given any set $X$, the empty set is the set of all $x \in X$ such that $x \neq x$. No numbers are required to make that definition.

It's also the set of all purple flying elephants. No numbers needed to make that definition either.

Sets precede numbers in the modern formalism. Although of course numbers preceded sets historically.
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June 7th, 2019, 02:08 PM   #8
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It's also the set of all purple flying elephants. No numbers needed to make that definition either.
Now wait a minute! I know three of them and they like the same brand of whiskey I do!

-Dan
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June 7th, 2019, 11:11 PM   #9
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Given any set X, the empty set is the set of all x∈X such that x≠x
Simple yet profound, Maschke.
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