I have a question about the empty set.
Is $0$ (zero) an element of $\varnothing$? And what is the cardinality of {0}?
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asked Oct 27, 2016 at 19:26
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11$\begingroup$ The empty set has no elements. $\endgroup$– DirkOct 27, 2016 at 19:27
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2$\begingroup$ Perhaps you should repeat in high voice the definition of the "empty set" ... $\endgroup$– DonAntonioOct 27, 2016 at 19:28
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2$\begingroup$ The cardinality of $\{0\}$ is $1$. $\endgroup$– user228113Oct 27, 2016 at 19:28
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2$\begingroup$ Funny enough, some set theorists may say that $\{0\}$ actually is $1$ (and that $0=\emptyset$). $\endgroup$– user228113Oct 27, 2016 at 19:30
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1$\begingroup$ @G.Sassatelli Most probably those set theorists would also give a set up of assumptions, axioms, etc. in which that would make some sense....or they are high, of course. :) $\endgroup$– DonAntonioOct 27, 2016 at 19:31
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2 Answers
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You have two boxes separate from each other. One box contains nothing. The other box has a piece of paper with the number zero on it. The first box represents $\{ \} = \emptyset$ while the second represents $\{ 0 \}$. Two different things. The first has no objects, the second has only one.
answered Oct 27, 2016 at 19:42
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No. The empty set is empty. It doesn't contain anything. Nothing and zero are not the same thing.
answered Oct 27, 2016 at 19:28
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2$\begingroup$ @fleablood Look at it this way: The set of all integers less than 1 and greater than -1 is $\{0\}$. The set of all integers greater than 1 and less than 2 is $\emptyset$. The two are not the same thing. $\endgroup$ Oct 27, 2016 at 19:39
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1$\begingroup$ Of course not. I was pointing that nothing is "nothing". Nothing and zero are not the same that but nothing and "nothing" are not the same thing either. $\endgroup$ Oct 27, 2016 at 19:56
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