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Searched it on Google and couldn't find it. Consider the following literal statement:

If there exists a supremum M for A, where A is a set contained within the one dimensional continuum R, then M may or may not be a member of A.

I wish to write it as a formal statement.

I'm not sure if a notation question qualifies for a valid question on this exchange, so please alert me if it doesn't.

Laplace
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    Since tertium non datur, the propsition $M\in A$ or $M\notin A$ means that $A$ is a (decidable, i.e. well-defined) set. Nothing is added. Instead you want to say there exist $A$ and $B \subset \mathbb R$ such that $\sup A \in A$ and $\sup B \notin B$. – AlexR Dec 03 '14 at 22:00
  • Thanks for replying to my question. I wish you had posted as an answer instead of a comment so I could give you accepted answer. Anyway, I like your solution. – Laplace Dec 03 '14 at 23:39
  • I'll do that for you. – AlexR Dec 03 '14 at 23:45

1 Answers1

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Since tertium non datur, the propsition $M\in A$ or $M\notin A$ means that A is a (decidable, i.e. well-defined) set. Nothing is added.

Instead you want to say there exist $A$ and $B\subset \mathbb R$ such that $\sup A\in A$ and $\sup B\notin B$. A specific example is $A = [0,1]$ and $B = (0,1)$ with $\sup A = \sup B = 1$.

AlexR
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