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Say, " $\psi \Rightarrow \varphi$ " is a theorem and $\psi$ is essential in the hypothesis.

I don't understand what's the meaning of essential.

Here's what i guess;

If $[\psi \Rightarrow \Phi] \bigwedge \neg [\Phi \Rightarrow \psi] \Rightarrow \neg [\Phi \Rightarrow \varphi]$, then we call $\psi$ is essential in the hypothesis.

Am i correct?

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2 Answers 2

up vote 8 down vote accepted

There is not a single formal definition of an "essential hypothesis"; it is an informal phrase used in mathematics exposition rather than part of the mathematics itself. It's easier to think of a theorem that has several hypotheses $A$ and $B$ and a conclusion $C$. There are two things people usually mean by an "essential" hypothesis.

  • If $A$ and $B$ implies $C$, but $B$ alone does not imply $C$, some people will say $A$ is an essential hypothesis. It is "essential" in the sense that it cannot simply be omitted from the proof. So for example the hypothesis of second-countability in the theorem that "every second-countable regular space is metrizable" is essential in this sense, because not every regular space is metrizable.

  • A stronger meaning is that the conclusion actually implies the hypothesis. For example, in the theorem that every second-countable regular space is metrizable, we know that any metrizable space is regular, so that hypothesis is not only essential in the sense of the first bullet, it is essential in the sense that if the conclusion holds it will also hold.

You have to use context to distinguish between these two meanings. My sense is that the first meaning is more common.

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Essential means that if the condition does not hold then the conclusion may not be true.

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