Definition of Strong and Weak form of a theorem

JFo

92
0
Could someone explain to me what it means for a theorem to be a strong(er) form or a weak(er) form of another theorem?

I've heard these terms used over and over, but never bothered to ask. If I had to guess at a definition, I'd say that if q is a theorem then we say p is stronger if p implies q. Similarly p is weaker if q implies p. Am I close?
 

matt grime

Science Advisor
Homework Helper
9,392
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Yes. Generally - aren't there always exceptions in mathematics.

As a general rule the strong version will need more hypotheses than the weak version, but correspondingly prove a result that is stronger (and implies the weaker version).

This is not always the case, as there is at least on situation where the strong and weak version actually are equivalent: induction and strong induction are the same, but phrased differently so that one seems like a stronger (more powerful) result.
 

JFo

92
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Thanks. I know it's mostly semantics, but I couldn't find a definition, or even a description, anywhere.
 

JFo

92
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So if we have a theorem, and are then able to show that we can drop one of the hypotheses (or replace it with a weaker hypothesis) and still obtain the same result, would that be considered "strengthening" or "weakening"

Intuitively I would think strengthening, but then I don't know how that fits in with the quasi-definition given above since we have less hypotheses but the result remains unchanged. Or is this a different meaning of strong/weak entirely?
 
316
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So if we have a theorem, and are then able to show that we can drop one of the hypotheses (or replace it with a weaker hypothesis) and still obtain the same result, would that be considered "strengthening" or "weakening"

Intuitively I would think strengthening, but then I don't know how that fits in with the quasi-definition given above since we have less hypotheses but the result remains unchanged. Or is this a different meaning of strong/weak entirely?
The theorem that assumes less in the hypothesis is "stronger" or "more general". I wouldn't worry too much about what these terms exactly mean though, since they don't carry much information.
 

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