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@fresh_42 @FactChecker @WWGD

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- Thread starter Leo Authersh
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@fresh_42 @FactChecker @WWGD

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The distinction between conjecture and hypothesis is not formally defined.

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Intuitive representation of the differences amongst Conjectures, Hypotheses and Axioms.

- conjecture -
**unproven**statement which is**expected to be true**by evidence like exemplary calulations or failed searches for counterexamples, e.g. ##NP \neq P##. The literally translation would be "a casting (throw) together (of facts, etc.)".##\\## ##\\## - hypothesis -
**unproven**assumption**without any expectations**whether true or false (however, often false), e.g. an assumption in an indirect proof that is supposed to be false. It is more of a**tool**in a reasoning than a statement on its own, e.g. https://en.wikipedia.org/wiki/Thesis,_antithesis,_synthesis. The literally translation would be "imputation" ##\\## ##\\## - axiom -
**unproven**statement which is**set to be true**by definition. It is somehow one of the fundamental bricks a building of conclusions is constructed upon. This leads to a true chain of conclusions and not necessarily to a true statement in an absolute sense. The attempts to find such absolute truths, which can be used as basis for all other calculations failed and had to fail. An axiom is the closest we can get to something absolutely true. The literally translation would be "tenet = it holds".

One could shortly write them by

$$

\text{ axioms } \Longrightarrow \stackrel{\text{use of hypothesis }}{\ldots \ldots} \Longrightarrow \stackrel{\text{ unknown way }}{\ldots \ldots} \Longrightarrow \text{ proof of conjecture or counterexample }

$$

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