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Media Center. An example of an axiom is that, given any natural number which is a primitive concept , there exists a larger natural number. This is self-evident, and not in serious doubt. Logic is then used to derive sophisticated results from axioms. Eventually, we are able to construct models, which are mathematical structures that satisfy a collection of axioms. Crucially, any statement proven from axioms, through the use of logic, will be true when interpreted in any model that makes those axioms true.
If there are six goats and eight trees, we will not be able to set up such a correspondence: no matter how hard we try, there will be two trees that are goat-free. The branch of mathematics that does this work is known as set theory. One can prove mathematical statements by first appropriately interpreting the statement in the language of sets which can always be done , and then applying logic to the axioms of sets.
Some set axioms include that we can gather together particular elements of one set to make a new set; and that there exists an infinite set. This prevented the Continuum Hypothesis from being disproven. Remarkably, some years later, Paul Cohen succeeded in finding another model of set theory that also satisfies set theory axioms, that does allow for such a set to exist.
This prevented the Continuum Hypothesis from being proven. It remains possible that new, as yet unknown, axioms will show the Hypothesis to be true or false. For example, an axiom offering a new way to form sets from existing ones might give us the ability to create hitherto unknown sets that disprove the Hypothesis. The uncertainty surrounding the Continuum Hypothesis is unique and important because it is nested deep within the structure of mathematics itself.
The proof lies in being able to disprove A hypothesis or model is called falsifiable if it is possible to conceive of an experimental observation that disproves the idea in question. Three types of experiments proposed by scientists Type 1 experiments are the most powerful.
Type 1 experimental outcomes include a possible negative outcome that would falsify, or refute, the working hypothesis. It is one or the other. Type 2 experiments are very common, but lack punch. A positive result in a type 2 experiment is consistent with the working hypothesis, but the negative or null result does not address the validity of the hypothesis because there are many explanations for the negative result. These call for extrapolation and semantics.
Type 3 experiments are those experiments whose results may be consistent with the hypothesis, but are useless because regardless of the outcome, the findings are also consistent with other models.
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