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galoisjr

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So anyway, this got me wondering what exactly constitutes scientific proof? Is there even such thing as absolute proof in science? Or are we doomed to only see evidence and not the entire picture?

The biggest problem lies in an area very familiar to us mathematicians: in set theory. Let's say we are given a theory. Now this theory is supported by a possibly infinite set of evidence. However, each single piece of evidence may or may not (more likely than not in my opinion) belong to the set of evidence supporting another theory also. In many, possibly infinite, circumstances, the single element of evidence may belong to a possibly infinite amount of sets each supporting a different theory. Also, given a theory with a possibly infinite supporting data set, there is also is more likely than not a possibly infinite amount of intersections with the data sets of other theories and each intersection may contain many, or, you guessed it, possibly infinite elements...

You can see how this gets old really quick...

Anyway not having looked at as much set theory as I would have liked to lately, I came to a question that I could not answer. And I really do not know if it is answerable...

Suppose you are given two sets whose intersection contains infinitely many elements. Does this restriction require one set to be a subset of the other? (if they are both subsets of each other then we have equality and thus the trivial conclusion) And also if not, can you think of an example?

For instance, an example of the affirmative would be that the integers and rationals have an infinite intersection and the integers are indeed a subset of the rationals.