Welcome to free math help online,
There are some cases in which assuming different sets of axioms will
give you different bodies of mathematics, and that can be fun. For
example, this is the difference between Euclidean and Hyperbolic
geometry; one simple axiom about parallel lines is changed, and it has
huge effects on what the resulting body of mathematics looks like.
Statements of type (2) are pretty tough, but we'll crack 'em. :-)
Statements of type (3) are things that can make you question your
place in the universe. An Austrian mathematician named Kurt Godel
proved in 1931 that there must be statements of mathematics that
cannot be proved or disproved. examples on math helper; The somewhat
irritating thing is that
we don't know which statements these are. The main thing that
mathematicians get out of this is that the forest of mathematics is a
complicated one indeed, and we'll never completely know its landscape.
learn more examples on online math forum.
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