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these questions are not obvious and were only cleared by 19th century number theory. See Bolzano weirstrass, or define reals as equivalence classes on convergi
by sebapi747 9y ago
these questions are not obvious and were only cleared by 19th century number theory.
See Bolzano weirstrass, or define reals as equivalence classes on converging series of rationals.
- adrianN 9y agoThe irrationality of sqrt(2) was known to Pythagoras.
- infinity0 9y agoYou missed the point of the post. Pythagoras could prove sqrt(2) is irrational but the proof already implicitly assumes it exists. "exists" in mathematics, basically means that it does not contradict the axioms of whatever system you're working under, or result in a logical inconsistency. For example, the set that is the subject of Russell's paradox, cannot exist in any reasonable system of set theory, and its definition shows that naive set theories are logically unsound.
- adrianN 9y agoNo, Pythagoras started from something that definitely exists: the hypotenuse of a triangle with two sides of length 1. He wanted to find out its length and to his great dismay he discovered that there is no rational number to express its length.
- sebapi747 9y agoYou don’t need Euclidean space to exist to define reals but you need reals numbers for distance to exist.