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Real Analysis!!!
Posted:
Aug 11, 2012 1:21 PM
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Language: First order logic + (=,e,<>,||)
e is the membership symbol.
= is the identity symbol
<> is the ordered pair symbol, a two place function symbol
| | is the cardinality symbol, a one place function symbol
Axioms: Identity theory axioms +
(1) Sets with the same elements are identical
For all x. For all y. ((For all z. z e x <-> z e y) -> x=y)
(2) A singleton is its element
For all x,z. ((Exist! y. y e x) & z e x) -> x=z)
(3) An element is singleton
For all x. ((Exist y. x e y) -> (Exist! z. z e x))
(4) There exist a set of all singletons satisfying phi
if phi is a formula in which x is not free then,
(Exist x. for all y. (y e x <-> (Exist!z. z e y) & phi(y)))
is an axiom.
(5) Ordered pairs with the same projections are identical
For all a,b,c,d. (<a,b>=<c,d> -> (a=c & b=d))
(6) An ordered pair of singletons is singleton
For all x,y,p.([(Exist! z. z e x) & (Exist! u. u e y) & p=<x,y>] ->
(Exist! m. m e p))
(7) Two sets have the same cardinality iff a bijection exists between
them.
For all x. For all y. ( |x|=|y| <-> Exist f. f: x-->y & f is
bijective )
(8) Each cardinal is a singleton
For all x. For all y.(y=|x| -> Exist! z. z e y)
(9) Every Dedekind Cut is Singleton.
(Dedekind cut is an ordered pair <A,B> where A,B are subsets of Q (the
set of all rational numbers) such that both A,B are non empty, A,B are
disjoint, and A union B = Q, and every element in A is strictly
smaller than every element of B, and A has no maximal element)
/
This will be sufficient to do all Real Analysis.
Zuhair
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