Search All of the Math Forum:

Views expressed in these public forums are not endorsed by Drexel University or The Math Forum.

Topic: Real Analysis!!!
Replies: 12   Last Post: Aug 13, 2012 7:28 AM

 Messages: [ Previous | Next ]
 Zaljohar@gmail.com Posts: 2,628 Registered: 6/29/07
Real Analysis!!!
Posted: Aug 11, 2012 1:21 PM

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