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  1. gufiguer

    What really are units? Why can we ignore them, like in class?

    Look on page 2
  2. gufiguer

    Defining the rule of an arbitray function

    It is not what you have asked, but a symbol for a variable in the "operator" ##\frac{d}{dx}## is just useful for doing long calculations in a draft at most. It is not rigorous, it is not used in the definition and in fact is long time dead. As a paradox, the same type of variable symbol is...
  3. gufiguer

    Set of all groups

    The answer is yes. For example, you can use ZF set theory with one more axiom: the existence of Grothendieck universes. This is equivalent to the existence of strongly inaccessible cardinals. More formally, the following two axioms are equivalent (i) For each set x, there exists a Grothendieck...
  4. gufiguer

    Free groups: why are they significant in group theory?

    We can build in this way, for example, tensor products of non-abelian groups, which are useful in homotopy theory.
  5. gufiguer

    Free groups: why are they significant in group theory?

    The structure of a free object F(X) depends uniquely on the cardinality of the set X. So, in a concrete category, there exists at most (up to iso) one free object for each cardinal number. In some categories some of these objects don't exists. In the usual algebraic categories, like groups...
  6. gufiguer

    Basis of module and Free module

    Search about "free objects" in a category (in category theory).
  7. gufiguer

    Good textbook on set theory?

    The book by Enderton is good too.
  8. gufiguer

    The metric space axioms

    Just continuing. The only metric on ##\varnothing## is ##\varnothing## and the only metric on a singleton X={x} is d={((x,x),0)}, ie, d(x,y)=0, for all x and y in X. If ##card(X) \geq 2##, then ##\varnothing## and {((x,x),0)} are not metrics for X as before, but in this case there are...
  9. gufiguer

    Role of classical logic in studying logic

    Beautiful philosophical discussion!! The human reason is behind all and Logic was born to describe it. If it was possible to forget Logic and to think without it, we would be no-humans. If there was a way to think without (classical) logic, then this way of thinking would be a part of logic and...
  10. gufiguer

    Is it worth studying mathematical logic?

    Mathematical Logic (ML) is a theory of "how to write logic", ie, it is a specific formal language theory and it mimetizes what you allready know on Logic. The human reason is behind all, behind the very modus ponens and behide every mathematical proof. Your problem is not with ML but with some...
  11. gufiguer

    Metrics and topologies

    For no more cofusing arise :)
  12. gufiguer

    Is the empty set always part of the basis of a topology?

    In the Aristotle theory of syllogisms it is a valid categorical syllogism that "all S are P" implies his subaltern "some S are P". But that theory don't consider the possibility that "there is no S at all", ie, this theory don't consider the idea of an empty set. The arguments showed in this...
  13. gufiguer

    More Rigorous Text than Rudin?

    Here is a philosophical approach (a formal approach is a little different) For sets you can read Jech, Halmos and Enderton. You can focus on ordinals. All textbooks on analysis that I know don't pay much attention on using the axiom of choice and the recursion theorem on ordinals. I feel sad...
  14. gufiguer

    The metric space axioms

    Let X be a set. A function [; d \ : \ X \times X \to \mathbb{R} ;] is a metric for X if, and only if, (i) d(x,x)=0, for all x in X (ii) d(x,y)=d(y,x), for all x and y in X (iii) [; (\forall x,y \in X) [x \neq y \to d(x,y) > 0] ;] (iv) [; (\forall x,y,z \in X) [d(x,z) \leq d(x,y) + d(y,z)] ;]...
  15. gufiguer

    Poor Symbol choice - Principles of Mathematical Analysis by Rudin

    Yes. That can be misleading at the beginning but is standard in textbooks. Some algebra textbooks adopt unusual symbols in the beginning, about elementary group theory, but return to the standard notation as soon as possible. On elementary ring theory the notation is the standard in most textbooks.