Recent content by gufiguer

Look on page 2 http://www.ime.usp.br/~tausk/texts/MathPhysics.pdf

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...

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...

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

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

The book by Enderton is good too.

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...

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...

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...

For no more cofusing arise :)

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...

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...

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)] ;]...

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.