[AntonioLao]

In mathematics, there are certain rules to follow and there are ones not to ignore. Ignoring them could result in confused mathematics while following does not guarantee complete understanding or complete knowledge of the underlying implication, for what is implied could not possibly or more likely surpass its own definition (assumption, axiom, and postulate). Mathematics is clearly a definitive science of absolutely exact logic based on the axiom of choice.

The applicability of this axiom is not just limited to mathematics. It could just as well apply to human relationships, economics, policy and decision makings, stock markets, governments, politics, international relationships, and even in the theory of biological evolution and extinction (watery creatures made to choose to live on dry land). However, without any rules or any laws (e.g. laws of survival), how can there be positive actions? Furthermore, at what point in the process, a positive action becomes negative or counter productive? Could there be absolute positive or negative actions? Or simply, that they are relative to a particular place and time that the actions were manifested? In a lawless society, who judges all the actions, major ones as well as minor day-to-day mundane activities, for example, to wake up in a gloomy rainy morning or to continue sleeping? In the military, this is not a choice but written in the training manuals of conducts and procedures as step by step flowcharts of how to optimize the winning combinations.

Now, returning to axiom of choice as used in mathematical analyses. The case for differentiable manifolds demands deeper investigations. The 1st rule or theorem is that the manifold must be a function. Next, the function must be continuous. Generally, it could still be piecewise continuous with intervals of discontinuities. The 3rd rule would be dimensionality. It could be differentiable in the 1st, 2nd dimension but not in the 3rd dimension. If it is not differentiable then it could be a dysfunction, many-to-many correspondence. How could a dysfunction subjects to an axiom of choice? Is it then ruled by an axiom of null choice? Again, in human affairs, it is often heard someone said: “…leave me no choice…” and it is also plausible the same thought occurs in the mind of someone before committing a grievous crime. In mathematics, a choice is equivalent to a degree of freedom and each dimension must have two degrees of freedom. Infinite dimensions would have doubly infinite degrees of freedom. However, absolute zero would possess no freedom as a zero-sum game or a game that every player loses, no one wins. So, why play the game? Playing or not playing uses an axiom of choice or an axiom of no choice. Nevertheless, the laws of survival are based only on one axiom.

[mkirkpatrick]

We all have choice Antonio,but I feel only in a limited degree,much like free
will,well it is"free" but your every action has a equal and opposite one!We are
held in a balance by Karma,we are in effect bound to those we have an
affinity with,and those we are repulsed by,so there is choice ,but somewhat
limited,until we are able to burn off all our debts.


kind regards michael

[hanzoganz]

The axiom of choice is an interesting subject when ones studies metamathematics. When doing maths, one simple decides if one has to adhere to one school of thougth or the other: admiting the axiom or not.
Maths have more meaning and are simpler to work with if one simple decides that the axiom of choice is true. Then one can forget about it. There are a lot of axiom in the fundaments of mathematics that have to be admited to be valid for one to be able to continue working.

The axiom doesn't really chooses between two options but, what it allows you to do, is choose a member from the set. It helps to handle objects in abstract spaces, it is an abstract function.

About diff geom: there could be functions that are not differentiable, a function is just a mapping that takes elements from one set and assigns them to another set. Differentiability is just an extra property added to functions.

[AntonioLao]

Originally Posted by hanzoganz It helps to handle objects in abstract spaces, it is an abstract function.

If it is not a function then what? A linear function is a one-to-one correspondence. A quadratic function is two-to-one correspondence. But what is a 2-to-2 or many-to-many correspondence? I called these dysfunctions.

[hanzoganz]

Being a 1-1 doesn't make a function a function. A functions is just a mapping that makes correspondences between elements of different sets, or from the set to itself, doesn't matter. So, functions is just a rule of correspondence.
Now, being 1-1 or what Bourbaki defined as inyective, is that correspondence you are talking about.
There is just one property from all that you can add to your function: inyectivity, surjective, biyectivity, homeomorphism, diffeomporhism,...

the definition of functions as 1-1 is just a restricted one made for classical math in the real line. Think about a unitary circle, x^2+y^2=1. It is a function, but not 1-1.

[AntonioLao]

Originally Posted by hanzoganz Think about a unitary circle, x^2+y^2=1. It is a function, but not 1-1.

I don't think the unit circle is a function x + y = 1 strictly by the vertical line test.

[hanzoganz]

That is exactly what I mean. The circle is a function but not a 1-1. so, te vertical line test is a very restricted way used only for R^2.

[AntonioLao]

Originally Posted by hanzoganz That is exactly what I mean. The circle is a function but not a 1-1. so, te vertical line test is a very restricted way used only for R^2.

Thanks. Is there any other tests for functionality?

[hanzoganz]

In R^2 that is the best. In general, if you want to prove that f is 1-1 you would have to prove that given x,y in your Field (R, C commonly), f(x)=f(y) implies x=y or, what is the same, x not = to y implies f(x)not=f(y).
That is your definition of 1-1, that each element is mapped to a different image and no two elements can be mapped to the same point. That's why the vertical line works.
It just happens that in topological spaces, where you don't have a measure, this can't be done.
Have a read to any book of functional analysis and in the first chapters they will state this things about mappings.
cheers

[AntonioLao]

Originally Posted by hanzoganz if you want to prove that f is 1-1 you would have to prove

On the contrary, what I'm trying to work on is to show many-to-many mapping. Do you think topology studies many-many maps?