Need (or not) for invoking axiom of choice in a certain case

[RockyMarciano]

It is still very unclear. What choice is there for the function? Given any curve from the family you compute its length. So the function is the map from those curves to real numbers assigning the each curve its length, no choice involved here. May be you can make it clearer on an example. Take the Euclidean plane and two point, consider all curves (let's say smooth) with those two points as end points. Each of them has a length and you have the function that maps each curve to its length i.e. the domain is the set of these curves the range is positive real numbers. What you have is a set of positive real numbers, does it have an infimum? Where in all this is the choice you are asking about?

I'm considering the set X of all curves between the 2 given points, and these curves as sets S of points, and the choice function f(S) as the one that assigns a positive real number to each S in X. I'm using this definition of choice function from Wikipedia:"A choice function (selector, selection) is a mathematical function f that is defined on some collection X of nonempty sets and assigns to each set S in that collection some element f(S) of S".

 

[martinbn]

I'm considering the set X of all curves between the 2 given points, and these curves as sets S of points, and the choice function f(S) as the one that assigns a positive real number to each S in X. I'm using this definition of choice function from Wikipedia:"A choice function (selector, selection) is a mathematical function f that is defined on some collection X of nonempty sets and assigns to each set S in that collection some element f(S) of S".

But in your case $f(S)$ is not an element of $S$, it is a real number, while $S$ consists of points of the manifold.

 

[WWGD]

Perhaps an example might help, would you say the following requires some form of choice or is enpugh with completeness? : "Given a sequence ($A_n$) of non-empty sets, choose $x_n ∈ A_n$ for each $n∈N$".
.

Yes, this is the actual statement, at least as I know it.

 

[RockyMarciano]

But in your case $f(S)$ is not an element of $S$, it is a real number, while $S$ consists of points of the manifold.

It is quite straightforward to identify the points with the real intervals(sets of reals) that the curves map to the points in the manifold, so S consists of real numbers. Or is this not licit?

 

[RockyMarciano]

@martinbn, @fresh_42, @WWGD could you please address #34? I'm not sure if curves can be identified with sets of real numbers in this context.

 

[fresh_42]

@martinbn, @fresh_42, @WWGD could you please address #34? I'm not sure if curves can be identified with sets of real numbers in this context.

You can label them with their lengths. I wouldn't call it an identification as it is a function from $\mathbb{R}^n \longrightarrow \mathbb{R}$ and information is lost. But in the next step, you build a limit to find the shortest of them, which is different from choosing one of them. If you'd chose one, then you will get any number or curve. Only the limit gets you the smallest or shortest. Its existence is guaranteed by completeness of $\mathbb{R}$ not by a process of choice. If we'd switch to $\mathbb{Q}$ we would still have a choice, but no limit.

 

[WWGD]

I'm considering the set X of all curves between the 2 given points, and these curves as sets S of points, and the choice function f(S) as the one that assigns a positive real number to each S in X. I'm using this definition of choice function from Wikipedia:"A choice function (selector, selection) is a mathematical function f that is defined on some collection X of nonempty sets and assigns to each set S in that collection some element f(S) of S".

You see, the thing is that, as Fresh mentioned, the (a?) problem with choice is that the selection of elements is " uncontrolled". In your case, there is, in a sense, a unique and well-defined method for choosing the length of the curve, if I understood your layout correctly --please correct me if I did not) so this does not, in this regard create a problem; the functional value is given by the inf. of all lengths, and we are not making AC choices regarding the curves, so I really don't see AC being an issue here.

 

[RockyMarciano]

You can label them with their lengths. I wouldn't call it an identification as it is a function from $\mathbb{R}^n \longrightarrow \mathbb{R}$ and information is lost. But in the next step, you build a limit to find the shortest of them, which is different from choosing one of them. If you'd chose one, then you will get any number or curve. Only the limit gets you the smallest or shortest. Its existence is guaranteed by completeness of $\mathbb{R}$ not by a process of choice. If we'd switch to $\mathbb{Q}$ we would still have a choice, but no limit.

You see, the thing is that, as Fresh mentioned, the (a?) problem with choice is that the selection of elements is " uncontrolled". In your case, there is, in a sense, a unique and well-defined method for choosing the length of the curve, if I understood your layout correctly --please correct me if I did not) so this does not, in this regard create a problem; the functional value is given by the inf. of all lengths, and we are not making AC choices regarding the curves, so I really don't see AC being an issue here.

Thanks. I'm myself convinced that there is no choice involved in the existence of the inf of the lengths. Sorry about creating this confusion in several of my posts. I am concerned just with the picking of the lengths from the countably infinite curves. This seems to hinge according to the above linked definition of choice function on whether one can consider each curve as a set S of real numbers, out of which a specific positive real number is chosen as a length by a choice function, and this is my actual question, can each of the curves between two points in the manifold be considered as a set of real numbers in this contex?.

 

[fresh_42]

This seems to hinge according to the above linked definition of choice function on whether one can consider each curve as a set S of real numbers, out of which a specific positive real number is chosen as a length by a choice function, ...

If you regard it as a choice, then even an ordinary function $f(x)=x$ is a choice and the detour on manifolds isn't needed at all. In your sense every assignment of values is a choice. You can think of it like this, but it again has nothing to do with the axiom of choice and it will cause exactly the same trouble everywhere as here: the rest of us call it function or limit, not choice.

... and this is my actual question, can each of the curves between two points in the manifold be considered as a set of real numbers in this contex?.

No. This is as if you considered every article in a supermarket as the price tag it's labelled with. You can do this if you're calculating the value of its inventory, but not if you're looking for (the cheapest) chocolate. Even if we only labelled chocolate, then the prizes would still be just one aspect of it. It is a function, not an identity. So if curves can be "considered as a set of real numbers", depends on what you want to do with it. In some contexts it will be sufficient and in many others it's not.

 

[WWGD]

Thanks. I'm myself convinced that there is no choice involved in the existence of the inf of the lengths. Sorry about creating this confusion in several of my posts. I am concerned just with the picking of the lengths from the countably infinite curves. This seems to hinge according to the above linked definition of choice function on whether one can consider each curve as a set S of real numbers, out of which a specific positive real number is chosen as a length by a choice function, and this is my actual question, can each of the curves between two points in the manifold be considered as a set of real numbers in this contex?.

But you see, we really don't need a choice function, since we have exactly one value and this value is selected, not by a random choice, but by a well-defined method. And, yes, a curve can be identified with a collection of Real numbers, e.g., through a parametrization; a point p in the manifold can be identified with t with C(t)=p , but then each curve is a continuum, but you are not really selecting any of the values t here; you are choosing from a collection of lengths, specifically the inf of all lengths. EDIT The process of finding an inf does not necessarily require AC that I can tell; e.g., inf(a,b)=b. And the collection of all curves is uncountable, e.g., for every pair of points (x,y) ; $x, y \in \mathbb R - \mathbb Q$ you can draw a curve passing through them.So it comes down to deciding if in this case, finding the inf requires AC, but I don't see how; you are selecting _one_ number from a collection of values ( the lengths of all possible curves between two points), and I don't see any choice being made.

 

[RockyMarciano]

I think I understand your point now. I only still have a doubt concerning how this applies to the Minkowskian case I mentioned at the beginning of the thread. I can see now how in the Riemannian case I'm not selecting the lengths as there is well-defined method to select lengths from curves and this is a given so there is no choice function here. But the integral giving the lengths in the Minkowskian case is not exactly like in the riemannian case, it includes a sign choice that is dependent on the previous choices to be consistently a timelike or spacelike curve, would this be an example of dependent choice?

 

[fresh_42]

This is a choice about the geometry or topology you want to perform your calculations in. The same way as if you chose real or complex analysis. It will change the set-up, possibilities and outcome, but it is a choice among different systems, not a choice within a given system.

 

[RockyMarciano]

This is a choice about the geometry or topology you want to perform your calculations in. The same way as if you chose real or complex analysis. It will change the set-up, possibilities and outcome, but it is a choice among different systems, not a choice within a given system.

Not sure what you mean by system here. Using timelike curves and congruences is a decision dictated within the spacetime notion "system", very specifically and well defined mathematically.

 

[martinbn]

I think I understand your point now. I only still have a doubt concerning how this applies to the Minkowskian case I mentioned at the beginning of the thread. I can see now how in the Riemannian case I'm not selecting the lengths as there is well-defined method to select lengths from curves and this is a given so there is no choice function here. But the integral giving the lengths in the Minkowskian case is not exactly like in the riemannian case, it includes a sign choice that is dependent on the previous choices to be consistently a timelike or spacelike curve, would this be an example of dependent choice?

No, it would not be such an example. If you want to study the geometry forget the axiom of choice. It only confuses you.

 

[fresh_42]

Not sure what you mean by system here. Using timelike curves and congruences is a decision dictated within the spacetime notion "system", very specifically and well defined mathematically.

Yes, but this is the same framework we discussed all over the thread. In your last post you've asked about the choice between Minkowski and Euclid, which is a choice of systems. Once you made this choice, we are back at post #1.

 

[RockyMarciano]

In your last post you've asked about the choice between Minkowski and Euclid, which is a choice of systems. Once you made this choice, we are back at post #1.

I didn't ask about any choice between Minkowski and Euclid, I asked about the integral of the metric in Minkowski incorporating a ± sign choice that must be consistently maintained at every point of the integrated path.

 

[martinbn]

What would be the situation where the sign is not only plus or only minus along the curve?

 

[RockyMarciano]

What would be the situation where the sign is not only plus or only minus along the curve?

You mean mathematically? I'd say only with dependent choice(where previous choices of plus or minus condition later ones) one can mathematically guarantee that such situation doesn't arise.

 

[martinbn]

No, I mean what would be a specific situation, and why would you be interested in it?

 

[RockyMarciano]

No, I mean what would be a specific situation, and why would you be interested in it?

As you can see below such situation(last sentence of quoted paragraph) is not considered, the reason is physical.
I'm not interested in it, rather I'm interested on how it is mathematically granted this situation doesn't arise, and I thought it was done by using dependent choice, but you claim it isn't, right?
https://en.wikipedia.org/wiki/Arc_length#Generalization_to_.28pseudo-.29Riemannian_manifolds

"The sign in the square root is chosen once for a given curve, to ensure that the square root is a real number. The positive sign is chosen for spacelike curves; in a pseudo-Riemannian manifold, the negative sign may be chosen for timelike curves . Thus the length of a curve in a non-negative real number. Usually no curves are considered which are partly spacelike and partly timelike."

 

[martinbn]

Even if it is partly timelike and partly spacelike, it will consists of finitely many pieces on which it is either one or the other.

 

[RockyMarciano]

Even if it is partly timelike and partly spacelike, it will consists of finitely many pieces on which it is either one or the other.

and your point is?

 

[martinbn]

There is no axiom of choice involved.

 

[martinbn]

What is your point in this thread?

 

[RockyMarciano]

There is no axiom of choice involved.

I can see this now, thanks.

What is your point in this thread?

I started the thread trying to convince myself that as I have frequently read the AC is not used in (pseudo)riemannian geometry. I chose an example where my intuition was quite confused and presented it here. Now that this example is clarified and after consulting some advanced books(Hawking&Ellis, Wald) on the field it turns out that there are a few examples(not the one I thought about first and discussed here) where actually the full axiom of choice is needed, not just some amount of it. So I'm a bit crossed with the insistence of many physicists that the AC doesn't come up in physics theories.

 

[martinbn]

And what are those examples?

 

[RockyMarciano]

And what are those examples?

Being just a math aficionado I'm sure I would do a terrible job explaining them(maybe I try in some new thread though), if you take a hold of say Wald just look up "Zorn's lemma" in the index and you are sent to an example dealing with the well-posedness of the initial value problem for the EFE.

 

[martinbn]

That is not related to your questions. The AC is used in the prove of the existence of the maximal Cauchy development, but logicians tell us that it is not needed. The way the prove goes one can prove that there exists a prove without the AC. Noone has given one thought, but we can be sure it exists.

 

[RockyMarciano]

The AC is used in the prove of the existence of the maximal Cauchy development, but logicians tell us that it is not needed. The way the prove goes one can prove that there exists a prove without the AC. Noone has given one thought, but we can be sure it exists.

I see, I would need to know who those logicians are and what they tell us, i.e where do they claim we can be sure such constructive proof exists?, do you have any reference at all or can point me to where I can find it?

 

[WWGD]

I see, I would need to know who those logicians are and what they tell us, i.e where do they claim we can be sure such constructive proof exists?, do you have any reference at all or can point me to where I can find it?

Eye of the Tiger, Marciano. Eye of the Tiger!