Geodesic Curves Covering Surfaces

[Hurkyl]

This is more a statement about the deficiency of ZFC than it is about the continuum hypothesis.

While that is literally true, the connotation is inappropriate.

CH really ought to be false.

This is an interesting opinion. Most often, strong feelings on these sorts of issues tend to favor either simplicity or reject nonconstructability. However, your opinion is contrary to both tendancies. Upon what do you base it?

 

[maze]

While that is literally true, the connotation is inappropriate.


This is an interesting opinion. Most often, strong feelings on these sorts of issues tend to favor either simplicity or reject nonconstructability. However, your opinion is contrary to both tendancies. Upon what do you base it?

I base the intuition on the hypothesized Freiling's axiom of symmetry from probability, which would imply CH is false.

 

[adriank]

I'm curious, how can you get this from Hahn-Mazurkiwicz?

Well, [0, 1]^ω is a Hausdorff space that is compact, connected, locally connected, and metrizable, so there is a (continuous) surjection from [0, 1] to [0, 1]^ω; and |R^ω| = |[0, 1]^ω|.

 

[SW VandeCarr]

I base the intuition on the hypothesized Freiling's axiom of symmetry from probability, which would imply CH is false.

I'm surprised that no one has responded to this. My response is intended to provoke some discussion (even at my expense.)

Freiling's axiom of symmetry (AX) proposes a set of functions A which map from the real number interval [0,1] to a finite set of countable subsets such that for every f in A and the arguments x and y, y is not in f(x) and x is not in f(y). It seems that these functions are behaving as inverse random variables.

A random variable is a function which maps from a finite set of countable elements E (the "event space") to the interval [0,1] where this interval is interpreted as the set of probabilities P (or "probability space"). Its clear that every element in E (or event) can have only one probability, but more than one event can have the same probability. What does this say about the validity of AX?

 

[Hurkyl]

The version on Wikipedia doesn't put a limitation on how many countable subsets you use.

Anyways, I don't think validity is the question you wanted to ask; it's not a theorem of ZFC, it's invalid in any model of ZFC+CH, and valid in any model of ZFC+AX.

I was content with maze's response -- I asked, and he answered, I didn't feel it important to press on. But my main reaction is simply that the criterion seems esoteric; it doesn't appear to obviously boil down to anything that I can imagine people having strong opinions about.

 

[SW VandeCarr]

The version on Wikipedia doesn't put a limitation on how many countable subsets you use.

Anyways, I don't think validity is the question you wanted to ask; it's not a theorem of ZFC, it's invalid in any model of ZFC+CH, and valid in any model of ZFC+AX.

I was content with maze's response -- I asked, and he answered, I didn't feel it important to press on. But my main reaction is simply that the criterion seems esoteric; it doesn't appear to obviously boil down to anything that I can imagine people having strong opinions about.

If AX were accepted, it would be a refutation of CH. People might have strong opinions about that. Should AX be rejected because it's 'esoteric'?

 

[Hurkyl]

I never said it should. Being esoteric just makes it hard to have opinions about.

The only times I've ever really ran into the CH are:
1. It let's you use $\aleph_1$ to refer to |R|
2. It simplifies the classification of real closed fields

Point (1) is highly superficial, and I don't work with real closed fields enough to have any string opinions about point (2). While AX is also related to CH, I have much less connection to it than these other two points.