Finite number divided by infinity

[A.T.]

What if all natural numbers are equally likely?

By this I take you to mean there is a valid (satisfies the Kolmogorov axioms) ... We can show that this is impossible.

What does it tell us, that we cannot assign a probability in this case, if we assume these axioms? Seems like a limitation of the theory build on these axioms.

 

[Matterwave]

What does it tell us, that we cannot assign a probability in this case, if we assume these axioms? Seems like a limitation of the theory build on these axioms.

First I would like to ask what is the purpose for which you would want a uniform distribution over all of $\mathbb{N}$? Or is it "just because"?

Kolmogorov formalized probability via his axioms in the 1930s and a vast literature of probability theory and things that depend on probability theory has been built on top. It also provides the axiomatic foundation for things prior to it like the law of large numbers and the central limit theorem.

You may formalize probability your own way, but before we embark on such an arduous journey, we might want to know where the intended destination is.

This path might have other travellers on it and I just don't know them. :)

 

[A.T.]

First I would like to ask what is the purpose for which you would want a uniform distribution over all of $\mathbb{N}$? Or is it "just because"?

I thought it's only real math, when it doesn't have an application.

Kolmogorov formalized probability via his axioms in the 1930s

100 years is not much, compared to how long it took mathematicians to do something about not having an answer for $1-2$ or $\sqrt{-1}$.

 

[Dale]

Not in QM. Only in classical statistical mechanics.

https://en.wikipedia.org/wiki/Born_rule

In QM, the sample space is determined by the measurement apparatus.

I am talking about the sample space of the probability distribution from the Born rule. There is a separate sample space for the measurement apparatus which is conditional on the first.

You claim you can get an arbitrary real number from identifying the position of a particle

What I claim is that physical positions are isomorphic with the reals. I.e. they are a continuum.

The wave-function is defined everywhere

And “everywhere” in this meaning is a continuum.

It's a question of the continuum being absolutely necessary, with no possibility of a theory that avoids it.

I am not making that claim.

But I do take the continuum model seriously because it is experimentally validated, and to my knowledge discrete models are not as well-supported experimentally.

It's not a question of the continuum being a working model.

So you do think that the continuum is a working model? Your arguments seem to indicate a stronger claim to me. Your arguments seem to indicate that you believe the continuum is not a working model.

Perhaps you are making a weaker claim than I thought, as I am making a weaker claim than you think.

 

[Dale]

What does it tell us, that we cannot assign a probability in this case, if we assume these axioms? Seems like a limitation of the theory build on these axioms.

It is a limitation, but sometimes such limitations are actually a good thing.

I have occasionally wondered if it would be worthwhile to define a probability measure on the hyperreals rather than on the reals. You could get rid of that specific limitation by doing so, but I am not sure that it would be a good thing to do. I am not aware of any literature on that topic, so I don't know what problems would be introduced by "fixing" that limitation.

 

[Matterwave]

I thought it's only real math, when it doesn't have an application.

Well, you could also give a mathematical application!

"I just want this to be" is also ok, but you would be the best person to build the theory in that case~

100 years is not much, compared to how long it took mathematicians to do something about not having an answer for $1-2$ or $\sqrt{-1}$.

There are other formalisms / axiomatizations of probability. They might give rigor to "a uniform distribution over the natural numbers". I'm only familiar with Kolmogorov's (it's the "main" one, like ZFC).

 

[pbuk]

Seems like a limitation of the theory build on these axioms.

Only in the sense that the fact that we cannot assign a value to $\frac10$ is a "limitation" of algebraic field theory. But the non-existence of a multiplicative inverse of 0 is necessary to provide consistency to $\mathbb R$ and $\mathbb C$.

What does it tell us, that we cannot assign a probability in this case, if we assume these axioms?

It tells us that the word "probability", as we have defined it, does not make sense in this context. In this case, it does not make sense because the phrase "uniform distribution over an infinite number of discrete events" does not make sense in the context of the Kolmogorov axioms (specifically it is not consistent with σ-additivity).

Although I am not familiar with the detail, work has been done on alternatives to σ-additivity, specifically "finitely additive probability" - see e.g. https://www.ma.imperial.ac.uk/~bin06/Papers/favcarev.pdf - which has I believe led to useful results.

 

[A.T.]

I don't know what problems would be introduced by "fixing" that limitation.

Yes, usually new problems arise. Just like introducing negative numbers, to cover the results of all subtractions, introduced the problem with roots of those negative numbers.

 

[PeroK]

What I claim is that physical positions are isomorphic with the reals. I.e. they are a continuum.

Here you make the claim that physical positions are isomorphic to the reals.

I am not making that claim.

And now you deny that claim.

You can't have it both ways. You can't say that a particle can be at any one of an uncountable number of positions - and thereby have an event of probability zero that actually happens. And, then concede that the set of positions could be finite or countable. If positions are uncountable, then they cannot then later to be found to be finite.

If you claim there are an infinite number of positions in spacetime, then there cannot possibly be only a finite number.

If you can really do what you claim, then that rules out finite or countable spacetime models.

Perhaps you are making a weaker claim than I thought, as I am making a weaker claim than you think.

I claim that we do not know whether spacetime is finite, countable or uncountable. I don't know why that seems to be so difficult to grasp. It might be continuous; it might be countable; it might be finite.

Note that in this thread, I seem to be the only one not claiming to be able to do things that have zero probability.

 

[pbuk]

Note that in this thread, I seem to be the only one not claiming to be able to do things that have zero probability.

I don't think that is true. Can you point to such a claim?

 

[PeroK]

I don't think that is true. Can you point to such a claim?

You're right. I let my frustration get the better of me.