Do you have an example of a mathematical theory of quantum field theory where the function is not commplete in the sense that makes things uncountable?

I'm not sure I understand the question, but I'm pretty sure that the answer is "no". I'm pretty sure that uncountability doesn't actually matter for any practical or even metaphysical purpose, but that's just my personal guess. I have no actual data to support it, and it's mostly above my pay grade. Does that answer your question?

The difference between a Hilbert space and an inner product space is that a Hilbert space is complete, i.e. has an uncountable number of points. Its all fine and dandy to say that in the history of the universe we can only come up with a finite number of numbers or even finite number of mathematical concepts, but all the current theories of QFT are on top of Hilbert spaces.

OK, but that is a consequence of the continuity of space, right? So you don't even need to go to QM. Basic Newtonian mechanics is all you need to get an uncountable continuum. But space might turn out to be quantized.

In any case, the answer to your original question is obviously "no". Why do you think this matters?

Weirdly it's not related to regular space-time that we seem to inhabit - the Hilbert space in which QM happens does not correspond to the space of special relativity. Space and time of specific things are both the result of applying operators to the Hilbert space. The Hilbert space completeness means you have nice properties as far as orthonormal bases and that sort of thing (venturing outside my knowledge of QM here). The dimensionality of the Hilbert space is related to how many distinct quantum states can exist; it is unknown to physics if the universe has a finite dimensional Hilbert space or infinite. Since starting this conversation, I have tried to find any example of a quantum theory based on something other than the mathematics of the continuum without luck (but I'm not well positioned to find such a thing, as I would guess it would come from some constructivist mathematics thing but even regular ZFC QFT isn't formalized yet).

I'm at the hairy edges of my understanding here as well, but AFAIK infinite-dimensional Hilbert spaces arise when modeling position-momentum states because it is assumed that space is continuous.

Well, distributions like the Dirac delta qualify. But they can also be viewed as approximations of a continuum.

Outside of that, maybe if you look somewhere like string theory?

Still, in general, the continuum is enough. And while it takes work, I'm pretty sure that nothing relevant to physics about the continuum cannot be said within a strictly Constructivist version of the continuum. So no, there is no physical theory from which we can justify the existence of more reals than rationals. The justification must be philosophical. And the answers that we accept will strongly depend on what our notions about what existence should mean.

Complete does not mean uncountable. Complete means that any sequence that looks like it is converging, actually is converging.

Related to Hilbert spaces, consider the set of analytic functions with period 1. Under the L2 norm, this is an inner product space. It is also uncountable. However as Fourier showed, the Fourier series for the square wave does not converge within the set of analytic functions. In fact it is trying to converge to a function that not only fails to be analytic, it fails to be continuous!

I see your goal. But I question how you're going about it.

First, I'm all for convincing people that the wave function is something real. If you believe that strongly enough, then you're pushed towards accepting the Everett Interpretation. Which makes far more sense to me than any other interpretation of QM.

But I would suggest that you get them there by the shortest reasoning that you can. With minimal detours. The more material you have, the more likely that someone with different preconceptions will find something to disagree about. Worse yet, people will lose track and abandon it.

I would also suggest that you simplify your writing. According to https://readabilityformulas.com/readability-scoring-system.p... your writing is college entry level. Readability is weird, we don't notice that it matters until we find ourselves unable to understand text. And then it is a wall. Meaningless phrasing choices can improve readability and broaden your audience.

Now for a more substantive criticism. But one of choice of topic, where you likely reasonably disagree with me. You are writing about the scientific method. But the style of your writing undermines the essential spirit behind the scientific method. What do I mean by that essential spirit? I mean the thing that Feynman was trying to explain in http://www.feynman.com/science/what-is-science/. It is the attitude that leads to statements like, "Science is the belief in the ignorance of experts."

Now you're telling them what to believe. You aren't telling them how they could think of it. You aren't encouraging them to disagree with you and question you. You're telling them what is true. I think that conveying the skeptical attitude is more important than convincing them of any particular fact.

Moving on, I see you've added a number of things to your article. They are significant improvements. But I would suggest an improvement to one in particular. Your arithmetic example.

Counting to negative numbers for debt makes sense. But you don't show how you get something that unexpectedly is useful. So I would make the chain that you lead people down to be counting, to positive integer arithmetic, to negative numbers for debt, to fractions, to the idea that we can put this on a line, to the question of whether we have everything on that line, to the reals. Just outline it. And then point out how we started with counting, and wound up most of the way to geometry...doesn't look like counting any more!

Finally, I think that high school students are capable of more philosophy than you do. In particular there are lots of youtube videos and articles that try to explain why uncountable means more. If they can see those explanations, why can't they encounter the idea that all such claims require additional assumptions? Thinking about how to connect reasoning and computation is something of interest. And leads to topics that are fundamental in such places as logic and CS.

Anyways, this is wrong and I don't know if you read it. But if you do, hopefully you found at least one thing that's worth thinking about.

Thanks for the feedback.

> But I would suggest that you get them there by the shortest reasoning that you can. With minimal detours.

Yeah, well, that's the challenge, isn't it. The target audience is so far away from QM that it's hard to fathom. These are people who believe (or at least seriously consider the possibility) that Jesus rose from the dead, the earth is 6000 years old, demons are literally real...

> Now you're telling them what to believe.

Am I? That is actually something I was trying very hard to avoid. Can you point out where I'm doing that?

The fact that the audience is so far is part of why I think that the attitudes of skepticism are more important than any particular set of facts.

But it is also important that you make it easy to follow your explanation without having to challenge emotionally held beliefs. For example, consider those religious beliefs. A big part of why we hold them is that trying to give up those beliefs involves accepting a lot of pain. Both from external causes, like the condemnation of your religious community, and internal ones, like the gut level belief that even thinking about rejecting Jesus puts you at risk of going to Hell. It is very hard to engage in quiet contemplation while feeling that you're risking eternal damnation.

As for where you tell people what to believe:

And indeed it is a counter-example to the idea that mathematical truths are grounded in actual objective reality, but that is not news -- we already established that with the example of negative numbers and imaginary numbers.

You claim to have shown that the ideas were not grounded in actual objective reality. But that isn't true. You only showed that they aren't grounded in the first way that you tried to ground it. Both of those examples are things that can easily be grounded in objective reality in other ways.

But the Banach-Tarski construction is different. It depends on the axiom of choice. Which in turn is a statement about a process that cannot be actually done in objective reality. And without being able to do that process, we're unable to do Banach-Tarski in objective reality.

I think that's an important difference. And it is one that is missed when you tell them that this is like that other thing which you just saw.

Thanks for the feedback. FWIW, I am very cognizant of the emotional toll of giving up religious beliefs, which is the main reason I've not been tackling those head-on.