What is the concept of the NFU

Set theories used in math-physical theories

Tl; dr: Physicists would care about which set theories to choose when there was an observable difference between the decisions. This is less likely than you think because "observable results" are always finite numbers, but somebody should (indeed) take care of them.

Almost all math physicists would agree that this is the only reason to care. For reasons of accuracy, you implicitly choose the most common set theory ZF (C) for reasons of time.


First, let's answer the following question:

Do we have to worry about the basics of set theory in physics?

Detour to the scientific method

Science is a social and personal endeavor in many ways (see my detour below). This is vital as it affects the way people think. In this section, however, I would like to quickly outline a few more "objective" ideas, as they are more or less generally accepted ideas: the scientific method.

The "objective" point of science that corresponds to this method is predicting outcomes. This could be past results, but it should work for future results as well. That is why science is "useful". Due to the complexity of life, this must be broken down into small units - experiments. The nature of the experiments is far from simple and certainly more than "observation" since our senses are not very good (think of optical illusions). To remedy this, we invent measures and measuring machines so that in the end the results of the experiment usually consist of a series of numbers on a computer screen. That is, all expenses are always predictable .

(Mathematical) physicists

Most physicists see math as a tool. They are amazed at the effectiveness in physics, but all they want to do is approach physical theories. If you don't work at Theories of Everything, you know you only work with effective theories work . An effective theory is by design only an approximation. Many mathematical physicists work on effective theories themselves and agree.

At this point, I would argue, whether we need to consider the bases of established theories depends on worldview. Here are the most common alternatives:

1st idea: Since everything we ever measure is calculable numbers and the number of excitations of quantum fields is also finite and calculable (due to the fact that the universe is considered finite and everything we think we know), a "true" theory of everything, only calculable numbers are actually used.

If we agree on natural numbers (and even on the set theories you usually cite), it means that the question of basics is irrelevant. But obviously we are using functional analysis and the like. Why this? Well, because it is much more convenient to approximate a finite but huge system by an infinite system. Things get a lot easier and we avoid a lot of technical effort. But at this point it doesn't matter which set theory we choose. The Banach-Tarski Paradox in ZFC? It's just an artifact of our approach and doesn't affect the real world. So why should we only use ZF or another system?

2nd idea: A true description of the world needs the continuum, but all we can ever achieve is one approximation of that true description, or even many different approximations that for some reason cannot be welded together. The basis of set theory may matter here - but to a very limited extent: It matters if and only if two theories are true observably different results . Again, since all experimental data we ever get consists of calculable numbers, that observable result must be in calculable numbers. I think most math physicists would consider this unlikely and therefore focus on other questions.

3rd idea: Some people believe that the universe is a mathematical structure that the continuum needs, and that structure can be found. At this point you need to take care of set theory. But again, you have to find a way to discover the "correct" basis of set theory through experimentation. It should be possible if you believe the structure can be found.

Result

In any case, a mathematical physicist will only concern himself with the basis of set theory if there are observable differences between two possibilities. Since all the data we can ever have is made up of calculable numbers, it seems like a long way to go, and therefore most people don't care. Some do - you might like the article "Set Theory and Physics" (Paywalled, but there are versions without a paywall that I don't link because they go straight to the PDF) and other articles by Svozil. Note that he tries to find observable quantities that differ depending on the result.


Does this answer your question? I tend to believe that no, since everything I've said so far applies not just to mathematical physicists, but to every physicist. The more refined question is likely:

Why don't mathematical physicists care that the basis of set theory is strict?

If you just want to be strict to have a consistent system, you can choose a consistent formulation of set theory. Consistency is nice, of course, because we like to believe that our world is consistent, so inconsistent theories would be wrong. To achieve this goal, you can choose whichever theory you want - and the most convenient way to do it is to do what everyone else is doing, which is ZFC.

However, this is not the only reason math physicists believe rigor is necessary. It might also help to understand the theory better. For examples, see the answers to this question: The Role of Rigor

For all of this, again, it is sufficient to use the most convenient formulation of set theory, as long as you choose one.


The last objection I can think of is:

But mathematical physicists seldom state which set theory they work with. Shouldn't they do that?

My answer to that would be that 99% of all mathematicians don't. Why should math physicists do this? Why don't mathematicians do it? Let me make a brief final detour:

Rigor in math:

Pure math is seldom rigorous. For example, when you read an advanced paper in functional analysis, you have many gaps to fill and the basics are rarely given. The closer you get to the basics of math, the more rigorous the work becomes (inherently), but even then, much of the evidence is routinely left out.

Math as a personal endeavor

The problem is that math is one Human is and therefore also a personal Endeavor, and for most people, it's a matter of Knowledge acquisition . However, unless otherwise forced, every person has a level at which they view things as "obvious". If you want to "know" something you have to make assumptions at some point or you will go down the rabbit hole and end up nowhere. Decartes landed famously on "I think that's why I am," but it's not even clear that this deduction is valid.

So you have to start somewhere and you can start with set theory. Most people, however, will not start there naturally, but will find things like "natural numbers" to be obvious (one could call this "naive set theory"). They won't really care whether or not this is actually consistent until they run into a contradiction. Other mathematicians like to pick just one phrase and work at higher levels, mostly the one they were taught at. As long as they gain "intuition" about what is going on, they see it as "knowledge" about the field of research. This is also why they leave out certain arguments in evidence - just thinking that they know how it would be done is enough for them (and they are wrong quite often!). This, as I understand it, is one of the main points of Bill Thurston's famous essay "On Evidence and Advances in Mathematics". There is also the question of time constraints: not everyone can start at the basics, otherwise nobody can build the higher theory.

Incidentally, the personal nature of mathematics is also the reason why many mathematicians feel uncomfortable with proofs produced by computers. Using computers to review evidence is fine, but to find evidence? What does the mathematician learn?

Mathematics as a social endeavor

Second, math is like anything else social endeavor . Both mathematicians and physicists often like to deny this, but research goals, the way in which research is carried out and the like are strongly influenced by social norms, I will probably do research in a similar way to my teachers. Of course, style differences will develop, but they will usually be incremental. The biggest changes in the way research is done are usually made by a very small minority of extremely headstrong thinkers like Hardy.

Otherwise, most of the changes in the degree of accuracy are due to the fact that something affects the outcome of the research: in the 19th century, convergence was analyzed more closely because, for example, it is easy to construct a class of functions that point to another Function f converges, but the arc length of functions fn does not converge. Since we intuitively think that this should not be the case, something is "off". Similarly, paradoxes have given rise to axiomatic set theories in the 20th century.

I am sometimes dismayed by this unfortunate lack of foundation, which is why I support algorithmic evidences and the like.