Does the definition of cardinality assume distinguishability?

[Stephen Tashi]

TL;DR Summary

If set S has cardinality 2, does the definition of cardinality imply the existence of an equivalence relation that distinguishes the two elements?

Physics speaks of a set S of N "indistinguishable particles", giving the set S a cardinality but forbidding any equivalence relation that can distinguish between two particles. Is this terminology inconsistent with the mathematical definition of cardinality?

Suppose $S$ is a set with cardinality 2. I want to show that there exists an equivalence $E$ defined on the elements of $S$ using only the definition of cardinality and without assuming such a relation exists. A constructive proof might be as follows:

Since $S$ has cardinality 2 there exists (by definition) a 1-to-1 function mapping the elements of S to those of the set $\{1,2\}$. Let $F$ be such a mapping. Define an equivalence relation $E$ as follows. Let $x$ be the element in $S$ such that $F(x) = 1$. Let $y$ be the element in $S$ such that $F(y) = 2$. By the definition of a function $x \ne y$ (because the same element cannot be mapped to two different numbers.) Define $E$ to be the set of ordered pairs $\{(x,x),(y,y)\}$ Since $(x,y)$ is not an element of $E$ then $x \ne y$ in terms of the equivalence relation $E$.

That seems straightforward, but the line " By the definition of a function $x \ne y$" assumes we have some equivalence relation that can be used to determine that $x \ne y$. Determining that fact is necessary to assure that $F$ is a function. We assume the aforesaid relation exists before we prove $E$ exists. So the mention of a 1-to-1 function in the definition of cardinality assumes the existence of an equivalence relation that we use to determine that two distinguishable elements are not mapped to the same element.

So the physics version of a set of N indistinguishable particles differs from the mathematical concept of a set with cardinality N. (At least that's my opinon.)

 

[maline]

In "pure" set theory, the elements of sets are themselves sets. In that case the issue of distinguishability is settled by axiom: if every member of A is a member of B and vice versa, then A=B. This is the Axiom of Extensionality. So there cannot be "indistinguishable" sets: sets can be distinguished by their members.

Some set theories allow "urelements", i.e. elements that are not sets. To determine the cardinality of a set containing urelements, we indeed need to add some rule that tells us when two of the urelements are equal and should only count once. But if we are told that $x\ne y$, then we can just take that as a fact; there doesn't need to be any "distinguishability" in terms of their properties. Of course, as you say, we are free to define functions that send $x$ and $y$ to different places, but that is an outcome from the fact that they are different, not a "reason" for the difference.

In physics, "particles" are not individual "entities" at all. The number of each type of particle is a quantum operator built from the fields, with the natural numbers as its eigenvalues. The eigenstate with particle number three is indeed not a "set of particles" with cardinality 3. It is just a state in the QFT Hilbert space, from which "particle destruction" events can occur three times.

If "things" called individual "particles" existed, then "indistinguishability" (in terms of measurable properties) would not be sufficient reason to not count permutations of the particles as different states for thermodynamics. To the extent that the name "particle $x$" has a specific referent, the statement "particle $x$ is on the right" can be true or false, even if no experiment can tell us that truth. What we need is that labelling particles should not even make sense; there is no "thing" to apply the label to.

 

[Stephen Tashi]

Some set theories allow "urelements", i.e. elements that are not sets. To determine the cardinality of a set containing urelements, we indeed need to add some rule that tells us when two of the urelements are equal and should only count once. But if we are told that $x\ne y$, then we can just take that as a fact; there doesn't need to be any "distinguishability" in terms of their properties.

I'd say that being told that $x \ne y$ is not being told anything unless an equivalence relation is assumed for defining the relation "=". Once that equivalence relation has been assumed we can define a property $P_y(x)$ by defining $P_y(x)$ to be true if and only if x has the property that x=y.

However, I understand the main point of what you said - namely that having N indistinguishable particles in a physical system is a concept distinct from having a set with cardinality N.

 

[maline]

You don't need an equivalence relation, you just need an equality relation. And the equality relation is a logical primitive - $x\ne y$ is treated as a well-defined proposition automatically. You do need axioms to tell you which things are or are not equal, but the equality relation itself is prior to such axioms. Logic just has a basic concept of "the same thing" vs. "different things".

 

[Stephen Tashi]

You don't need an equivalence relation, you just need an equality relation. And the equality relation is a logical primitive - $x\ne y$ is treated as a well-defined proposition automatically.

"Automatically?". I have a general idea of what you mean, but how semantics relates to syntax isn't clear to me. I can define a formal language in which "$x = y$" is a well formed formula. To the rules of that language, I can add the requirement that there is a set of constants and for each pair of constants $x,y$, the wff $x = y$ is assigned a (constant) truth value, True or False.

That approach doesn't mention properties or even define the concept of properties. Is that what you mean?

 

[maline]

Sure, that's one way it could be done. I was thinking more of the way you can always say "for all $x,y$ such that $x \ne y$, there holds...".

 

[PeroK]

Physics speaks of a set S of N "indistinguishable particles", giving the set S a cardinality but forbidding any equivalence relation that can distinguish between two particles. Is this terminology inconsistent with the mathematical definition of cardinality?

The physics itself is not concerned with the particles as entities, but the states that the system may occupy. If we have two distinguishable spin 1/2 particles, the the composite system has four basis states. But, if the particles are indistinguishable, then we have only the one anti-symmetric state. All the physics is predicated on the state. Writing down a two-element set $\{p_1, p_2 \}$ is not part of the mathematical formalism. The formalism begins by defining the relevant Hilbert space.

More generally, particle number is a (real) eigenvalue of the number operator.

 

[Stephen Tashi]

The physics itself is not concerned with the particles as entities, but the states that the system may occupy. If we have two distinguishable spin 1/2 particles, the the composite system has four basis states. But, if the particles are indistinguishable, then we have only the one anti-symmetric state.

I have the impression that classical statistical mechanics effectively uses a similar idea. The usual presentation of classical statistical mechanics is to declare that since we are in the case where N particles are indistinguishible, therefore we analyze them by assigning each distinct combination of N identical balls in K cells the same probability. I don't think the "since...therefore" reasoning has any substance. It seems to me that what's actually being done is to define "indistinguishable" particles to be those particles where that method of analysis gives the correct answer. So the "since...therefore" reasoning is just a proof-by-definition.

 

[PeroK]

I have the impression that classical statistical mechanics effectively uses a similar idea. The usual presentation of classical statistical mechanics is to declare that since we are in the case where N particles are indistinguishible, therefore we analyze them by assigning each distinct combination of N identical balls in K cells the same probability. I don't think the "since...therefore" reasoning has any substance. It seems to me that what's actually being done is to define "indistinguishable" particles to be those particles where that method of analysis gives the correct answer. So the "since...therefore" reasoning is just a proof-by-definition.

What's the point of asking the question? You get an answer and then you say "physics doesn't say that, physics says what I imagine physics says". What you imagine physics says may very well be nonsensical and self-contradictory. But, that doesn't matter, because it's only your own straw man you are questioning.

 

[Stephen Tashi]

What's the point of asking the question? You get an answer and then you say "physics doesn't say that, physics says what I imagine physics says". What you imagine physics says may very well be nonsensical and self-contradictory. But, that doesn't matter, because it's only your own straw man you are questioning.

If you're referring to my general behavior on the forum, I feel no need to offer any apologies. If you are referring to the specific topic of how "indistiguishable" particles are defined in classical statistical mechanics, the problems of defining them have been debated by other forum members (whom you may regard more highly than me). e.g. https://www.physicsforums.com/threa...e-particles-obey-boltzmann-statistics.939086/