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- 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

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.)

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.)