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I am self-teaching myself basic set theory, and have not been able to find an answer to a confusion I have; namely, the meaning of the equality sign in the definition of a totally ordered set (TOS)

I have found some sources defining a TOS in terms of $\geq$, and not using = , although I think this is generally understood to be unpacked in terms of the fundamental relations $> , =$ as $a\geq b \iff a>b \cup a=b$.

But what is the meaning of $=$ here? In a totally ordered set, it seems impossible for it to be the case that neither $a>b$ nor $a<b$ without $a,b$ being the same element- if that is what $=$ is to mean?

Abstractly, to me $=$ always meant object identity. Whather one has an ordered set, a totally ordered set, or just a set, or even just two objects not in a set, $a=b$ is meaningful as saying 'the object here symbolised in type by the letter a is the same object as that symbolised in type by the letter b'

I appreciate that $>$,$<$ are also just abstract relations, and imputing some meaning in the way of 'comparing magnitudes' to them is doing things the wrong way round. i.e. the notion of magnitude arises because we have a set for which such realtions hold.

However it is not clear to me that element equality is non-trivially combinable with <,> . I should think that sources should describe this more clear (links to pages I have referred to below).

While writing this, I realised that perhaps the notion of -partially ordered set' is exactly that which I am looking for. i.e. there are elements $a,b$ for which neither $a<b\ ,\ a>b\ , \ a=b$. Perhaps it is just my prejudice that makes me feel like there should be a relation for comparing 'magnitudes' of sorts, without claiming object identity. What if, for instance, one equipped all of the elemnts of an ordered field with some additional property, and duplicated some of the elemnts of the set but gave the cpies a different new property so that they were no longer the same? It feels like there is something perverse about this. Changing the size of the set at will.

Fundamentally, I am trying to get to the bottom of what $=$ means, and whether this meaning is contextual oor not.

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Equality means what you thought it does: the things named by the expressions on the left and right of the "$=$" symbol are the same thing. In a total order, your intuition that "it seems impossible for it to be the case that neither $a>b$ nor $a<b$ without $a,b$ being the same element" is also correct. In fact that's either one of the clauses in the definition of "linear order" (or an easy consequence of the definition depending on exactly which formulation of the definition you're using).

The definition includes this sort of "obvious" information explicitly, because it's needed when you want to prove things about linear orders. There are plenty of relations that don't satisfy this requirement, for example the relation $\subsetneq$ between sets. And in a proof, the information you use needs to have a real justification, not just something like "it seems impossible" or "implicit in the notion of ordering" or anything like that.

Finally, you asked "what if, for instance, one equipped all of the elements of an ordered field with some additional property, and duplicated some of the elements of the set but gave the copies a different new property so that they were no longer the same?" I wouldn't call this "changing the size of the set at will"; I'd call it replacing the set with a different (larger) set.

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