User blog comment:BlauesWasser/Why Zero Shouldn't be considered a cardinal/@comment-30754445-20180501091716/@comment-27513631-20180507190859

When I talk about structures, I'm using the term in the algebraic sense. For example, a monoid is a structure that has a nullary operation \(e\) and a binary operation \(\cdot\) such that \((a \cdot e) = a\), \((e \cdot a) = a\) and \(((a\cdot b)\cdot c) = (a \cdot (b \cdot c))\). \(\mathbb N\) has multiple structures that satisfy these definitions - \((\mathbb N, 0, +)\), \((\mathbb N, 1, \times)\) and \((\mathbb N, 0, \max)\) are probably the most common.

When we talk about a set, the structure of the set is usually clear from context, and will vary (the way to extend one structure to another is not unique). As (order) topologies, \(\mathbb N \iso \mathbb Z\), but as monoids, \((\mathbb N, 0, +) \not\iso (\mathbb Z, 0, +)\).

The other key concept I'm using is Identity of Indiscernibles. This is the driving force behind equality - it states that two objects are identical if they're indiscernible. This, to me, seems evidently true.

Now, we generally consider structures up to isomorphism. If we don't - for whatever reason - we're simply considering structures with more structure (think generalising \(\mathbb N, 0, +\) to \(\mathbb N, 0, +, 1, \times\)) up to isomorphism instead (this may include \(\in\), for instance).

Now, if we combine indiscernability of identicals with an algebraic definition of structures, and note that isomorphic structures are internally indiscernible, then we arrive at the following principle: Isomorphic structures are equal. This is incompatible with taking a structure to just be a set of sets of sets of etc. with certain properties, as that contains too much information (\(\in\)) which prevents this from being true.

Sorry for not directly answering your post - but I hope this has cleared up some of what I'm thinking? Could you say what I haven't cleared up too, please?