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Well, the definition of order-type is wrong: order-preserving bijections need not be isomorphisms in . You might just say \lq\lq isomorphism” (in ) instead.
It may be curmudgeonly to ask, but what motivates you to create such a page?
Ah, my mistake, I’ll edit when I get home and thank you for catching it. The motivation was twofold, the notion came up in my studies recently but I couldn’t find a clear definition — Wikipedia had a page (https://en.m.wikipedia.org/wiki/Order_type) for order types but it only discussed what it meant for two posets to “have the same order type”, not what the order type itself was as an object, so I figured the nLab should have a page with clearer definitions.
Seeing that you use capital letters elsewhere, I’d appreciate it if spell my name the same way I spell it, beginning with a capital letter – thanks.
One way you define an order type is as an equivalence class, which I think is fine and practically speaking all one really needs to say. It seems to me that instead we might as well be defining the general concept of isotype (such an article doesn’t exist yet) for any category , as a connected component of the core groupoid or however one wishes to say it.
In the first paragraph of the discussion section, you mention that all sets are in bijection with some cardinal, assuming the axiom of foundation. I assume you mean something along the lines of Scott’s trick, which Scott used as a way to define cardinality as a formal object without using the axiom of choice. But very soon afterward you say that cardinals are special types of ordinals. Asserting that every set is in bijection with a (special type of) ordinal is saying that every set can be well-ordered, which is equivalent to the axiom of choice. It just sounds to me, from the way it’s worded, that the discussion is conflating two distinct notions of cardinal number: one using the axiom of foundation but not the axiom of choice, the other using the axiom of choice but not the axiom of foundation.
I don’t really follow what you’re driving at in the final sentence of the discussion.
My apologies Todd, will do from now on (I had just gotten home and rushed the edit).
You are correct that I’m conflating two notions of cardinal which I did not intend to do, I’ll delete the part about cardinals being special types of ordinals — the sentence at the end was likely a naïve version of the ‘isotype’ notion you describe, I could try to reflect on it and delete the comment/start a new article but perhaps I should leave it for someone more well versed?
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