# Is my 'proof' that any strict total order is not total correct?

I am wondering whether a strict total order is total.

As I understand it, a strict total order is (among other things) irreflexive.

But for a relation $R$ to be total, it needs to be true that for any $a$ and $b$; either $aRb$ or $bRa$.

So, if we pick $a=b$, then we must have $aRa$ for any $a$ for it to be total, making it reflexive, rather than irreflexive.

So, it seems that every strict total order is not total.

Is this correct? It seems ... counterintuitive ...

For all $a,b$ either $a<b$ or $b<a$ or $a=b$.
• Yes, thanks! OK, so we are using a different notion of 'totality' when talking about total orders as opposed to when we are talking about strict total orders? Why not use something like: For any $a \not = b$: either $aRb$ or $bRa$ as the condition for both kinds of total orders? Also: since strict total orders don't have the condition of 'totality' as defined for total orders, can't I still argue that strict total orders are not total orders? By the way, I am using the Wikipedia page on total orders here; maybe that's my cardinal sin :) Apr 16, 2018 at 17:30