Axiom talk:Hilbert's Axioms

Work to be done
Obviously, this is not the final version, but this is the form of the axioms I found in my source. I understand that every axiom should go to a separate page which will be done in the near future. I will probably need to introduce a shorthand notation for primitive relations, because any other presentation I find online so far is very verbal. I've seen betweenness relation in Tarski's axioms, and I think others should have their shorthand notations as well. Also the names of individual axioms and groupings have not settled yet. Some sources provide better suggestions than others.--Julius (talk) 20:24, 22 August 2022 (UTC)


 * Well overdue, and big thanks for taking it on. --prime mover (talk) 21:48, 22 August 2022 (UTC)


 * I guess these axioms are for planar Euclidean geometry? Good to make that explicit. &mdash; Lord_Farin (talk) 09:31, 23 August 2022 (UTC)


 * Nice catch. Planar geometry indeed.--22:33, 23 August 2022 (UTC)


 * Incidentally, before we travel too far down the road, I moved this page to the Axiom namespace. I think it better belongs there. --prime mover (talk) 18:38, 23 August 2022 (UTC)

Presentation
I wonder whether it would make the page prettier if we were to make the axioms heading-3 rather than heading-2? --prime mover (talk) 09:36, 26 August 2022 (UTC)


 * Whatever suits you better. At the moment I am focused on rewriting everything with fewer words and making it look more like Tarski's axioms. It seems that various sources introduce their own notation, and the number of axioms seems to vary as well. Hence, mapping their ideas on what I have here is not so trivial.Julius (talk) 22:13, 27 August 2022 (UTC)

$p$ is a point
Depends on what we want to do. For a simple description this is sufficient. But if we ever plan to make strict connections with the first- or second- order logics then we need this notation.--Julius (talk) 22:21, 27 August 2022 (UTC)


 * Yes, absolutely, But I would say we need it in describing the connection, not in the axioms itself. So only in the translation to 1OL/2OL does it become relevant, or at least that's how I see it.
 * We just have to be careful about applying set-theoretic machinery because a priori this translation might not be 100% faithful. &mdash; Lord_Farin (talk) 08:52, 28 August 2022 (UTC)


 * Oh, I only now saw the edit you made. In my view adding the "sets" convolutes the exposition because instead of just working with points and lines, we have to talk about sets and stuff. Before you know it you will assume stuff like $l = \{p: I(p,l)\}$ and write $p\in l$.
 * If we want to discuss point and lines, then we should not bother with sets. That comes when we want to interpret planar geometry in terms of set theory. But not before that moment.
 * So I would really like to suggest that the exposition be modified to exclude the $\in$ symbol and "set" word. &mdash; Lord_Farin (talk) 08:59, 28 August 2022 (UTC)


 * Let's see how I see this problem. We are trying to write down an axiomatic system which is a sort of digest of abstract notions pertaining to the Euclidean geometry. For that we have to write down basic notions of various elements and their relations. We can write this with words or with symbols whose meaning we explain somewhere earlier. At the moment we are not considering any connection to 1OL or 2OL, so any symbols or notions we introduce here are independent of standard axiomatics. So even if here we use a word "set", until we show that this maps to the standard set, the term should have no problem existing within the context of these axioms (except for the confusion using the same string of symbols denote apriori two different things). However, you suggest avoiding using the notion of sets and, by extension, the entire set builder notation. Most likely this can be done. Just at the moment I am interpolating between several texts, and set theory for me was a good starting point to condense everything. I will see what I can do. But along the way I may need to introduce some new notation so please bare with me.--Julius (talk) 18:04, 28 August 2022 (UTC)