Talk:Membership is Left Compatible with Ordinal Addition
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I have added the terms "base case", "inductive case", and "limit case" to Transfinite Induction/Schema 2. Should I add links for base case, inductive case, limit case, etc. to their own pages?
Also, it against house style to use the eqn format to create the equations where some lines are all on the left side (see below)?
| \(\ds \neg x < \varnothing\) | \(\) | \(\ds \) | Empty Set has No Members | |||||||||||
| \(\ds \) | \(\implies\) | \(\ds ( x < \varnothing \implies ( z + x ) < ( z + \varnothing ) )\) | Consequentia Mirabilis |
--Andrew Salmon 23:18, 27 July 2012 (UTC)
- I don't know about "against house style" - but IMO it looks untidy. I may have a go later at getting it to look a bit smarter. Oh, and I know it sounds fussy, but the "compact" style of putting the entire equation on one row is not house style. Space is not an issue, this is electronic media not paper (I have spent my entire professional career kicking the fundaments of computer programmers in my sphere of influence for writing source code which is compact to the point of unreadability) and wide layout is always better than this. This is how we like it:
| \(\ds \neg x < \varnothing\) | \(\) | \(\ds \) | Empty Set has No Members | |||||||||||
| \(\ds \) | \(\implies\) | \(\ds ( x < \varnothing \implies ( z + x ) < ( z + \varnothing ) )\) | Consequentia Mirabilis |
... except it will be restyled something like:
| \(\ds \) | \(\neg\) | \(\ds \left({x < \varnothing}\right)\) | Definition of empty set | |||||||||||
| \(\ds \) | \(\implies\) | \(\ds ( x < \varnothing \implies \left({z + x}\right) < \left({z + \varnothing}\right) )\) | Consequentia Mirabilis |
- While I'm about it, I can't see how you get from A to B by using Consequentia Mirabilis. This would be saying "$(\neg x < \varnothing \implies x < \varnothing) \implies x < \varnothing$". We are assuming that $\neg x < \varnothing$ where (it is assumed that $<$ means "subset inclusion" - a link needs to be made here or $\subsetneq$ needs to be used instead, incidentally), but we are not as a consequence proving that "indeed there actually is an ordinal $x$ such that $x \subsetneq \varnothing$". In fact I'm not quite sure whether I follow the logic of the above at all. (I need to think about it but this won't be immediate - my time is short this morning, the sun is shining on a Saturday for the first time since -5000 BC and my wife and I are going to take advantage of it, I'm just waiting for her to get ready.)
- Take a look at the existing Principle of Mathematical Induction for how the original concept was structured. Just bunging the vital terms that need to be defined in an afterthought called "remarks" does not feel right. Yes I know, I know what a "base case" is (although I prefer "basis for the induction" - we try to avoid colloquialisms here) and you know what a "base case" is, and I've seen (but strongly disagree with) the argument that says "if you don't know what a base case is then you have no business reading my proofs", but there's more to it than that. Transfinite induction talks about a "limit case" which very importantly needs to be explained very carefully, as it's subtle.
- It is also worth pointing out that on this page, that while yes there's a link (it's there, it's not prominent) to Transfinite Induction, it is not clear that this link contains the definitions of "Base case", etc. barf. Look at examples of proofs using Principle of Mathematical Induction, and see the preferred technique. Separate, individual links for each of these three concepts, linking back to the specific definitions.
- I'm sorry to go on about this in great depth - you'd rather be putting proofs up than spending all your time fussing around tarting up the style - but this site: it is what it is. A great deal of effort has been expended in making it this way. It looks (I like to think) smooth and effortless because we've worked hard to make it so.
- But do not worry. When a "tidy" or "MissingLinks" template is added to a page, it is not an instruction to you necessarily that you must tidy your page!. Not at all. It's a placeholder to indicate that "this page could do with some tidying up to get it into house style" and "some links are missing here". If you care to go through and bring them up to house style (the rules are straightforward and have a reason) then it's up to you. It is possible to learn that style (I know this because several of the writers here are fluent in it) but if you don't want to, then take note that the "tidy" template will appear. --prime mover 08:44, 28 July 2012 (UTC)
- As far as the "Consequentia Mirabilis" mistake, I got confused, thinking the page was the (also valid) propositional law $\neg \phi \implies ( \phi \implies \psi )$