# Talk:Systems of Logic Based on Ordinals

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Field:  Foundations, logic, and set theory

## Hypercomputation

I removed the sentence

This work eventually became the basis for the concept of hypercomputation towards the end of the 20th century.

due to undue weight. Hypercomputation (in the sense of the linked article) mostly a fringe topic. It's fair to say Turing's thesis led to a lot of development in recursion theory and reflection principles. I'd rather let one of the logic editors figure out how to write that up though. 64.160.39.217 (talk) 09:34, 30 April 2012 (UTC)

Maybe so, maybe not - and will probably not matter much either way. Yes, hypercomputation is somewhat far out, but so is much of this type of theory. I knew a mathematician at a research organization who often recounted the story that was criticized for being too theoretical because he dealt with way out hypothetical issues. Then he said he went to talk to one of the leading logicians in a university one day, and the professor told him of his work "I sometimes think of practical issues too", considering his far out work "practical"... So I am not sure how to call this. I will ask for a more informed opinion. History2007 (talk) 12:24, 30 April 2012 (UTC)
What is the exact link between Turing's paper and hypercomputation? I will look at the paper again, I may have forgotten exactly what is in it. But most "hypercomputation" schemes aren't based on ordinals in any way. On the other hand, "hypercomputation" in the sense of generalized recursion theory is a mainstream subject. "Hypercomputation" of the sort that tries to disprove the Church-Turing thesis is a fringe topic, but our article on hypercomputation is written more broadly. — Carl (CBM · talk) 12:45, 30 April 2012 (UTC)
Frankly I hardly recall my reasons for adding that. It seems that it was a year ago that I started this page, and I just remember that I did so just because it was his thesis and I thought it deserved a page. The first sentence in the history section of the hypercomputation Wikipage refers to the thesis, but now I have forgotten the details. If the link gets deleted, I do not mind, I just thought you might know off hand. If you have to spend effort on it, it may not even be worth it. Thanks. History2007 (talk) 13:40, 30 April 2012 (UTC)
Now I remember. The issue was that of using an oracle-based system - that was the connection. History2007 (talk) 17:15, 30 April 2012 (UTC)
I personally think it would be reasonable to say "the use of oracle machines has been mentioned in relation to hypercomputation" with a reference to Copeland's paper. — Carl (CBM · talk) 17:52, 30 April 2012 (UTC)
That would be worth having in a more complete article. IMHO it's still out of balance in this tiny stub, though I guess it's an acceptable compromise if anyone really wants it. 64.160.39.217 (talk) 17:58, 30 April 2012 (UTC)

(edit conflict) Turing's paper is connected to hypercomputation in that it discusses a Turing machine with an oracle for the halting problem (p. 18 of the scan), and maybe the transfinite iteration of such oracles. There is an historical article about the thesis here. I do believe (I guess mainly from typing "hypercomputation" into search engines) that the term hypercomputation is mostly used by people postulating physical realizations of hypercomputersm, at least at the level of thought experiments. 64.160.39.217 (talk) 17:27, 30 April 2012 (UTC)

Yes, and as here there is literature on it in semi-respectable sources. History2007 (talk) 17:32, 30 April 2012 (UTC)
The first result there, "Hypercomputation: computing beyond the Church-Turing barrier", is the sort of thing that the IP is referring to. Copeland's original paper [1] does not only talk about oracle Turing machines (which he also calls "coupled Turing machines"), it also talks about other models of hypercomputation. He goes back to oracle machines in section 4, but they are not the main focus of his paper. The IP is right that the term "hypercomputation" is not called that in normal computability theory books (they call oracle computation "relative computability"). I do think that the term "hypercomputation" is mostly used by people who are looking for physical ways of obtaining non-Turing-computable functions rather than by people studying generalized computabiity theory. — Carl (CBM · talk) 17:52, 30 April 2012 (UTC)
Yes, it looks that way, although Syropoulos refers to Copeland. I wonder if a disambig page of some type is needed anyway, and a way to explain the two sides of the hypercoin. History2007 (talk) 18:11, 30 April 2012 (UTC)
I think it could be described at the hypercomputation page. It's one of those things that is clear on reviewing the literature but is hard to literally source, which makes it hard to write about here. And I don't think there have been a lot of people motivated to work on the hypercomputation article in general. — Carl (CBM · talk) 18:18, 30 April 2012 (UTC)
Yes, that would probably be the easiest way. Martin Davis overviews the relationship of Turing to oracles here and that may be an item to add here at some point, not just about Copeland, but about how Davis views that aspect of the dissertation - which he considers secondary anyway. But Davis considers the implications of the dissertation as important.
And yes, not enough people to write about hypercomputation, but then that applies to many parts of Wikipedia. The logic articles are generally in much better shape than others (say computing), however, so I should not complain really. History2007 (talk) 18:33, 30 April 2012 (UTC)
Anyway, I added the Davis quote just because it was there. Thanks for your input. History2007 (talk) 19:56, 30 April 2012 (UTC)
Did Davis's article actually mention the polynomial hierarchy (my browser doesn't show the google books scan)? It looks out of place. Those (i.e. the poly. hierarchy) are all computable classes. It might be better to just generically mention relativization or use Feferman's summary of the paper's introduction of oracles ([2] 5th page of pdf). 64.160.39.217 (talk) 21:41, 30 April 2012 (UTC)
He specifically mentioned it. History2007 (talk) 04:17, 1 May 2012 (UTC)
Well, ok. It still comes across as a little bit odd to me. Thanks. 64.160.39.217 (talk) 06:28, 1 May 2012 (UTC)

## Should this and related articles be in the category "Ordinal numbers"?

Should Systems of Logic Based on Ordinals, ordinal logic, and hypercomputation be placed into Category:Ordinal numbers? They might be of interest to people studying ordinals as an application. JRSpriggs (talk) 20:12, 30 April 2012 (UTC)

To be upfront with you, I neither use, nor endorse the Wikipedia category structure. I have for long wished that it would use the formal discipline of ontologies, rather than a somewhat haphazard user selected graph structure and wish that it would base itself on Wordnet. Yet, in an imperfect world, I think we can add this and ordinal logic to that category, but not hypercomputation. But CBM would probably b a better judge of that. History2007 (talk) 21:10, 30 April 2012 (UTC)
Per History2007, I wouldn't add hypercomputation to the category either, but the other two are ok. In fact maybe ordinal logic and this article should be merged. 64.160.39.217 (talk) 21:30, 30 April 2012 (UTC)
Merge does not make sense given that this is about a thesis, the other a concept. History2007 (talk) 04:19, 1 May 2012 (UTC)
OK, I will add those two as you suggest. JRSpriggs (talk) 05:26, 1 May 2012 (UTC)
Thank you. History2007 (talk) 09:07, 1 May 2012 (UTC)
re merging: the content of the thesis is the topic that the other article is about, so there's not much point to two articles. CBM would know more, but I think "ordinal logic" doesn't really exist as much of a topic under that name any more (it has been absorbed into a more general body of theory). I think Torkel Franzén's book "Inexhaustibility" discusses the general subject. 64.160.39.217 (talk) 05:32, 1 May 2012 (UTC)
I think you need to read WP:Note. That is the applicable guideline here. And per this the topic is clearly notable, and should no be merged. Regarding recent research, Phlogiston is no longer in vogue among physicists, but has a page given WP:Note. History2007 (talk) 09:07, 1 May 2012 (UTC)
WP:Note says it's permissible to have two articles, not that it's required or a priori likely to be worthwhile. I'm mostly thinking about what a fully-fleshed-out article about the thesis would look like. It would say a bit about the background and publication history, then have a longer exposition of the content, then a little more exposition about how the results in it connected up to later work by others. Then I think about what a fleshed out "ordinal logic" article would contain, and it's just about exactly the same stuff, maybe with minor organizational differences. Are you volunteering to write those two almost-identical articles? We don't have many articles about individual math papers other than really epoch-making ones. We usually just write about the mathematical topic and cite the paper in the math article.

This particular thesis had a throwaway mention of "oracles" that became an important topic in CS and logic, mostly because of work by Kleene as I understand it. But it was mostly(?) about a rather different subject (transfinitely iterated consistency statements) that is pretty cool but not of that much consequence in terms of the amount of further work that it directly inspired. Phlogiston is not comparable since it was a widely held theory of physics for decades or centuries. Turing's work on the Entscheidungsproblem was far more influential than the ordinal logic thesis was. 64.160.39.217 (talk) 17:53, 1 May 2012 (UTC)

No sorry, that is not how WP:Note works. If the topic Ordinal logic is notable, it should have its own article. And per this the same applies to this article. And I do not know why this long discussion is necessary. This is getting to be longer than Turing's dissertation now. There are many more articles that need help. I think energy should go into fixing those, not writing another dissertation on this talk page. History2007 (talk) 17:59, 1 May 2012 (UTC)
If you really insist both articles should be written, then please get busy writing. It's just not impressive for you to go around making these decrees and expecting other people to do the work. Regards, 64.160.39.217 (talk) 21:13, 1 May 2012 (UTC)
My apologies for being so lazy. Yesterday I only started 3 new articles - and would have probably done more on the Matrix completion article if I had not been distracted here. I will try to work harder.... But seriously, that is not how Wikipedia works. I started San Cassiano (Venice) with just one or two sentences and someone else built it into a nice page. But then someone else started Computer cluster as a less than adequate article and I rewrote 90% of it. We all cooperate for whatever needs to be done... History2007 (talk) 22:28, 1 May 2012 (UTC)