Talk:Transfer principle

message: This page needs clean-up, especially the example.
I got zero knowledge transfer from this article. Thumbs down. AJRobbins (talk) 07:46, 3 January 2008 (UTC)
replies: text: The discussion of the transfer principle at hyperreal numbers seems much better to me.  If there is no objection, I will replace this article with a redirect to that section. -- Dominus (talk) 14:53, 22 February 2008 (UTC)
replies: text: Done. -- Dominus (talk) 06:01, 18 August 2008 (UTC)

text: My solution was suboptimal, but I think the present state of affairs is an improvement.  I quite agree that the topic is worth a separate page, and I hope someone writes one.  -- Dominus (talk) 13:53, 18 August 2008 (UTC)
thread_title:  untitled 

message: The current lead seems more appropriate to a wider article on hyperreal numbers rather than the more specific topic of the transfer principle.  Furthermore it would probably be described as "editorial" (meaning largely polemical) by CSTAR and Arthur Rubin and is definitely in need of pruning.
Furthermore, I am concerned about the mention of first order logic in the second paragraph.  Is it correct that the order needs to be first?  Please comment.  Katzmik (talk) 08:07, 5 September 2008 (UTC)
thread_title:  lead 

message: As I mentioned elsewhere, there are two types of transfer principle: the one generally described here, referring to hyperreals or internal set theory, and the Jech/Socor and Pincus transfer theorems on transferring properties from models of ZFU (ZF with Urelements) to models of ZF.  Unfortunately, it's been over 15 years since I've formally worked with the latter, and I'm not sure I could do it justice.  My mother's book Consequences of the Axiom of Choice makes a reference to transfer results in note 103, according to Google Books.  — Arthur Rubin  (talk) 12:41, 5 September 2008 (UTC)
replies: text: There are also other uses of "transfer" (something called a transfer homomorphism) in number theory which I know absolutely nothing about. This is mentioned in André Weil's book on number theory. --CSTAR (talk) 17:19, 5 September 2008 (UTC)
replies: text: I don't think that's a transfer principle.  — Arthur Rubin  (talk) 02:25, 6 September 2008 (UTC)
replies: text: It looks like eventually a disambiguation page will have to be created.  At the moment perhaps the page could be improved somewhat.  It currently lacks a decent lead.  Michael Hardy suggested some material that goes in the right direction.  Could you please set me straight concerning the technical point I asked about?  Namely, the second paragraph of the lead currently claims something about restricting logic to first order, which seems to contradict what I have read in Keisler and Robinson.  Katzmik (talk) 11:07, 7 September 2008 (UTC)
thread_title:  Focus 

message: Thanks for the edits, they are very helpful.  I see in the statement and examples section that transfer holds for quantification over internal sets.  If this does not go beyond first order logic, I don't know what will.  Is the comment in the second paragraph of the lead correct? Katzmik (talk) 13:04, 8 September 2008 (UTC)
replies: text: I don't think it goes beyond first order set theory, which can be confused with second order logic.  — Arthur Rubin  (talk) 13:27, 8 September 2008 (UTC)
replies: text: You will notice that the lead wants to restrict to first order logic, which would seem to indicate the inaccuracy of the lead.  Could you comment somewhat on the distinction?  Katzmik (talk) 13:30, 8 September 2008 (UTC)
thread_title:  Statement and examples 

message: Apparently what you are saying is that a second order statement such as the mean value theorem (or completeness, or B-W) can be replaced by uncountably many first order statements.  Then one can apply transfer.  The upshot is that we write virtually all of real analysis in terms of first order, given an ample supply of ink (and at least a continuum of scribes).  Is this a correct summary?  If so, the current lead is very misleading.  It opens with a discussion of giving up properties (commutativity, associativity), and then proceeds to mention first order logic, as if what is being given up is higher order logic.  Now there are two possibilities.  The first one is that the author of this lead really meant that one has to give up statements traditionally viewed as second order (see examples above), hence the discussion of the quaternions and the octonions.  In this case, he is making a fundamental error, since NSA without the mean value theorem would be of limited interest.  The second possibility is that he did not actually mean to give up anything, but rather replace the higher order statements by large amounts of ink as above, in which case the lead is very misleading.  Hopefully I have managed to formulate my objection in a comprehensible manner by now.  Any reactions? Katzmik (talk) 11:53, 11 September 2008 (UTC)
replies: text: Oh dear, I seem to have spread confusion here. What the original statement seems to have been trying to say is that when we extend the reals to the hyperreals, there are certain second-order properties of the reals which do not hold for the hyperreals. The most obvious of these is the statement that non-empty bounded-above sets have least-upper-bounds. This is a second-order statement about the reals which is not true about the hyperreals. The equivalent statement about the hyperreals which is true (by a version of the transfer principle) is that every internal non-empty bounded-above set of hyperreals has a least-upper-bound. In general, a second-order statement about the reals has to be weakened to allow only internal sets and functions for it to be guaranteed true in the hyperreals. Here I use 'internal' to mean 'thing which is the natural extension of something defined on the reals'; I hope this is what the word means. My comments about infinite families of statements were an artefact of my logical background and the way I reached the above conclusion, and can safely be forgotten about. Algebraist 14:31, 11 September 2008 (UTC)
replies: text: I added some material to the lead, let me know what you think.  Katzmik (talk) 14:44, 11 September 2008 (UTC)
thread_title:  Bcrowell's edit bis

message: I would like to re-word the sentence:
Kanovei and Shelah[1] have found a method that gives an explicit construction, at the cost of a significantly more complicated treatment.
To something like: 
Kanovei and Shelah[1] have found a method that give a construction that eliminates the need of being given an ultrafilter, at the cost of a significantly more complicated treatment.
Does anyone object? Thenub314 (talk) 18:53, 30 October 2008 (UTC)
thread_title:  Question 

message: Can someone provide a reference to give more details on Robinson's quote on "Henkin's sense"? Tkuvho (talk) 15:47, 28 January 2014 (UTC)
thread_title:  Henkin 

message: The paragraph "Thus, the notion of an irrational number is meaningless to even the most powerful floating-point computer" is inexact, since computers can process irrational and transcendental numbers symbolically. 
It probably means that floating point hardware designed for arithmetic manipulation is restricted to a limited number of bits, but that cannot be generalized to any existent or possible floating point hardware, and cannot be generalized to symbolic representation and manipulation of numbers in computers.  — Preceding unsigned comment added by 186.59.57.227 (talk) 15:54, 12 March 2015 (UTC)
thread_title:  Generalizations of the concept of number 

message: I'm not an expert on the hyperreals or transfer principles but some of these sections make no mention of the transfer principle and just focus on explicating the structure of the hyperreals. I agree with what people have said re: merging, namely that it's a bad idea and that this topic deserves its own page, but even the lead mentions that transfer principles exist for things other than the field of hyperreals, which is not the impression given by the rest of the article. Unless anyone raises concerns fairly soon, I'm deleting a lot of the content on hyperreals, and leaving a bit of it under one consolidated heading. Hopefully someone with more knowledge on this specific topic can add something about other examples at some point in the future. Bromar71 (talk) 19:50, 10 June 2015 (UTC)
replies: text: I hereby raise concerns. The absence of material of transfer principles in other areas is a weakness that should be corrected by adding material on those areas, rather than by deleting existing material. Tkuvho (talk) 08:14, 11 June 2015 (UTC)
thread_title:  Deletion of (lots of the) hyperreal content