Talk:Transfer principle

untitled

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)

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)

Done. -- Dominus (talk) 06:01, 18 August 2008 (UTC)

OK, I've looked at it. Definitely a lousy article. But I think the topic is worth a separate page, since the transfer principle has broader applicability than the topic of the other article. Maybe I'll come back and write something here at some point. Michael Hardy (talk) 06:16, 18 August 2008 (UTC)

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)

lead

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)

Focus

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)

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)

I don't think that's a transfer principle. — Arthur Rubin (talk) 02:25, 6 September 2008 (UTC)

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)

Statement and examples

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)

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)

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)

crucial ability

The ability to speak in a more extensive language than that to which the transfer principle applies is certainly crucial. However, perhaps one could examine the extent to which one can actually claim that the language is the same so long as one applies it to internal objects. Some statement of that sort, it seems to me, is the crux of the matter, combined of course with the fact that one can connect the two models externally using objects such as the stardard part. If we don't manage to say something of that sort in the lead, we will continue getting comments of the kind that graces the beginning of this page, and perhaps justifiably so. Once we make a case in support of the assertion that the language is the same to a well-defined extent, we can go on to point out that at the set theoretic level one cannot expect to get it all, e.g. completeness as Keisler points out in his epilogue. Am I hopelessly confused? Katzmik (talk) 14:15, 8 September 2008 (UTC)

That that language is the same seems pretty simple.

$\forall x\,\exists y\,s.t.x+y=0$

is written in that language. It's the same regardless of whether it applies to the standard universe or the nonstandard one. So your concern is escaping me. Michael Hardy (talk) 16:34, 10 September 2008 (UTC)

Bcrowell's edit

Through a tedious "history" search, I discovered that the edit that creates confusion was added on 13 march '05 by Bcrowell. It is interesting that on his userpage he notes that "Many articles seem to reach a certain level of quality, and then gradually degrade due to random, disorganized, uncoordinated edits." I think this is exactly what happened here. The comments concerning "repeated generalisations" accompanied by "giving up properties" seems to suggest that this is what happened with the hyperreals. Namely, that in order to construct the hyperreals one needed to give up statements from higher order logic. A sentence later in the section says something about higher order logics as well, but the impression given by the earlier paragraphs is as I outlined. Before I make any changes, I would like to get some input. Katzmik (talk) 11:58, 9 September 2008 (UTC)

I'm a little confused by your statement

in order to construct the hyperreals one needed to give up statements from higher order logic

The hyperreals can be "constructed" (non-constructively...gasp) by an ultraproduct. Why does this need "higher order logic", since everything takes place within ZFC which is 1st order. IST is also a first order theory.--CSTAR (talk) 14:01, 9 September 2008 (UTC)

Here is the point I am making. Consider, for example, the Bolzano-Weierstrass theorem. This is a theorem about sets. Therefore the formulation of the theorem requires quantification over sets. Thus a language that includes the B-W theorem would have to include quantification over sets. Now suppose one develops *R in such a way that the transfer principle only applies to first order logic. This would apparently mean that the B-W theorem cannot be stated in *R. Note that this does not merely mean that B-W cannot be proved in *R. It means that B-W and *R inhabit different universes of discourse. You can have either one or the other but not both. Such a restricted version of hyperreal analysis would be of limited utility, even as far as calculus is concerned, as most people assume that B-W is an integral part of calculus. Having said all this, I reiterate that I have no serious training in logic and it could be that I am making a basic error. In such case I would greatly appreciate a clarification. Katzmik (talk) 14:14, 9 September 2008 (UTC)

I don't really understand what this discussion is about, but fwiw, here is what I understand the transfer principle to say: if p is a sentence in the language of the real numbers (plus, times, less than, zero, one) augmented by finitely many extra function and relation symbols, and p is first-order in the sense that all variables range over the reals (so we can introduce new function symbols but not quantify over them), and if we interpret the new function and relation symbols in p to be any functions and relations we like on the real numbers, then p is true in the real numbers with these interpretations if and only if p is true in the hyperreals, with variables interpreted as ranging over the hyperreals, and the new function and relation symbols interpreted as referring to the natural hyperreal extensions of their real versions.

/me pauses for breath.

This statement is immediate (for the ultrapower construction) from Łoś' theorem. Statements such as B-W or the completeness of the reals cannot be stated in the first-order language of the reals, and so the principle doesn't apply directly. Indeed, if it did, then the hyperreal construction would collapse, since the reals are determined up to isomorphism as the complete ordered field. However, one can (for example), take a specific non-empty subset A of the reals that is bounded above, and since A has a sup in the reals, the natural extension of A must have a sup (in fact the same sup) in the hyperreals. Algebraist 13:07, 10 September 2008 (UTC)

Doesn't every internal Cauchy sequence converge in *R? Katzmik (talk) 13:32, 10 September 2008 (UTC)

Yes. That's an example of the sort of thing my last sentence is about: we have a specific function (well, partial function, but who cares?) on R which converges to a given value, and by the transfer principle, its hyperreal extension converges (in the order topology) to the same value. This is because the statement that a given function converges to a given value as its argument goes to infinitity is a first-order statement. Algebraist 13:37, 10 September 2008 (UTC)

Does every Cauchy system in *R² converge as well? Katzmik (talk) 13:51, 10 September 2008 (UTC)

I don't know what that means, but as long as it's the natural extension of a Cauchy system in R², then probably. Algebraist 13:55, 10 September 2008 (UTC)

I don't really understand what you wrote about individual functions, it surely has to do with my lack of background. I assume B-W and completeness can be stated in a suitable version of hyperreal analysis, that they are true when interpreted internally, and that they cannot be stated in first order logic. Are you with me so far? Katzmik (talk) 13:59, 10 September 2008 (UTC)

I'm not comfortable with talk of internal things, but I think such talk means the same as my complicated statement about individual functions. The point is that once you've fixed an individual function (or set, or whatever), your statement becomes first-order, and so can be transfered to the hyperreals. Thus your statement is true for all functions on the hyperreals which arise from real functions (I think this is what is meant by internal function). So by transferring one function at a time, you can transfer a statement of the form {(for all real functions, sets, whatever) (something firstorder happens)} to one of the form {(for all internal hyperreal functions, sets, whatever) (the same thing happens)}, even though these statements are not themselves firstorder (at least over numbers; they can of course be interpreted as statements of set theory, which is firstorder over sets). Algebraist 14:12, 10 September 2008 (UTC)

I have a feeling I am going to have to learn some basic material here, but I don't quite understand how one could transfer them one at a time. If one could, why can't one reformulate B-W and completeness as first order, one at a time? Katzmik (talk) 14:15, 10 September 2008 (UTC)

You can. But that only gives you completeness for sets of reals (aka internal sets of hyperreals), whereas the full second-order version of completeness would be the (false) statement that every set of hyperreals that is nonempty and bounded above has a sup. Algebraist 14:25, 10 September 2008 (UTC)

Keisler in his book defines a real statement on page 907 and then goes on to say that the completeness axiom is not a real statement. In my ignorance I thought this means that it is necessarily a second order statement. If it can be reformulated in first order, what do we miss exactly by restricting to first order? Katzmik (talk) 14:35, 10 September 2008 (UTC)

The completeness axiom cannot be reformulated as a first order axiom. What you can get is an infinite collection of first-order statements (in the language of set theory augmented with names for arbitrary subsets of the reals) expressing the least-upper-bound property for all internal sets. This infinite collection fails to express the true meaning of the completeness axiom, since it doesn't apply to any non-internal sets. Algebraist 14:43, 10 September 2008 (UTC)

(unindent, edit conflict X4)

Let me try to explain the "individual functions" argument, as I understand it, although I wouldn't have presented it that way.... Consider the statement "every Cauchy sequence converages".

$(\forall f:\mathbf {N} \to \mathbf {R} )[[(\forall \epsilon >0)(\exists n)(\forall n_{1},n_{2}>n)(|f(n_{1})-f(n_{2})|<\epsilon )]\implies [(\exists x)(\forall \epsilon >0)(\exists n)(\forall m>n)(|f(m)-x|<\epsilon )]]$

(Notation:

x and ε are reals (or hyperreals, as appropriate)

n, m, n₁, and n₂ are integers (or hyperintegers, as appropriate)

This is first-order except for the f. I think what Algebraist is trying to say is that for any sequence f, the sequence *f is *Cauchy if and only if f is Cauchy, and *f *converges if and only if f converges, by the transfer principle. (Also *f *converges to a real x if and only if f converges to x.)

My statement of the transfer principle would imply that the statement is true for all internal f, whether or not a "transfer" from a specific standard sequence f. In other words...

(with all unspecified quantifiers running over ^*R or ^* N as appropriate)

$(\forall \mathrm {internal} f:{}^{*}\mathbf {N} \to {}^{*}\mathbf {R} )[[(\forall \epsilon >0)(\exists n)(\forall n_{1},n_{2}>n)(|f(n_{1})-f(n_{2})|<\epsilon )]\implies [(\exists x)(\forall \epsilon >0)(\exists n)(\forall m>n)(|f(m)-x|<\epsilon )]]$

(I'll add line spacing and remove this comment when I get the chance, so the formula doesn't extend over the edge of the page. — Arthur Rubin (talk) 14:57, 10 September 2008 (UTC) copyedited 14:44, 11 September 2008 (UTC)

Yes, that is what I have been trying to express, hampered by the fact I'm making it up as I go along. In summary: we can transfer second-order statements from the reals to the hyperreals, as long as all second-order quantification is restricted to only internal objects. Algebraist 15:03, 10 September 2008 (UTC)

This is made somewhat moot by noting that the proof that there is a one-to-one function from $\mathbf {R} ^{\mathbf {N} }$ (the set of sequences of reals) to $\mathbf {R} \,$ can also be transferred. — Arthur Rubin (talk) 15:14, 10 September 2008 (UTC)

True (though I hadn't realised it). Better stick with the least-upper-bound axiom as our example then. Algebraist 16:14, 10 September 2008 (UTC)

Arthur, please don't write things like

for all internal *f, whether or not a "transfer" from a specific sequence f.

To denote it *f means it is a "transfer" from a standard object called f. Michael Hardy (talk) 16:30, 10 September 2008 (UTC)

I see your point, but this is a notational convienence, to indicate where the variables operate. The alternative is to have a different typeface for variables over the reals or hyperreals, but that's difficult to operate. In any case, there's little confusion here. I'll try to do better in the future. — Arthur Rubin (talk) 21:11, 10 September 2008 (UTC)

Is that better? — Arthur Rubin (talk) 14:44, 11 September 2008 (UTC)

Reply to comment of User talk:Katzmik) posted 14:14, 9 September 2008 (UTC). Sorry to take so long to respond. You are correct that quantification over sets is required, but this doesn't make it a higher order theory. For example, there are no type distinctions between sets of integers and integers. In ZFC all variables range over the entire set-theoretic universe. If one had a weaker no-standarad analysis, with limits on the range of quantification, the resulting theory would be less interesting. In fact, you can make the transcendental extension 'R[t] into an ordered field in which the indeterminate t is infinite and 1/t is a non-zero infinitesimal. But this is pretty much useless for a development of calculus. I don't know if I've addressed any of your concerns.--CSTAR (talk) 14:48, 11 September 2008 (UTC)

Bcrowell's edit bis

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)

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)

I added some material to the lead, let me know what you think. Katzmik (talk) 14:44, 11 September 2008 (UTC)

Question

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)

Henkin

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)

Generalizations of the concept of number

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)

The edit of 14:04 14 September 2008 replaced a helpful discussion of the history of numbers with a subjective, misleading, and ahistorical viewpoint (c.f. the ancient discovery of the irrationality of the square root of 2). I propose at least a partial restoration of the previous version of this discussion. --Mosher (talk) 14:57, 6 April 2021 (UTC)

Deletion of (lots of the) hyperreal content

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)

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)