Talk:Compactness theorem

message: For instance, it follows that if something is true for any field of positive characteristic, it must also be true for fields of characteristic zero (so, for real, complex or rational numbers too).
:I moved this from the main page, since the statement is strictly untrue. For example, it is true for any field with positive characteristic that there exists a non-trivial finite subfield with the same characteristic, but this isn't true for fields of characteristic 0. I'm sure something similar and more accurate can be phrased; but I don't know enough to say it. Chas zzz brown 08:31, 27 November 2002
::Well, the statement in question is not a sentence. You cannot express it by the first order predicate calculus formula. I have reverted with this clarification.
::Actually, you are right, here is an example: there are x,y,z,t different from zero, s.t. x^2+y^2+z^2+t^2=0 - this holds true in every characteristic p, but not in rational numbers. This is a valid sentence. There is however always SOME field of characteristic zero for which the statement holds. (it is enough to have fields of arbitrary large characteristic) I have added corrections, the original statement actually goes the other way around and is called robertson principle, and the other statement is its contraposition. User:18.252.2.32 06:02, 28 November 2002‎ 
:::It solve by wiwo function and also this type of conjecture also solve from this like falting theorem  also fermat last theorem Ajitendra kumar jain (talk) 03:01, 13 October 2022 (UTC)
thread_title: section1

message: Can't find Robertson's principle in this context. Is this rubbish?
User:David Martland
Try 
http://www.media.mit.edu/physics/pedagogy/babbage/texts/rt.html
if you dont believe
:Shouldn't it be Robinson (as in Abraham Robinson)? --Zundark 21:31 Nov 28, 2002 (UTC)
:As noted at the top of the referenced page:
:: This is a transcription of relevant notes from the class 18.511 taught by Prof. Sacks in the Spring of 1998, organized and reinterpreted. Homework problems starting with problem 9 are solved in vitro. Notation is indecipherable.
:...so there could very well be many typos. Chas zzz brown 22:25 Nov 28, 2002 (UTC)
Well, I found this Robertson on that site - I never heard about the name for the principle before, it seems it is really a misprint. But the statement is certainly true (it is a straightforward application, as fields of characteristic zero have axioms p<>0 for each p prime, and hence if a sentence is inconsistent with theory of fields of characteristic zero, there must be a finite subset of this set of axioms inconsistent with a statement so for every field of characteristic p greater than largest prime in that set it will be inconsistent too) - it is perhaps too simple to be some big principle, but it seems to be very illustrative for compactness theorem.
G.E. Sacks indeed calls it Robinson's principle; see e.g. p.630 in The Differential Closure of a Differential Field,
Bulletin of the American Mathematical Society vol.78:5 (1972).
http://www.ams.org/bull/1972-78-05/S0002-9904-1972-12969-0/S0002-9904-1972-12969-0.pdf 
130.83.2.27 (talk) 12:26, 1 January 2010 (UTC)
thread_title: section3

message: I think an important application of compactness theorem is nonstandard analysis, but this article provides no link to related pages. For example, the sentence from the application section:
So, for instance, there are nonstandard models of Peano arithmetic with uncountably many 'natural numbers'.
I think nonstandard models of Peano arithmetic here may be linked to the corresponding section of the page, i.e. Peano arithmetic#Nonstandard models. Also we should mention nonstandard analysis somewhere.Tony Beta Lambda (talk) 12:37, 22 January 2012 (UTC)
replies: text: It is mentioned in the third application.  Tkuvho (talk) 19:13, 22 January 2012 (UTC)
thread_title: section5

message: The content of this article is rather very thin, important specifications and examples are missing. It does not present an example of a set of sentences with an associated model. Also the proof by Gödel could be detailed. Could someone add this type of details?--93.122.249.165 (talk) 22:45, 28 August 2019 (UTC)
thread_title:  Absence of important details and examples 

message: How is the method useful but not effective?  Rather: efficient?  — Preceding unsigned comment added by 2A02:8106:225:1100:983D:F6F8:ACA1:8324 (talk) 22:30, 4 February 2022 (UTC)
replies: text: The word "effective" has a specific technical meaning effective method, for example effective descriptive set theory. 67.198.37.16 (talk) 01:24, 29 November 2023 (UTC)
thread_title: useful but not effective