Talk:Löwenheim–Skolem theorem

message: The notation on this page used 'sigma' as the arity function, whilst the article on 'signature' used 'sigma' as the name of the signature itself and the arity function was called 'a' -- I've changed the notation in this article to conform with the article signature Zero sharp 04:21, 24 September 2007 (UTC)
thread_title:  Change of notation for signature: <math>\sigma \rightarrow  a</math> 

message: I think the article on the upward L-S theorem should be merged here. — Carl (CBM · talk) 16:14, 16 November 2007 (UTC)
replies: text: I agree. --Hans Adler (talk) 23:00, 16 November 2007 (UTC)

text: Also agree, not clear why there were ever two (articles, not theorems). Zero sharp (talk) 05:08, 22 November 2007 (UTC)
replies: text: I merged them. Please feel free, as always, to rearrange and fix my prose. — Carl (CBM · talk) 01:37, 25 November 2007 (UTC)
replies: text: I am sorry Carl, I forgot to leave a note that I was rewriting this article in user space. Now you have obviously put quite a bit of work into merging the two original versions. I have replaced the article with my version nevertheless, but feel free to revert. --Hans Adler (talk) 10:18, 25 November 2007 (UTC)

text: If you do revert, the complete text (and quite a lot of other stuff) is also here: User:Hans Adler/Model theory and universal algebra. --Hans Adler (talk) 10:22, 25 November 2007 (UTC)
thread_title:  merge 

message: Gregory Moore writes in Zermelo's Axiom of Choice, page 251f:
:Not until Gödel [1930] did mathematical logicians carefully and uniformly observe the distinction between a syntactic notion such as consistency (no contradiction can be derived) and the curresponding semantic notion of satisfiability. [...]
:Löwenheim studied sentences in which quantifiers ranged over relations as well as sentences in which the quantifiers were restricted to individuals. In addition, he permitted his sentences to contain denumerable strings of quantifiers and of relations.  Despite this wealth of expressive power, he formulated a theorem which revealed a severe limitation in the strength of first order logic:  if a first order sentence A is true in every finite domain, but not in every domain, then A is false in some denumerable domain. [...]
:Löwenheim's  [1915] proof [...] relied on a distributive law which stated that a sentence \forall x\, \exists y\, A(x,y)  is true in a domain if and only if {  \exists} y_\lambda \, \forall x \, A(x, y_\lambda(x)) is true there, where \lambda ranges over all individuals of the domainl. (To Löwenheim, the quantifier \mathbf \exists y_\lambda was a possibly uncountable sequence of individual existential quantifiers, one for each individual in the given domain.) [...]
: Skolem [retained] countably many quantifiers in a given sentence, [but] circumvented Löwenheim's second order semantics through the introduction of new first order relations -- in effect, Skolem functions. Yet Skolem continued to obscure the boudnary between [...] consistency and satisfiability by stating L's result in the following form: 
::A 1st order sentence is either contradictory or else satisfiable in a countable domain. 
:[...] What he actually proved was 
::If a first order sentence is satisfiable in a set M, it is satisfiable in a countable subset of M. 
:[...] By 1922, Skolem [...] criticized Zermelo's system [...] Furthermore, he proved without using AC a weaker version of the LS theorem: 
::If a countable family of sentences is satisfiable, then it is satisfiable in the domain \mathbb N of natural numbers. [...] concluded  that Z's system could be satisfied in \mathbb N [...] known as the Skolem paradox. 
So there are several issues here: 
# Single sentences, or countable theories
# strictly first order, or allowing infinitely many quantifiers, and/or second order quantifiers
# Do we want just any countable model, or a countable submodel of a given model?
Concerning the axiom of choice, we know now that 
* Clearly some version of the axiom of choice is needed to get a countably infinite  substructure of any infinite structure (never mind elementarity -- ZF does not prove that any infinite set contains a copy of the natural numbers)
* No AC is needed if we want to show that any countable consistent 1st order theory has a countable model.  The usual (Henkin) proof of Gödel's completeness theorem gives such a model, and does not use AC. (Neither does Gödel's original proof, I think.)
Aleph4 (talk) 11:14, 12 December 2007 (UTC)
thread_title:  AC and Skolem 1923 

message: Substituted at 18:28, 17 July 2016 (UTC)
thread_title: Assessment comment

message: The  of this article very similarly matches text on page 160 of the book "A Functorial Model Theory: Newer Applications to Algebraic Topology " by Cyrus F. Nourani. At present At least one similar sentence existed in this article before the publication of the book. Needs more investigation. Jason Quinn (talk) 13:29, 27 February 2020 (UTC)
thread_title:  Copyright infringement or violation?