October 1, 2009
The goal of this note is to present a proof of the following fundamental result. A theory is said to be satisfiable iff there is a model of
Theorem 1 (Compactness) Let be a first order theory. Suppose that any finite subtheory is satisfiable. Then itself is satisfiable.
As indicated on the set of notes on the completeness theorem, compactness is an immediate consequence of completeness. Here I want to explain a purely semantic proof, that does not rely on the notion of proof.
The argument I present uses the notion of ultraproducts. Although their origin is in model theory, ultraproducts have become an essential tool in modern set theory, so it seems a good idea to present them here. We will require the axiom of choice, in the form of Zorn’s lemma.
The notion of ultraproduct is a bit difficult to absorb the first time one encounters it. I recommend working out through some examples in order to understand it well. Here I confine myself to the minimum necessary to make sense of the argument.
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October 1, 2009
From Tampa Bay Times, October 1, 2009, page T10.
A student’s letter to tbt*
Editor’s note: On Monday, tbt* published a story describing President Obama’s desire to increase the number of days U.S. children go to school each year. A junior at a Pasco high school e-mailed the following response — with the subject line “why dose nobody ask the students’’ — to tbt*. It is printed here, unedited:
Dear “tbt” editor of which it may concern I’m a student of pasco coun ty a junior to be precise and pleas do not mistake this for a Dear Abby seg ment I am not a 40 year old women concerned about her felines. I am just appalled as a member of the student body that in ever y in other words the only article I could find of president Obahmas plan to extend the school year, school days, and e ven rumored around school of mak ing us work weekends that they have not a single students opinion on the subject.
more so that they believe that the school boards “research” should stand as good evidence of the im provement in our “higher achieve ment”. a very intelligent man once said if you obser ve you have therefor changed the out come (now notice I didn’t abbreviate that phrase as I did the other stated phrases that’s because it’s not exact not as an in sult to any of the readers but some would use this against the students to further there pointless war against other country’s grade average).
at any rate not only do I belie ve it to be unreliable research but I also believe it dose not properly address to how our mind will adjust to the added stress. now personally I don’t ge t what in heavens name these peo ple they put in these articles have to do with the students spending more time in school it’s not like there stuck in the classroom ding the e xtra work the teachers prepare to compensate for the e xtra time. are they siting first row in this teenage bomb they’re building.
the plan reducing summer well first why don’t they come clean that they all ready reduced it we’re only on vacation for two months give or take a few days and for years they lied to our faces saying we have three months vaca.
to sum it all up I just wish he pub lic the pollutions the school board the reporters would address us on things that affect US not them now granted there may have been a seg ment on the news some were that addressed students about it but if there was did you even pay attention. and let me end this by saying the subduction of certain rights in order to maintain the learning enviorment making your usage of the 1 st amend ment right lesser then you would have in normal society is the same as removing them at the front step!
(I would like to stay anonymous)