@BillyRubina - There are some different notions of tameness, but stability is one of the big ones. You'd find definitions and examples in something like Marker's Model Theory: An Introduction would have information about stability and some of the other notions of tameness; though it's not really what I'd call introductory if you're not already a little familiar with models of first order logic.
I'd check out something like Rautenberg or Ebbinghaus et. al for something more introductory that will still cover how compactness/the Lowenheim-Skolem theorems work.
And those would prepare you to read a more involved text like Marker's.
(I know Rautenberg is one that user21820 also recommends, because they're the one who introduced me to it!)
@BillyRubina Indeed, MaliceVidrine has provided you the outline of a proof of LS. You should actually work through the technical details, because if you don't understand the proof then you won't know how ridiculously little control you have over the models that LS produces.
And yes I recommend Rautenberg's "A Concise Introduction to Mathematical Logic". It really lives up to its name for conciseness; it packs a lot in a single book! =)
But you really need undergraduate mathematics background to learn from Rautenberg. If you do not know basic FOL (i.e. how to use FOL in normal mathematical reasoning), then you should start with learning that first, otherwise Rautenberg may be inscrutable.
But I see from our conversation history that you should have learnt basic FOL, so I think Rautenberg is a good book for you to try!
Anyway, back to the question of models of different sizes, here is a simple example: Let T be the theory with a single predicate-symbol P and axioms that force elements satisfying P to be infinite and force elements satisfying ¬P to be infinite too. That is, ∃x,y ( x≠y ∧ P(x) ∧ P(y) ), ∃x,y,z ( x≠y≠z≠x ∧ P(x) ∧ P(y) ∧ P(z) ), ... and similarly for ¬P. Then T has exactly one countable model up to isomorphism, but has 3 non-isomorphic models of cardinality aleph(1).
As for PA, there is a theorem that only the standard model (ℕ,0,1,+,·,<) of PA is computable, even though we can use compactness to construct uncountably many non-isomorphic countable models of PA! That is how bad models produced by compactness can get. Models produced by LS are worse.
I am curious about the following (excuse me for the stupid question): What doing mathematics in those other models look like? For example, I guess I know what doing math with PA looks like but what doing math in these other models "look like"?
@BillyRubina By definition, all models of PA satisfy PA... so if you only use axioms of PA then what you prove is true in all of them. But since PA is syntactically incomplete (see the starred post on the incompleteness theorem), there are arithmetical sentences that are true for ℕ but false for other models of PA.
You cannot 'do math' in models. You can only (as I did) talk about which sentences are true in which models.
A system of equations can be understood as a set of equalities over a given FOL structure. It is the structure that tells you what operations are allowed, which allows you to form terms possibly with free variables, and each equality is simply a formula of the form "$t=u$" where $t,u$ are terms o...
When I was solving system of linear congruences (n variables, n equations), like this:
$AX \equiv b \pmod p$
I was told that ordinary Gaussian Elimination works if $p$ is prime. And I figured out that when $p$ is prime, integers $\pmod p$ form a field, otherwise it doesn't form a field, but a r...
@Threnody In case you are misreading, that thread does not imply that it works only over fields.
By the way, if you see a problem in my post, let me know. Currently there are 3 downvoters who refuse to explain why they downvoted, but I don't see an actual problem.
@Threnody Well, I guess The_Sympathizer's answer has a simplified version of mine, but I think the same downvoter downvoted that answer as well, so who knows why.
@user2103480 Haha.. I have always liked graph theory. =)
how long do you take breaks from serious studying?
I've found myself totally unwilling to do any exercises to prepare for my next semester haha. which is pretty different from last semester's break where I did a lot of computability/category theory/statistics
@user2103480 You may not want to learn from me. When I was a student, I never ever learned ahead except for reading Spivak's Calculus. That was the only textbook I ever read systematically through.
Instead, what I did was to explore whatever I wanted all the time, including during class while waiting for the lecturer to finish some easy proof that I knew I could construct myself.
The end result was that my knowledge was full of holes, but when I actually took courses those holes were not hard to fill in at all.
for example, my set theory lecture barely covered cofinality, so I went through some 40 pages of further set theory to be able to follow in the models of set theory class
3/4 of my courses next semester will be based on stochastic analysis. Which is also harsh since I had to learn the rudiments for a course last semester, but dont know the topic in depth. But I'm more scared of topology II
But the stochastic analysis things will hopefully be ok since I don't have to switch the mode of thinking as much as in other semesters
In germany, teaching is done by PhDs, postdocs, professors. There are also permanent researchers at institutes like MPI that teach, or affiliate professors (probably comparable to adjuncts, but with a decent salary)
but it's hard to get a job as an almost pure lecturer
When I was solving system of linear congruences (n variables, n equations), like this:
$AX \equiv b \pmod p$
I was told that ordinary Gaussian Elimination works if $p$ is prime. And I figured out that when $p$ is prime, integers $\pmod p$ form a field, otherwise it doesn't form a field, but a r...
