Using Godel's completeness theorem, it can be shown that the compactness theorem is equivalent to the ultrafilter lemma. The compactness theorem can also be proven using ultraproducts and Los's Theorem, but it requires reference to the full axiom of choice. My question is: are there any known proofs of the compactness theorem that make no reference to syntactic provability, that use no choice principles stronger than BPIT?
Apologies if this has been asked somewhere before, but I didn't see what I was looking for after several pages of Google results.  I was reading Jech's The Axiom of Choice and was introduced to the equivalence of the ultrafilter lemma and the compactness theorem for FOL. I gather that these are also equivalent to the completeness theorem. I am primarily interested in trying to understand the "ultrafilter lemma $\implies$ completeness/compactness" direction. Jech's own proof takes an odd route through "binary messes" and I don't really follow the last couple of steps. (It's on pp.17,18 of the 2008 Dover reprint, if one is interested).  I'm wondering if anyone knows a good reference for, or can provide, a readable proof that the ultrafilter lemma implies completeness or compactness, with a preference for completeness. Alternatively, if you have Jech's book on hand, helping me bridge the concluding paragraph of his $\textrm{(ii)}\implies\textrm{(iii)}$ step with the preceding steps would clear up much of my puzzlement. (Though in this case I can't offer any help in saying what, specifically, is missing in the way of a bridge).
The shop is five minutes away.     According to the dictionary, away is an adverb. An adverb modifies a verb. In the above example, what word does away modify? Why is away not an adjective?       The shop is five meters tall.     tall is an adjective in the second example. It modifies five meters (does it?). So why is away an adverb? PHP function substr() error Does anybody know which files I need to save in order to have a simple backup?   Reason for my question is, that my Galaxy Android smartphone lost all calendar data without any interaction. I found tons of apps, tons of sync stuff, but NO simple list of files to save. I do NOT want to save my data remotely on Google.  There must be at least one local file (database?) on my phone containing the synced calendar file (on my server, using ICSSync) How do I join two lists in Java? Explicit example of a compact manifold of dimension $>2$ with strictly negative sectional curvature I am looking for examples of compact manifold of dimension $&gt;2$ with strictly negative sectional curvature (for dimension 2 it is well-known). Can anybody please help? The reminder email says:     We noticed you have not accepted our invitation to join Stack Overflow Careers. Only 3 days remain before it expires [...]      But please act soon: this invitation expires in 14 days [...]   Is it 3 days or 14 days now? In the movies, the pack of Uruk-Hai capture Merry and Pippin, and are bringing them back to Saruman as quickly as possible.  Why didn't they use Wargs to carry the Hobbits instead of packing them on their backs?    Or, better yet, why didn't the Nazgul fly down and take them on their Fell Beasts? A (not necessarily unitary) $R$ module $M$ splits into $B\oplus C$ where $B$ is unitary and $RC=\{0\}$. I try to prove the following formula:  $$n!=\prod_{k=1}^n \operatorname{lcm}(1,2,...,\lfloor n/k \rfloor)$$  I noticed that  $\upsilon_{p}(\operatorname{lcm}(1,2,...,\lfloor n/k \rfloor)) = s$ iff $\lfloor n/(p^{s+1}) \rfloor &lt; k \leq \lfloor n/(p^{s}) \rfloor $ Let $K=F[x]/(p(x))$, where $F$ is a field, $p(x)\in F[x]$ is an irreducible polynomial.  What are all the maximal ideals in $K[y]$?  Attempt: I have an attempt, but not entirely sure it is correct.  Since $(p(x))$ is a maximal ideal in $F[x]$, this means $K$ is actually a field.  Thus $K[y]$ is a PID.  So maximal ideals in $K[y]$ are $(g(y))$ for irreducible polynomials $g(y)\in K[y]$? How to 'sum' a standard deviation? I have a monthly average for a value and a standard deviation corresponding to that average. I am now computing the annual average as the sum of monthly averages, how can I represent the standard deviation for the summed average ?  For example considering output from a wind farm:  Month        MWh     StdDev January      927     333  February     1234    250 March        1032    301 April        876     204 May          865     165 June         750     263 July         780     280 August       690     98 September    730     76 October      821     240 November     803     178 December     850     250   We can say that in the average year the wind farm produces 10,358 MWh, but what is the standard deviation corresponding to this figure ? Simple examples of uncorrelated but not independent $X$ and $Y$ Any hard-working student is a counterexample to "all students are lazy".  What are some simple counterexamples to "if random variables $X$ and $Y$ are uncorrelated then they are independent"? I've been going through  and have a question on the inductive proof for the . Recurrent relation:  $J(1) = 1$ $J(2n) = 2J(n) - 1$ $J(2n+1) = 2J(n) + 1$  Closed form hypothesis:  $J(2^m + l) = 2l + 1$, for $m \ge 0$ and $0\le l&lt;2^m$  Inductive Proof (for the even case):  $J(1)=1$ $J(2^m+l)= 2J(2^{m-1}+l/2)-1=2(2l/2+1)-1=2l+1$  How do the authors get from $2J(2^{m-1}+l/2)-1$ to $2(2l/2+1)-1$? Reason behind choosing the compensating resistor for input bias current in op amps I found the following explanation online on determining the values used for the compensating resistor:  In either case, the compensating resistor value is determined by calculating the parallel resistance value of R1 and R2. Why is the value equal to the parallel equivalent of R1 and R2? When using the Superposition Theorem to figure how much voltage drop will be produced by the inverting (-) input's bias current, we treat the bias current as though it were coming from a current source inside the op-amp and short-circuit all voltage sources (Vin and Vout). This gives two parallel paths for bias current (through R1 and through R2, both to ground). We want to duplicate the bias current's effect on the noninverting (+) input, so the resistor value we choose to insert in series with that input needs to be equal to R1 in parallel with R2.  Although it is a 'crisp' and simple explanation, I don't understand how it is true, since the current flowing through R2 won't directly go to the ground, but will probably enter the op-amp at the output (op-amp as a current sink). Also, it might go through a few more resistances before going to the ground, since Vout isn't directly connected to the ground. Where can I find people for hire with LaTeX skills and designer skills? Fine-tuning "Open safe files after downloading" Suppose $\displaystyle P(x,y)$ a polynomial in the variables $x,y$.   For example, $\displaystyle x^4$ or $\displaystyle x^3y^2 + 3xy + 1$.  Is there a general method which allows us to evaluate the indefinite integral     $$ \int P(\sin x, \cos x) \text{d} x$$   What about the case when $\displaystyle P(x,y)$ is a rational function (i.e. a ratio of two polynomials)?   Example of a rational function: $\displaystyle \frac{x^2y + y^3}{x+y}$.    This is being asked in an effort to cut down on duplicates, see here:   and here: Finite group for which $|\{x:x^m=e\}|\leq m$ for all $m$ is cyclic. Let $G$ be a finite group. For each positive integer $m$, if $x^{m}=e$ has at most $m$ solutions in $G$, $G$ is cyclic.   What I have thought is that $n=\sum_{d\mid n}\phi(d)$ can be used to solve this and showing that $|G|$ order element in $G$ exists is enough. What is a "callback" in C and how are they implemented? From the reading that I have done, Core Audio relies heavily on callbacks (and C++, but that's another story).    I understand the concept (sort of) of setting up a function that is called by another function repeatedly to accomplish a task.  I just don't understand how they get set up and how they actually work.  Any examples would be appreciated. How to find a generator of a cyclic group?