Notes for Wed. February 29

Due to an aberration in the spacetime metric brought on by the fact that today is February 29, the notes from today are being posted before the notes from Monday.  Here are the notes and here is the TeX.

Includes:

• An application of the Łoś-Vaught test
• A taste of computability
• Skolem's Paradox
• An introduction to the completeness theorem for uncountable languages.

Notes for Feb 24

Here are the notes and tex file for February 24. If you see any errors, omissions, or improvements let me know.

Scribe Notes for February 22

Here is the pdf and here is the tex file. Enjoy!

Scribe notes from February 17, 2012

Hi all,

Here are the scribe notes from Feb 17, 2012. (TeX code is available here.) In class, two notations for $\mathcal{M}$ were given: one without tilde and the other with tilde. The tilde notation works well for simple symbols such as $\tilde{b}$, $\tilde{c}$, and $\tilde{t}$. But, I think, it is not for long terms such as  $\widetilde{f(t_{1},\ldots,t_{k})}$   (\widetilde{f(t_{1},\ldots,t_{k})}) . 

Have a great weekend!

Best regards,
Tomoya Sato

Scribe notes from February 15, 2012

Here is the pdf, and here is the TeX.

Notes: Monday, Feb 13

The PDF is here. The TEX is here.

Scribe notes from Feb 10, 2012

Here are the scribe notes from Feb 10, 2012. (TeX code is available here.) They introduce Frege-style systems for first order logic (aka Hilbert-style systems), as well as sequent calculus systems for first order logic (both LK and LK_e systems).

Bob Chen emailed to ask "What is a resolution derivation? We couldn't find the term in the 'Handbook of proof theory' (there it only mentions resolution refutations)."


My answer:  A resolution derivation is the same as a resolution refutation, except not requiring the final clause to be the empty clause.   In particular, if there is a resolution refutation of a clause $C$ from  a set of clauses $\Gamma$, then $\Gamma \vDash C$ holds.   (We proved this fact in class as part of proving the soundness of resolution.)


Sam

Announcement: Homework II is now due on Friday, February 10.

For Math 260 students (UCSD, Winter-Spring 2012).

This blog is intended for discussions accompanying the Introduction to Mathematical Logic course, during Winter and Spring 2012.  

It will be set up so students can create new posts, and anyone can view or answer them.  For this, if you are attending the course, please send me your preferred email for blog posting.  It is probably best if you have a gmail address, but I expect it will work also if you do not have a gmail address.    

Any discussions related to the course is fine, including discussions about, and hints for, homework problems.

It is a bit counter-intuitive how to use Blogger: To view a particular post including all of its comments, you must either click on the post title in the sidebar on the right side of the screen, or on the "Comments" link at the bottom of the past.

You can use LaTeX commands in either posts or comments, which are implemented with MathJax and should work in almost all up-to-date browsers.  Note that enclosing math in single $'s does not work.  However, you may use "$\backslash($" and "$\backslash)$" for inline math and double dollar signs or "$\backslash[$" and "$\backslash]$" for displayed math.  Examples:  $(y+\sqrt z)^{-1}$ and  $\sin^2 x^2$.  And, a displayed equation is: $$\frac 2 3$$
Another displayed equation is here:
$$\forall x \exists y (x\le y \land y\le x \leftrightarrow x=y) .$$

Sam Buss
sbuss@math.ucsd.edu
sambuss@gmail.com