Ultraproducts

The video presents a proof of the completeness theorem of first-order logic based on the ultraproduct construction. The compactness theorem is then applied to prove the Loewenheim-Skolem theorem.

[aidanprattewart]

Fantastically clear, especially for someone not particularly versed in logic! thanks

[nikitasmarkantes]

It is amazing how such hidden gems as this lecture shed light to platonic dialogues such as Cratylous and Parmenides and Republic and Timaeus and Theaitetus.

[kodfkdleepd]

I have the book Model Theory by Cheng and I'm trying to work through it. It's quite dense. The first chapter talks about semantics, syntax, and theories. Maybe you can explain why these distinctions are made as it seems to me to all be the same(and which it is since he proves they are the same). It seems to me that the distinctions add unnecessary complexity through obfuscation rather than serving any actual theoretical purpose.

At best all I can come up with is that historically semantics, theories, and syntax were thought of differently because the theories connecting them were not well developed. When I learned logic it was more from the syntactical approach and that is typically how I break things down. About all I have gotten out of the distinctions is that there are typically 3 approaches to think about propositional logic which as only helped me in realizing why some forms of thinking seem "off" to me... and it is because they are semantic approaches.

Other than that I can't seem to reconcile why one should go to the trouble of making the distinction, maybe you have some good reason why it is done? I haven't worked through model theory much but I've always had in the back of my head that there was far more to theories and Chengs book seems to make it explicit(and I guess that is what model theory is all about). But it seems overly formalized in a way that feels very pedantic without real purpose(I haven't worked though the book yet though so I'm hoping there is more to it). I do have an extensive background in logic but more from a mathematicians point of view and along the lines of simply learning mathematical theories, category theory, basic logic courses, computation/computer science, etc. I always feel a little uneasy in when logicians present their material. I've seen the same thing but the way they go about it always feels like they haven't learned abstract syntax tree's, category theory, and the like(obviously they know something about it but the language jargon seems to be geared towards something else). Of course maybe the point is not to be circular but pretty much all life is circular in some sense.

For example, Models are defined as simply a subset of a language of sentence symbols. Then one makes a relation between models and "truth tables" by assigning values. What's the point? It's it solely for language? I grew up with truth tables and I automatically translate models in to truth labels. It seems perfectly natural to do this... but if there is a distinction being made it makes me think I'm doing it wrong. It really makes no sense to me to say something is "true in a model" and "true in a truth table" as different statements(but only the 2nd is natural to me). Now if I assume historically it wasn't know they were equivalent then fine, I get it but I then rather learn model theory that doesn't make the historical distinction since it's much more work to track the unnecessary differences while trying to learn the technicalities of model theory. (learning something should be as easy as possible, not unnecessarily complicated)