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.
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.
? What do you mean? Do you mean abstractly? As far as I know none of those things really use model theory in any way except, of course in the sense that all thinking is model theory(but then no need to pick out any specific instances as if they are one of just a few). The Greeks were heavily in to logic, of course, and our modern logical theories are just expansions on it... but all thinking is based in logic so it wasn't just the Greeks doing it(but they did it well and made a big stink about it which I think some of us which we could get back to).
I mean nothing. I mean something that coexists in abstract but also exists in the negation of abstract. And platonic dialogue is guiding me in that pursuit. Thank you for getting into trouble to reply ..
@@nikitasmarkantes5046 lol. Well, negation of abstract would be concrete. So you are saying something that is both abstract and concrete. Technically the terms are duals/opposites so if something is in both then it transcends those concepts. By excluded middle such things cannot exist. But maybe you do not take excluded middle as tautological? If you are referring to feeling something deeper in the logic of the ancient Greeks then, well, simply study logic, category theory, math, and model theory... you will see the tree that has grown from the seeds they planted. Model theory is pretty much the culmination of understanding the ramifications of their logical system and it transcends "logic" in to all fields of thought. Logic = intelligence = thought. It might be difficult to comprehend how such at theory can say so much about everything when it seems to say so little and only about logic but once you realize our brains are logic machines then it is natural to see how all things our brains do(which basically is everything because without our human brains we are animals) is based in logical theories in some form or fashion. Maybe these "hidden gems" you are speaking of are tautologies or invariant which are special things that exist in theories that act as "ultimate truths" which "theories" are built around. Basically things as close to "god" as we can get.
May be these hidden gems are invariant may be they are not maybe they are both and nothing...may be they are ultimate truth may be they are not. Thank you so much for responding. I totally agree with your recommendation to study category theory and model theory. These are the trees from the seeds they planted. Agree. May be they are the seeds may be they are the trees. Who knows what precedes what.... may be it transcends everything may be not as rule of excluded middle proves. But what will happen if ουσια partakes both, transcending everything including its own negation. To an uninitiated and naive as me this childish approach is what fascinates me. Thank you again for replying...
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)
Fantastically clear, especially for someone not particularly versed in logic! thanks
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.
? What do you mean? Do you mean abstractly? As far as I know none of those things really use model theory in any way except, of course in the sense that all thinking is model theory(but then no need to pick out any specific instances as if they are one of just a few). The Greeks were heavily in to logic, of course, and our modern logical theories are just expansions on it... but all thinking is based in logic so it wasn't just the Greeks doing it(but they did it well and made a big stink about it which I think some of us which we could get back to).
I mean nothing. I mean something that coexists in abstract but also exists in the negation of abstract. And platonic dialogue is guiding me in that pursuit. Thank you for getting into trouble to reply ..
@@nikitasmarkantes5046 lol.
Well, negation of abstract would be concrete. So you are saying something that is both abstract and concrete. Technically the terms are duals/opposites so if something is in both then it transcends those concepts. By excluded middle such things cannot exist. But maybe you do not take excluded middle as tautological?
If you are referring to feeling something deeper in the logic of the ancient Greeks then, well, simply study logic, category theory, math, and model theory... you will see the tree that has grown from the seeds they planted.
Model theory is pretty much the culmination of understanding the ramifications of their logical system and it transcends "logic" in to all fields of thought. Logic = intelligence = thought.
It might be difficult to comprehend how such at theory can say so much about everything when it seems to say so little and only about logic but once you realize our brains are logic machines then it is natural to see how all things our brains do(which basically is everything because without our human brains we are animals) is based in logical theories in some form or fashion.
Maybe these "hidden gems" you are speaking of are tautologies or invariant which are special things that exist in theories that act as "ultimate truths" which "theories" are built around. Basically things as close to "god" as we can get.
May be these hidden gems are invariant may be they are not maybe they are both and nothing...may be they are ultimate truth may be they are not. Thank you so much for responding. I totally agree with your recommendation to study category theory and model theory. These are the trees from the seeds they planted. Agree. May be they are the seeds may be they are the trees. Who knows what precedes what.... may be it transcends everything may be not as rule of excluded middle proves. But what will happen if ουσια partakes both, transcending everything including its own negation. To an uninitiated and naive as me this childish approach is what fascinates me. Thank you again for replying...
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)