Straight from the wiki page on the LS theorem:
> ... no first-order theory with an infinite model can have a unique model up to isomorphism... In general, the Löwenheim–Skolem theorem does not hold in stronger logics such as second-order logic.
Since it is established that the complete ordered field (the real numbers) is unique up to isomorphism, it seems to follow that the complete ordered field axioms are not a first-order theory.
Intuitively, this makes some sense to me, since the completeness axiom discusses not reals, but sets of reals; however, I'm having trouble making the distinction precise.
For instance, the completeness axiom requires some kind of set theory to state. The theory we use to construct the field is first order, so why doesn't LS apply here?
Alternatively, we could extend the language of the ordered field axioms (which are first order) to include just enough set theory (also first order) to be able to state the completeness axiom and form sets of reals, much in the same way that the NBG comprehension schema is axiomatized.
Another approach would be to forget set theory altogether and opt for an 'axiom schema of completeness'. Consider a set that is defined as all the reals x that satisfy some first order formula P(x). We could formalize the statement 'P(x) is nonempty and bounded implies P(x) has a least upper bound'. The schema would consist of one axiom for each formula P(x).
The alternative axiomatizations I've described seem to avoid the 'second-orderness' of the complete ordered field axioms, which I can't reconcile with what I know. I feel as though I'm fundamentally misunderstanding something here. What might that be?
I know that L-S fails for 2nd order logic with standard semantics, but that a form of it does hold for Henkin semantics. What is it about standard semantics that makes L-S not apply? In particular, what part of the proof works for Henkin semantics and does not work for standard semantics?
This is as much a question about perspective as it is about logic itself, but you all seem like the people to go to still. In the SEP article on Infinitary Logic the author writes,
> On the other hand, the upward Löwenheim-Skolem theorem in its usual form fails for all infinitary languages. For example, the L(ω1,ω)-sentence characterizing the standard model of arithmetic has a model of cardinality ℵ0 but no models of any other cardinality. However, all is not lost here, as we shall see.
The author's perspective there seems to be one of lamenting that the upward LST theorem fails in infinitary logic, and so excitedly goes on to rescue a weaker form of it. I think of LST theorem being the result that doomed early hopes of finding a categorical theory for mathematical foundations. In that light, I'd imagine seeing upward LST failure being a good thing, rather than a bad thing.
Can someone try and explain to me this author's point of view?
Logic textbooks generally approach the Compactness and Löwenheim-Skolem theorems in one of two ways.
The first way is to prove the soundness of some proof procedure and the Gödel completeness theorem, and then use these results to derive the model theoretic theorems. This is what the authors do in "A Friendly Introduction to Mathematical Logic".
The second way is to prove the model theoretic results first and use them to prove soundness and completeness far more briskly. This is the approach that Boolos et al opt for in "Computability and Logic".
Basically, I want to understand them the second way: I would like to see a proof of Compactness and Löwenheim-Skolem that doesn't presuppose Gödel's completeness theorem.
Unfortunately, the presentation in "Computability..." is (in my opinion) very poor. I can't motivate what they're doing and I don't understand how the model theoretic results follow.
So my humble request is that somebody point me in the direction of proofs of Compactness and Löwenheim-Skolem that (a) do not presuppose the completeness theorem, and (b) are proved in as simple a manner as possible.
Thanks!
My girlfriend has always cited the Löwenheim-Skolem theorem/paradox as her favorite. I've seen some documents made into artworks by printing them on nice paper and placing them in a nice picture frame. I want to do this for her, for christmas, with a cool historical/"antique" document on Löwenheim-Skolem.
I need to find a single page that has some nice words on it and a clear mention of the theorem. I was searching online for the original paper on the theorem but wasn't able to find anything.
Maybe somebody here can help me with some ideas of a page of a document on Löwenheim-Skolem that would look nice in a frame (and have some historical significance), or perhaps with a link to the original paper where it was first mentioned.
Thank you!
EDIT: Thanks to AngelTC I've found something I want to use. But as I know no german and nothing about old notation and terminology I am unsure if what I took is a actually an image of some version of the theorem. Could somebody verify for me that this is actually (some version of) the Löwenheim theorem?
http://imgur.com/JyUdp4y
I found this explanation but I'm still unsure. I wish I knew what "Aleph_null-valid" means.
http://imgur.com/a/ca2Gd
Hi, I posted this yesterday in /r/askscience, but it's still in the spam filter. So I'm trying my luck here:
Lindström's First Theorem says, that every regular logical system that has more expressive power than FoL loses either the Löwenheim-Skolem or Compactness theorem. I'm having a bit of a problem understanding why the LST is a property of a logical system that is desirable. Wouldn't a system where you can differentiate between structures of different infinte cardinalities be more useful? Is there an important theorem or property that follows from it?
I have heard from a few people that the Lowenheim-Skolem Theorem is deep and possibly just as interesting as Godel's Incompleteness. One supposedly surprising implication is that there is a countable structure which satisfies all the same ordered field sentences that the reals satisfy. Perhaps I misunderstand the significance of this but it seems to me somewhat predictable. Or at least, if you had asked me before I had learnt the result, I'd probably say I had 60% confidence it would turn out that way. After all, the set of sentences in a language with countably many symbols is a pretty big restriction, regardless of the size of the structure used to interpret the sentences.
Perhaps the countable structure which is implied to exist is itself interesting in some way that I haven't learned?
In plain and simple english as I am struggling to make sense of it. Thank you
Hey there, I'm looking at the limitations of first order logic and am having trouble understanding the relevance of Skolem's Theorem as it's presented in the book: "There exists a nonstandard model for the theory of natural numbers - a model not isomorphic to the natural number structure. Moreover, there is such a model whose domain is denumerable".
Could anyone explain to me why exactly this is a limitation on the expressive capacity of first order language to define structures? So long as a theory describes the structure completely, why should it matter that it also has non-standard models?
Thanks!
I'm confused about finding a Skolem form of the following formula:
F = (∀x)(P(x)→(∀y)((∀z)Q(z,y)→¬(∀z)R(y,z)))
So following the algorithm I've done these transformations:
F = (∀x)(¬P(x)∨(∀y)((∃z)¬Q(z,y)∨(∃z)¬R(y,z)))
= (∀x)(¬P(x)∨(∀y)((∃z)¬Q(z,y)∨(∃m)¬R(y,m)))
= (∀x)(∀y)(¬P(x)∨(¬Q(f(x,y),y)∨¬R(y,g(x,y))))
= ¬P(x)∨(¬Q(f(x,y),y)∨¬R(y,g(x,y)))
But the solution on the textbook gives a different answer which is:
F = (∀x)(∀y)(∀z)(∀v)(¬P(x)∨¬Q(z,y)∨¬R(y,v)))
= ¬P(x)∨¬Q(z,y)∨¬R(y,v)
I can't quite understand. What's wrong with my own solution? And why here the zz is replaced with a constant instead of a function like g(x,y)g(x,y) or f(x,y)f(x,y) ?
I hope this doesn't sound too confused or chaotic because I'm writing this in a hurry...
For context:
In Models and Reality, Putnam tries to show that if we assume external realism to be true, the language-world relation will be indeterminate. He does so by using model theory of first-order logic. Very roughly, the argument goes like this:
Let T be a first-order theory that consists of the statements that can be made about the real world. T will include formulas that need an uncountable domain to be satisfied. But by the downward Löwenheim-Skolem Theorem, for every uncountable Model of a theory A, there is a countable Model for A. So, If we take T, there must be some countable model of T. He concludes, and I don't know from the top of my head how exactly, that language is indeterminate if we assume realism.
The Issue I am having:
If Putnam uses model theory for his argument, he implicitly uses some form of set theory, be it weaker than ZF or not. If we give formulas meaning in FOL, we need to do so by some operation like an assignment function. But clearly, we need to assume some sort of primitive reference relation to give meaning to things like functions and sets. And if we say, well, model theory can be built using first-order syntax, do we not again need some form of reference in our heads so that we know, what we are even doing when we manipulate and derive formulas?
For example: In my mathematical logic textbook, one of the axioms of the formal system, in which we can derive formulas syntactically in, is "x=x". So we assume the derivability of this formula. But we chose this formula because of how we think logic should work. We implicitly have some form of reference to know what it means for x=x to be true or else it couldn't have been chosen as an axiom for derivations. Of course, there are many ways how one could make a decent syntax for first-order logic but I cannot imagine one, that doesn't have some primitive intuitions built into it.
My question is this:
Do you happen to know any literature that puts this foundational issue of logic into better words than I have? Something that discusses how logic gets put together by human minds and that discusses which notions might just be irreducible in logic? Thank you in advance, I am really appreciating the help I've gotten on this sub so far.
Skolem's Paradox is the conjunction of the downward Lowenheim-Skolem theorem and Cantor's Theorem as applied to infinite sets.
>Skolem went on to explain why there was no contradiction. In the context of a specific model of set theory, the term "set" does not refer to an arbitrary set, but only to a set that is actually included in the model. The definition of countability requires that a certain one-to-one correspondence, which is itself a set, must exist. Thus it is possible to recognise that a particular set u is countable, but not countable in a particular model of set theory, because there is no set in the model that gives a one-to-one correspondence between u and the natural numbers in that model.
This is from the Wikipedia article on the paradox. So is the idea that the countable model thinks u is uncountable, but there is exists some model that thinks u is countable? If so, won't the paradox be generated anew in that new model because that model will have an uncountable set (per the model), but there is always some other model that thinks its countable?
If the above is correct, then this seems to imply that no set is uncountable in an absolute sense (i.e. in all models of ZFC). Is that correct?
If there are other resolutions to the paradox, I'm interested in hearing those as well.
In my textbook the downward Löwenheim-Skolem theorem is stated as follows:
If Σ ⊆ Sen(ℒ) has a model, then Σ has a model 𝒜 such that |A| ≤ |Sen(ℒ)|. In particular, if ℒ is countable, then Σ has a countable model.
Now, the definition for a Structure 𝒜 is stated as follows in my textbook:
An ℒ-structure 𝒜 for a language ℒ consists of
(i) a non-empty set A called the universe of 𝒜
(ii) an n-ary relation r^𝒜 ⊆ A^n for each n-ary r ∈ R(ℒ)
(iii) an n-ary function f^𝒜:A^n → A for each n-ary f ∈ F(ℒ)
My question is now:
What does it mean for a model to be countable? Or, for that matter, what does it mean for a structure to be countable? Does it mean that the Union of all the sets defined in my definition of a structure is countable? Does it mean that only the universe is countable? I can’t seem to find the answer in my textbook, maybe I’m blind lol.
My boss asked what I was doing with my measurements and calculator - I proudly showed her. The new bulbs are a perfect “X” in the dining room. Thank you to the math teachers of Lake High School. 🤘
Basically a theorem that says “all but some number of cases” satisfies the theorem
Howdy, y’all! First time poster here. :)
So, I learned about the FToA this semester, in my Pre-Calc class, but my textbook honestly did kind of a bad job explaining it. (Didn’t even have any sort of proof for it, for one thing.[Edit: Turned out to be good reasons for this, lmao.]) This video helped clear it up a bit for me, but I’m still having trouble understanding it. Like, what is it about this theorem, in particular, that makes it “the link between algebra and geometry” (to paraphrase that vid, iirc), what is is that makes this theorem in particular so important? I get why the Fundamental Theorem of Calculus gets the title, but idk, I feel like I’m just not quite getting or grasping something here, and not understanding why this is so important to algebra.
Edit: I appreciate all the responses! Y'all definitely have given me a better perspective on it, and what I should further study to better understand it.
It me, I'm the 8th grade math teacher. It's always bothered me that we can't easily "see" the solutions that the FTA tells us to expect for polynomial equations if those solutions are in the complex numbers rather than the real numbers. So when I introduce the FTA to students, I usually talk about it in terms of real solutions and how it tells us to expect up to some number of real solutions depending on the degree of the polynomial. This video is my attempt to connect that idea to the actual statement of the FTA relating polynomial degree to complex solutions, and to visualize where the real solutions "go" if we, for example, vertically shift a third-degree polynomial from three to two to one real solution.
Perhaps not at the level of most of the commenters here, but I'd love to know if I'm making any egregious missteps in how I discussed the math.
I also said at least one thing that, after the fact, I realized I wasn't really sure about. Would you say a mapping of a polynomial function's complex inputs to its complex outputs makes a two-dimensional surface in four-dimensional space?
As Omicron cases surge, I’ve seen people question how reliable COVID-19 tests are.
People often look at the Sensitivity or Specificity numbers, when in reality it doesn't give them the information they want: How likely is it that I don't have COVID?
Using Bayes Theorm, I took a stab at calculating how likely it is for an individual that tests negative to actually have COVID.
https://preview.redd.it/jwerh6sbjwb81.png?width=600&format=png&auto=webp&s=66838fb392e043029c30e69aa30dc580492c5114
This is my first time writing anything technical! So feel free to give me any feedback.
Edit: added a graph.
Why does the quadratic, cubic, and quartic formula work, while no quintic formula can exist?
Hey guys, as you have seen the title I am planning to do my IA on proving the fundamental theorem of calculus but I am skeptical as I don't think there's much personal engagement but I remember my teacher saying that since it is Math AA HL if the math rigor is there then personal engagement would just follow. I would appreciate it if anyone has any piece of advice or tips on how I might be able to add personal engagement to my IA. Thanks again :)
As I understand it, the CLT states that for a sufficiently large sample size "n", the sampling distribution of the mean from a given population will approximate a normal distribution. Also as the sample size grows, the better the mean approximation becomes, according also to the law of large numbers. What However, when watching a lecture about confidence intervals, the lecturer explains that an assumption when creating creating confidence intervals is that the population distribution is normally distributed, and if it is not "you should use a large enough sample and let the CLT do the normalization magic for you"; and then proceeds to work off of a single sample. In another video, a different lecturer creates a sampling distribution of the mean and states that we can create a confidence interval from the sampling distribution. So which one is correct and what is the difference. Is the sampling distribution explanation of the CLT only a "proof" of what the theorem dictates and extracting a single large sample is enough? Or am I missing something?
After reading and rereading the wiki I can't figure it out how to do it.
Also, is this the best idea, compared with grinding 49 mysteries and turning them into it?
In my textbook the downward Löwenheim-Skolem theorem is stated as follows:
If Σ ⊆ Sen(ℒ) has a model, then Σ has a model 𝒜 such that |A| ≤ |Sen(ℒ)|. In particular, if ℒ is countable, then Σ has a countable model.
Now, the definition for a Structure 𝒜 is stated as follows in my textbook:
An ℒ-structure 𝒜 for a language ℒ consists of
(i) a non-empty set A called the universe of 𝒜
(ii) an n-ary relation r^𝒜 ⊆ A^n for each n-ary r ∈ R(ℒ)
(iii) an n-ary function f^𝒜:A^n → A for each n-ary f ∈ F(ℒ)
My question is now:
What does it mean for a model to be countable? Or, for that matter, what does it mean for a structure to be countable? Does it mean that the Union of all the sets defined in my definition of a structure is countable? Does it mean that only the universe is countable? I can’t seem to find the answer in my textbook, maybe I’m blind lol.
