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Synopsis


Hypernaturals are an alternative model of arithmetic that uses constant sequences of the same number to create a new structure. Using an ultrafilter, an equivalence relation is added and truth is characterized by being true in most of the structures according to the ultrafilter. Wash's Theorem states that a formula is true in a model if it is true in a large set of interpretations, while Hypernatural Primes are numbers that satisfy a first order statement and are larger than the N-star. Consistency is the only way to determine whether a model reflects truth.

Short Summary


Ultrafilter is a model of arithmetic which uses hypernaturals, constant sequences of the same number, to create a new structure. An equivalence relation is added using an ultra filter, and truth in the new structure is characterized by being true in most of the structures according to the ultra filter. This non-standard model is called Star n. Addition works point-wise and if two Hyper Naturals are swapped out, the result is still the same. An example is given with two numbers A and B, where one is even and one is odd but it is unknown which is which.
Ultra filters are used to define ultra products, which are products of a collection of L structures. Interpreting constants and function symbols in a structure involves taking the equivalence class of sequences which are equivalent under a specified relation. For a constant symbol, this means interpreting it in all of the structures and making a sequence out of it. For a function symbol of arity n, this means applying it to n objects and taking the equivalence class of the resulting sequence. The interpretation of a relation symbol of arity n in this model is a truth value of 1 if the set of indices of the ith elements of each sequence are equal.
Wash's Theorem states that a formula is true in a model if and only if it is true in a large set of interpretations of the formula, determined by looking at the set of indices where the predicate holds of the element of the sequence within the ith structure. This theorem is used to determine the truth of a formula in a model. However, this theorem is incorrect as Phi of X bar is not necessarily a formula. Two options are proposed to interpret formulas in a metatheoretic theory: adding an absurd amount of constants to name every single object in the domain of the structure, or using valuations. Both options are unsatisfying as the point of the theorem is to characterize truth in the model.
This theorem states that the interpretation of a given L formula in a structure with a given valuation is true if and only if for all models in the ultra product, the formula is true. The proof is done by induction on the formula, and the atomic formula is true if the set of objects yields one. Negation, conjunction, disjunction, implication and universal quantification are all true if the associated index sets are large and in the ultra filter. The theorem can be specialized to finite and co-finite sets, allowing general claims to be made about any Ultra product.
Hyper Naturals are a model of arithmetic consisting of n copies of the natural numbers quotiented by an ultra filter. Each numeral in the language corresponds to an equivalence class of sequences, with the representative being the constant sequence of that number. The compactness theorem states that for any first order formula in the language of arithmetic, if it is true for one evaluation, it is true for all evaluations, demonstrating that first order logic can be used to prove statements in arithmetic. This contradicts the idea that the incompleteness theorem implies, as it shows that there is more weirdness in mathematics than the standard model allows for.
Incompleteness theorem states that first order logic cannot express all truths about the natural numbers, as induction must be used to prove a proposition for all natural numbers, which is impossible in a non-standard model. Completeness, however, gives us the hope of being able to prove all true statements, as it states that if a theory is consistent, then there is a model in which all of the statements in the theory are true. Even if one were to allow for uncomputable axioms, it is still not possible to categorize the models of arithmetic due to the restriction of first order reasoning.
First order logic cannot accurately describe real world events, so completeness must be given up. Hypernatural primes are numbers that satisfy a first order statement and are larger than the N-star, a model of true arithmetic. To find the spectrum of this object, the hypernaturals must be changed into a commutative ring by adding inverses. This theorem will then carry through, and any product of two hypernaturals that is equal to the hypernatural prime will satisfy that it is prime. Consistency is the only way to determine whether a model reflects truth.

Long Summary


A family of structures indexed by natural numbers is used to create a new structure, where objects are taken from each structure. An equivalence relation is added using an ultra filter, and truth in the new structure is characterized by being true in most of the structures according to the ultra filter. The set of natural numbers case and hyper Naturals are discussed informally, and then formally. The non-standard model created is called Star n.
Ultrafilter is a model of arithmetic which uses numerals like zero, successor of zero and so on. These numerals are interpreted as hypernaturals, which are constant sequences of the same number. The ultrafilter identifies hypernaturals which are mostly the same, even if they differ in finitely many positions. An example of a hypernatural which is not equal to any standard hypernatural is 0, 1, 2, 3, 4, 5, 6, and so on.
Hyper Naturals are exotic objects which can be interpreted using numerals. Addition works point-wise and if two Hyper Naturals are swapped out, the result is still the same. An example is given with two numbers A and B, where one is even and one is odd but it is unknown which is which. This is because the outcome depends on which set in the Ultra filter is designated as large, either the set of even numbers or the set of odd numbers.
Ultra filters are used to define ultra products, which are products of a collection of L structures. Two sequences are identified if they agree on a set that is in the ultra filter. This set is determined by whether the elements in the sequence and the indices behind the sequence are even or odd. Ultra filters contain the co-finite filter, which allows questions to be answered about the model.
Interpreting constants and function symbols in a structure involves taking the equivalence class of sequences which are equivalent under a specified relation. In the example of the hypernatural structure, all of the structures are the same, known as an ultra power. For a constant symbol, this means interpreting it in all of the structures and making a sequence out of it. For a function symbol of arity n, this means applying it to n objects and taking the equivalence class of the resulting sequence.
Interpreting a function symbol F in a structure MI, F is applied pairwise to each element of a sequence AI1 to AIN. Taking the equivalence class of this sequence, a representative of this class is formed. This representative is a set of n sequences, where the ith element of each sequence is selected. The interpretation of a relation symbol of arity n in this model is a truth value of 1 if the set of indices of the ith elements of each sequence are equal.
Wash's Theorem states that a formula is true in a model if and only if it is true in a large set of interpretations of the formula. This is determined by looking at the set of indices where the predicate holds of the element of the sequence within the ith structure. If the set of indices is co-finite, then the formula is true in the model. This theorem is used to determine the truth of a formula in a model.
Let Mi be a family of first-order language structures, each with an associated ultra filter on I. There is a theorem stating that an ultra product entails a relation symbol, such that a formula Phi of X bar is true in the structure Mi if and only if the set of all indices such that the formula is true is in the ultra filter. However, this theorem is incorrect as Phi of X bar is not necessarily a formula.
The speaker proposes two options for interpreting formulas in a metatheoretic theory. The first is to add an absurd amount of constants to name every single object in the domain of the structure, and then interpret the constant as the particular object. The second is to use valuations, which is more in line with the canonical story of logic. Both options are unsatisfying as the point of the theorem is to characterize truth in the model, and switching out the model every time is less satisfying.
In order to build theories, one must talk about valuations which are functions from the set of variables to the domain of the meta theory. This theorem states that for any L formula, Phi, the interpretation in this structure with valuation V is one if and only if for all MI with valuation VI, Phi is one. The proof is done by induction on the formula.
The atomic formula is true under an interpretation if and only if the set of objects yields one. This can be exchanged with a set being large. Negation is true in the ultra product if and only if the set of models at which the projected object has the property is not in the ultra filter. For conjunction, truth of both psi and theta means that the associated index sets are large and must be in the ultra filter for it to be true.
Ultra filters are sets of large sets which can be used to characterize truth in a model. The intersection property states that if two sets are large, then their intersection is also large. Using this property, cases for disjunction, implication, and universal quantification can be obtained by combining the previous connectives. This theorem can be specialized to apply to finite and co-finite sets, allowing general claims to be made about any Ultra product.
Hyper Naturals is a model of arithmetic consisting of n copies of the natural numbers quotiented by an ultra filter. Each numeral in the language corresponds to an equivalence class of sequences, with the representative being the constant sequence of that number. Valuations which send variables to things in hyper Naturals will split into Vis for each of the copies of n, and truth in this model is determined by a large set. Examples of hyper Naturals are the constant sequence of a natural number and the sequence of all the natural numbers.
The speaker discusses the concept of hyper natural numbers and how they can be used to create larger objects than the standard natural numbers. They explain that when the compactness theorem is used to introduce a new constant, it can create an object which is not equal to any of the named objects and is larger than all of them. This can be seen as an indication that there is more weirdness in mathematics than the standard model allows for.
The compactness theorem states that it is not possible to write a first order logic and semantics such that soundness and completeness are achieved. The theorem 2.0.6 states that for any first order formula in the language of arithmetic, if it is true for one evaluation, it is true for all evaluations. This contradicts the idea that the incompleteness theorem implies, as it shows that first order logic can be used to prove statements in arithmetic.
Incompleteness theorem states that it is impossible to axiomatize true arithmetic with computable axioms. However, even if one were to allow for uncomputable axioms, it is still not possible to categorize the models of arithmetic due to the restriction of first order reasoning. This means that even if one were to achieve the 'Holy Grail' of true arithmetic, the uncountable objects of star n would still be a model of the theorems.
Incompleteness theorem states that first order logic cannot express all truths about the natural numbers. To prove a proposition for all natural numbers, induction must be used. This is impossible in a non-standard model, as there is no way to categorize the natural numbers with first order formulas. To work around this, an additional object is added at infinity, so that everything is the successor of something. However, this still cannot be expressed in first order logic.
Completeness is an important concept in first order logic, as it states that if a theory is consistent, then there exists a model in which all of the statements in the theory are true. Without completeness, the truth of certain statements cannot be proven, as the non-standard model case allows for weird numbers which mess up the game and allow it to be true that a formula says there's a proof that there's no proof. Completeness gives us the hope of being able to prove all true statements, as it states that if a theory is consistent, then there is a model in which all of the statements in the theory are true.
Completeness and non-standard models of first order logic show that consistency is not sufficient to talk about real phenomena. This leads to a philosophical realization that a consistent system does not necessarily describe any real world events. To formalize sentences that cannot be expressed in first order logic, completeness must be given up. This puts the spotlight on consistency as the only way to determine whether a model reflects truth.
Hypernatural primes are numbers that satisfy a first order statement and are larger than the N-star, a model of true arithmetic. Euclid's proof of the infinitude of primes in an object is a hypernatural prime that is divisible by every hypernatural prime. To find the spectrum of this object, the hypernaturals must be changed into a commutative ring by adding inverses. This theorem will then carry through and there is an operation of multiplication on the hypernaturals. Any product of two hypernaturals that is equal to the hypernatural prime will satisfy that it is prime.
Hypernaturals are a type of number that are larger than all standard hypernaturals. A prime hypernatural is a special type of hypernatural that is larger than all standard hypernaturals, but not larger than all of them. This weirdness can only be identified by looking at the meta-theory, where we can see the standard hypernaturals and the weird prime hypernatural. The compactness theorem can be proved in a direct way, usually using an ultrafilter. This example is a good overview of the concept of hypernaturals.

Raw Transcript


um but the plan for today is okay we I introduced this Ultra filter Gadget last time and uh this is this lecture is the promised payoff of that where I make something cool with it um and so if I outline the plan for today what we're going to do is uh we're going to take an an indexed family of structures um and so suppose they're sort of indexed by natural numbers in one M2 M3 uh maybe there's infinitely many of them uh and I said structures and I guess for continuity with uh the the seminar so far we have been saying interpretations um I guess I'm going with structures because I'm it's sort of adhering to the notes that I'm deriving my notes from and also I will also be using I as the symbol for the index set so uh please tell me to re-explain notation if it's not clear what something is but yeah structure is just an interpretation so a a a domain and sort of ways to interpret the terms um in a given first sort of language and so what we're going to do is uh in order to make the promised uh non-standard model is what in general we can take an eye indexed family of structures and we're going to make a new structure which in a way projects into uh all of the all of these structures um so the objects are kind of like uh like an index sequence of objects so if you like you take an object from from structure one run from structure two one from structure three and so on um and the uh this won't work on its own what we need to do is uh we need to quotient this thing like put some kind of equivalence relation on these things um using the ultra filter and then we're going to characterize truth in the sort of big model we've created out of all of these um smaller structures uh we're going to characterize truth in that structure by kind of like in a vague way like being true in most of the structures where most is sort of according to the ultra filter um and then once we've done that I'm going to show you the the sort of the set of natural numbers case um and we'll talk about the hyper Naturals um and so well the easiest path there is to kind of just skip to the end uh and then we like and do it sort of uh informally and then I'll show you the formal way to do it um but hopefully this will give you a picture to have in your head as we go so I'm going to informally tell you about the hyper Naturals and so just like the picture above uh in in this example so this is the the sort of non-standard model I'm going to make it's going to be called star n and the kind of things that are in Star
n well uh the point of it is that it's supposed to be a model of arithmetic and so in particular in the language of arithmetic you have numerals which are like there's zero and there's the successor of zero and the successor of this successor of zero and so on and those are your numerals uh not one that you have the numeral zero there uh will can you remind me do we when we do when we write a numeral is is the line over it or below it oh let's pull it okay so this is the numeral zero means uh that symbol above and then numeral one is s Open Bracket zero close bracket and that's two and so on and so uh in order for this to be a model okay what is the object uh what is the hyper natural object which is which uh the those numerals are interpreted as well uh for example two is going to be uh it's going to be something like this it's just kind of a constant sequence of twos that's a canonical representation for the hyper natural two um but the whole point of the ultra filter is to now uh identify this with any other hyponatural which is uh mostly the same as this hyper natural so for example uh this is equivalent to say let me just replace finitely many of the uh of the the numbers in the sequence so say I just replaced a couple of those and then from then on it's all just twos so uh the idea here is that yeah as I said only finally many of these things aren't the same as uh the corresponding position well I guess I'll point out the ones that aren't the same it's this one and I guess past that it doesn't line up anymore um yeah in those positions they disagree but in co-finitely many positions they're exactly the same and so that's that's the equivalence relation on the hyper Naturals is you take your Ultra filter and like on the indexing set in in this case the indexing set is also just the natural numbers and you use it to identify hyper natural which are mostly the same um so so there there's your there's a standard hyper natural which is just the it's just an interpretation of one of the terms of the language but the whole point is now we have these extra exotic things which are not equal to like an interpretation of some numeral sorry the most obvious weird one I can come up with would be this number I'll call K which is just 0 1 2 3 4. 5 6 and so on um and so clearly this is not equal to any like standard hyper natural because it's if you take uh if you take any of these high of the of the sort of numeral hyponaturals or like anything it's equal to and compare
like pairwise the equality between the sort of normal Naturals in the sequence uh you will only have uh I guess finitely many or like a a code large amount of agreeance between them um and so this is a truly exotic um hyper natural um someone's got the microphone unmuted I think it's I think it's you then uh he might not be at his computer okay I'll just proceed um okay any questions about this already no not at the moment okay great so um just a I've shown you how to interpret like some numerals and maybe some uh uh one of the weird objects let me show you how Edition works so it works sort of Point wise so if I write out uh K plus 2 all I have to do is add them up in the corresponding position so this would be 2 3 . 45 6 7. and so on um and so the question you should ask is if I uh if I swapped out for like a hyper natural representation like this one that was equal to the first one and then did the addition do I end up with an equivalent thing and yes you do so in this case if I say use this thing um to add to to my number K instead of the other one uh I'd end up with two four five nine hundred and five and then it would just proceed 8 9 10 as normal and so on and so again it's it's only co-finitely many positions of this infinite sequence in which these two representations of this hypon natural are different so that makes them the same hyper natural um so beyond that uh there's like here's some examples where I can't say uh okay let me try these two numbers so one of them I'll call a which is just two three two three two three and so on and the other one is three two three two three two and so on [Music] and the kicker here is I don't know which of A and B is even um or odd but I know that exactly one of them is because if I um okay yeah I was talking about equality between objects but as I was saying before truth um of hyper Naturals is about Truth at most in most of the the sort of substructures and you can see exactly half the time in this hyponatural a you get an even number but it's at all of the even positions in the uh in the sequence and it's odd in all the odd positions of the sequence and so the question of whether or not uh a is an even number is going to boil is an even hyper natural is going to boil down to uh whether it's going to boil down to which sequence uh sorry which set in your Ultra filter you designate it to be large is it is it the set of even numbers as large or the set of odd numbers is large um because those are the sort of indices
that correspond to evenness or oddness in these numbers any any questions about that before we get into the performance stuff foreign but I think I just want to say the formal stuff and I think I will start clicking so I think you can just move on okay as they go part of the confusion is probably because there's two like evenness and oddness shows up in two different ways in both the members of the sequence and the indices behind the sequence but maybe that every third number even and the rest are odd or the uh yeah so again that will entirely depend on uh it'll depend on the ultra filter um which in some sense for particular sets especially the infinite ones it will be kind of impossible to answer that question in general because as I said it'll depend on the ultra filter but uh because of the ultra every Ultra filter contains the co-finite filter we can still sort of uh there's enough uh like for the for the questions which can be answered by oh the well the code finite set is in the ultra filter uh we'll be able to grasp those things about the the model um okay let's Press On Okay so definition 201 so we take some uh I indexed collection of alt of L structures so L is L first order language and then we take an ultra filter so this is my calligraphy you okay now we're defining the ultra products the ultra product is written like this it's written as a product over I of all the Mis and then we're quotienting well this is just the notation what is quotient by the ultra filter mean that's what I'm going to Define so first I have to tell you what the domain is so like the underlying set foreign and it's just the normal set product and I guess sorry I will freely abuse notation here by identifying the model Mi with the domain of the model like there's extra stuff that's in the model that's not part of the domain as in interpretations of other symbols but I just I'm just I just mean the domain here uh quotiented by yeah [Music] yeah uh where um so I when I just I'll write like circular uh sorry I mean round braces with some index when I mean indexed by uh capital I but I'll I'll just omit that part of it so two I'm going to call them sequences but like I don't know I could be uncountable if you wanted it to be um so we identified two sequences if and only f they agree on a large set and what I mean by that is this set uh such the AI is bi that set that kind of pre-image in a way uh is in the ultra filter you um and so for example on the last board I was identifying two different
representations of two by saying well I I sort of I changed uh I I like we start with the entire like uh okay this set that I've written down here in that example in the previous board where I identify the two versions of two of the hyper natural 2. uh the set of indexes such that they're equal as co-finite because there's only finally many of occurrences aren't equal and as I approved last time every Ultra filter contains the co-finite filter meaning it has all the co-finite sets so although I know nothing else about the ultra filter I still know that um you always Identify two sequences if they're only co-finitely different okay so next up I need to tell you how to interpret the constants function symbols and relation symbols so if C is a constant symbol uh okay this again I'm I'm sort of using the notation of the notes I got this from but we will probably write it a bit differently in previous seminars so when I when you put the model as the superscript um that's the the interpretation of this symbol uh in in that in in this uh in this structure and so uh to the interpretation of C is you take you interpret C in all of the Mis and you make a an i sequence out of that and then you take the equivalence class of all of those um of yeah the equivalence class of sequences which are equivalent under this relation uh to to that sequence um so for example we when you take the constant symbol zero uh I'll write for this in a different color if you take interpret zero in like the in the hyper natural uh structure as before you end up with uh actually I should write start in was equal to this thing because it's it's just interpreted as zero in every single structure I'll point out that here in the general case I'm allowing each of each structure am I to be a completely different structure um but the example we'll be interested in for hyper Naturals uh so yeah in the hyper natural case all of the structures are just the same one um that ends up being called an ultra power for obvious reasons okay now to interpret a function symbol so if f is a function symbol of arity n then can I fit this here I'll go to the next line f m so the function that the function symbol F is interpreted as in this structure m is the one which takes um n of these objects so I'm going to write it like this ai1 dot dot dot let me close that equivalence class up to a i n these eyes don't have anything to do with each other that's just sort of that's a sequence uh inside itself if you say if you apply a function
to those objects you end up with the object which is uh pairwise apply uh we're making a sequence of things and each of those is going to be we'll interpret F in the structure MI and apply it to uh the object ai1 up to a i n and here the i's are linked so it's like you're this is one I'm making one sequence of things here and I'm fixing I across all of these AIS applying uh the ice interpretation of f amongst the structures to it and I make a sequence out of that and then I take the equivalence class of all them so the canonical example of this is what I did on the previous board with adding K and 2 is I'm sort of pair wise I'm interpreting the function symbol Plus as the regular Plus in the standard model of natural numbers where you just add up the numbers as normal and then yeah applying that pairwise to each element of the sequence and then taking the equivalence class of the result resulting thing I end up with is that clear I hope so I think so but I don't understand the notation but I feel like I'm just I'm not getting there sorry this AI one is a sequence and then you're taking an equivalence class of that sequence which is fine so you've got your father sequence of equivalence classes of sequences and then you're taking a representative of H of these equivalents classes so you've got n sequences in your hand and then how do you form ai1 after a i n so from uh you fix I and then you go to each of those sequences and pick the if element of this of each sequence and you pull that one it doesn't make sense so the ith element of the sequence so ai1 is a sequence uh sorry I'm just trying to avoid too much notation about me that's hurting it ai1 if I write it with brackets like that that's a sequence where um like it it's I'm just uh I'm just neglecting to write this to write I and I sorry so in this when I write this notation oh I is not a free variable there oh okay so a i J yeah is the if element of sequence J um a i j is the ith element yeah actually yeah yeah but the eyes as they appear here are kind of they're a captured variable that I'm just not writing the the extra bit here for um I can write oh okay let's see I can write this here yeah does that make sense yeah that's fine cool okay now relations for ah is a relation symbol of arity n then the interpretation of ah in this big model is same as before that's equal to one like the the truth value one if and only if uh similar thing to the equality thing so a set of all indices such that when you look at the ith
structure um an interpret r and then apply that to the ith term or the ith object of all the sequences that's equal to one a large amount of the time so that's how I determined that uh I guess so if this set if we were talking about the evenness of of a from the previous board uh if you apply the sort of evenness predicate um well I guess this is strictly about relation symbols but it's going to work similarly for like any formula when you apply the evenness predicate you look at this uh set of indices where you have a where like the the predicate holds of the element of the sequence within the ith structure um when you construct a set of the all those indices if you end up with a large set according to the ultra filter then that ends up being that's that means truth that that means it's true in the in this model that makes sense so that's what you were saying earlier that whether two three two three two three is even odd depends on the choice of ultra filter yeah yeah okay but if you end up with a co-finite set as as you there's the sort of amount of sets where you have where your predicate or your formula is true um in each of the iPhone structures then then there's no question of which Ultra filter you're using because yeah the finance that is guaranteed to be in the ultra filter yeah that makes sense all right cool new board making progress um Okay so uh in the notes I was following and in this lecture it's left as an exercise to show that the equivalence relation all that uh yeah sure that defines an equivalence relation and that uh the interpretations of times and uh relation symbols is well defined so this is just talking about that um when we when we when we Define when we Define the interpretation we would sort of taking a ref a representative amongst the equivalence class of sequences um so it's it's fairly straightforward to verify that but um but I won't be doing that now so um the theorem is uh this is uh I meant to look up the translation again sorry not the the pronunciation but I believe it's washes theorem I'm gonna say like that um spelled with fancy L and O and then a s with an accent and then I believe that's what I would have said and it makes absolutely nothing and that's what I would have said um I think there's an apostrophe I guess because it's an S like it's oh man that's quite confusing washes theorem right I don't know is that a nurse yeah I'm gonna say it's an ass okay so cool I'm gonna write this theorem which is essentially lifting the relation
symbol definition to a theorem about any formula um and what's going to happen is I'm going to write something that's actually nonsense and then I'm going to say why it's nonsense and then we're going to fix it um so look out for that so uh actually I want to write this I was just going to write let Mi be a i indexed family of L structures where L is your first order language um and then again U and Ultra filter on I so that's the same as in our previous definition so with that out of the way um let Phi of X bar which is just again notation for like a tuple of things uh and this is in this case it's a triple of variables I'm just using the notation from the notes and then I'll I'll give my sort of modifications for it after let 5x be a first order uh formula in the free variables hex bar and uh let's uh AI bracket bracket bar be a tuple of objects from the ultra product so let me write that so here um again I is is a captured variable just it's just about the the sequence AI inside this thing and the bar is denoting a triple a finite Tuple of things so the I has nothing to do with the triple um to let that be Drupal of objects from the ultra product s well it seems but two different locations for keyboards inside the same thing but there's a sequence Ai and then you're taking a tuple of equivalence classes of sequences machine rotating with a bar yeah is that a finite people what can it be arbitrary so it's necessarily fine okay um yeah I'll I didn't sort of try and massage this notation too much because I end up writing it differently in the next bit uh let me just get through this so am I where should you so from so some Tuple of objects from that Ultra product so I I just I could have just said like I could have just listed them out and instead of saying troop level of objects but whatever um all right here's the theorem uh so this Ultra products um entails this thing or in other words this is true in this structure if and only if this set of all indices such that the such that this formula is true in the structure MI is in the ultra filter um so again just compare this with the definition of the relation symbol so it's all it's doing is just lifting that to like take any formula um but okay here's the problem I mean can anyone point it out yet as to what's sort of nonsense about this I mean that's exactly equivalence class but I feel like that's not it no so the problem is that uh this is not a formula like this is not necessarily all right this part is kind
of nonsense and so is this part because these things are not these things are not like symbols uh like I'm supposed when I say a structure or an interpretation uh makes this formula true or entails this formula it's supposed to be a formula but what I've done is I've like sort of started with my language I've said all right here's some structures with domains um which like interpret theories uh or interpret formulas over that language um and here's some objects like in in the domain of those structures and then I'm and then I'm uh I'm you I'm using those objects as if they're already part of the language but they're not um so it's it's strictly they're not a uh they're not formulas so like unless you already believe in some kind of sort of mixing together of your like metatherian Theory uh I think this is sort of in bad taste and there's a sort of more correct way to communicate this idea that kind of properly uh like describes the kind of spirit of doing versatile logic um yeah yeah I see a point but you basically you can just replace the thing on the interior with a constant symbol or something and then interpret that constant symbol AS this particular object right like that's that's the intention behind this notation right but so I I so well I was gonna list like two options one in which uh when you state washes theorem you first like you take your your index family of structures and then you like add an absurd amount of like constants each to name every single object um and then like and then I can actually write this and it actually makes uh I guess it sort of counts um or I guess you were saying like on a uh on a case-by-case basis per object like decide to or like like I guess you augment the language once put your constant symbol there and say all right this time I'm going to like like because then this time I'm going to interpret the constant as this object um but I guess I know that's a little unsatisfying because of the fact that like the point of watch this theorem is to characterize truth in this model but if you're like switching out the model every time for like a slightly different one I think that's less satisfying um and I think you can do something else to sort of facilitate that uh which is my the second object the second option I wrote down which is just using valuations um to do this uh because it's kind of more in line with the canonical story of all right first of all the logic you start with some some cannibal decidable collection of things that you're gonna use as your
symbols you build theories like name the things you need to name and then uh then you do semantics where as The Meta Theory you talk about how your the syntax of personal logic and the the your first order Theory how it relates to the the sort of real objects in your in your meta Theory um sorry you can't do semantics without the meta Theory so let the meta Theory do its job rather than like just shoving like chunks of the meta Theory into like the first oral language I think is a really gross way of doing things yeah I agree yeah sure so let me let me restate it which is I think it's a cleaner way of doing things um so how you do it is you talk about valuations which I haven't really done yet so just to remind you uh I'm going to take evaluation for each of the each of the Mis and I know in the notes it's a Greek letter Nu but I can't just I just can't bring my brain to recognize it as anything other than the letter V so I'm just going to say V um so VI is a valuation which which is a function from the set of variables to the domain of MI so take that I indexed family evaluations uh and then the whole valuation is the one which takes uh for any variable into the domain of the ultra products which sends a variable X to you just pairwise well not powers index-wise uh apply each valuation and then you get a sequence indexed by I and then you take the equivalence class of those sequences so um okay let me just continue and state the theorem well any uh for any what is it L formula uh Phi let me use VAR Phi here curly file whatever it's called and then in the notation we've been using in the seminars you sort of subscript with the valuation so interpretation in this structure with valuation V of this formula is one if and only if again m i with valuation v i sorry for all the subscripts uh of Phi is one the set of all those I's is a large set sorry here you are letting the medicine Theory do its job in relating Theory objects to your formula and yeah the the idea is you want to say the formula about the objects and uh you letting you're letting the meta Theory do it by uh whenever a variable shows up in Phi the valuation VI gets applied to it um yeah okay so that's that's my restatement of it um we can talk about it more later but I need to press on for now so join me up here okay so the proof is by induction here let me try and do it as timely manner as I can so proof uh by induction on the formula so in our base case uh yeah or I'll write it up so I won't write out the surrounding text
I'll just write down write down the meat of it so if you've got an atomic formula that's just relation symbol with a bunch of terms and you apply Envy of that formula that thing is one if and only if uh interpretation of I applied to all the objects yields one that's just normal threshold logic definition and then by the definition of the model we can exchange this with a set being large uh so the I uh model MI of r applied to the iPhone model and my with VI of T1 and the same thing with TN uh set of all I's such that that's equal to one is Lodge um and that's by yeah the definition from before but this is just exactly what it means to say I'm just rewriting inside the set here that's just uh the interpretation of this formula in the ith model and that's exactly what we want for that inductive case um I won't do the equality because uh it's a bit too similar to the previous one but the next one is kind of a critical moment so uh let me just excuse that case uh okay now if Phi is a negation then we have uh not PSI is true in in the ultra product with the valuation V if and only if um Phi is false um but that by definition all by the inductive hypothesis means it's not the case that uh this set is large meaning it's small am I VI of PSI so the set of models at which the projected objects object is has uh has the property Phi PSI I guess um is not in the ultra filter but that's the same thing as if that's not in the ultra filter then the complement is in the ultra filter that's this is where we use the the ultra filter Axiom so not equal to one I equal to zero so this set must be in the ultra filter by the ultra filter condition um and then that's the same as Mi VI of not PSI U for one uh wait oh yeah in the Ultra filter and that's exactly what we wanted um did you follow that I imagine that's exactly what you're going to use the ultra filter so it makes sense no surprises negation this complement um okay next one is where we'll you can make it an equally educated guess as to when the what we will do with conjunction uh so truth of Psy and Theta well that means that size true and Theta is also true and that's the case uh that's the case e uh is by the inductive hypothesis that the associated index sets that I've written analogous versions of already is large and I'll just sort of skip ahead to ah I'll just write it out in my VI PSI equals one is in u and my iPad is so hot it's like I'm writing on a barbecue yeah and I did that every time I like shoot as well actually I probably should turn down the
quality hadn't actually done that all right thanks um no I'm just bend my hand instead yeah so the birthday sets are large and okay I can explain both of these directions of this upcoming uh equivalence it's the same as the intersection being uh being in the ultra filter foreign and so the forward direction is just applying the the intersection Axiom for filters if you have two large sets you take their intersection you end up with a large set and then the other direction is the superset Axiom so if the intersection is in the filter then uh the two sets that you intersected they're both supersets of that intersection so they're also large they're also in the filter um okay and then you just proceed uh you squash those two sets together and I'll just skip to this part where it's m i v i PSI and theta equals one um and again that's exactly what we need um okay there's an ex there's a case for exists X PSI [Music] um I will skip it because there's some like it takes some thinking uh but I don't think anything interesting really happens um like there's not much Ultra filter interaction uh yeah all right sorry I'll skip that one and try and on some more interesting notes but um uh but then after I do the uh I have to walk my iPad character over close to the board because it rendered out um after I do this case uh then I claimed the cases for disjunction implication and uh Universal quantification can be obtained foreign to uh formulas using only the previous connectives and that's that's the proof by induction any questions about that that makes sense gee I guess maybe the question to ask is do you believe it that um that this is maybe you don't have a solid sense of it but that uh truth in this model is indeed characterized by truth at most of the the projected the projection models yeah as long as we just as long as we understand most to be like as in southern Ultra filter I'm still not sure how I feel about the intersection property but module like that I believe yeah as I said last time like Lodge kind of like becomes weirdly opinionated at the sort of infinite sets which are not co-finite uh yeah territory but it uh it makes sense for the the finite and co-finite sets and that's where we will be sort of making all of our general claims about any Ultra product um okay so a corollary of this um I'm gonna try and make it uh specialize that theorem to make it a bit more palatable look at the old cam move was it still over on the other two bullets and now it's moved
so a corollary is um or let me grab my notes so let's go super specialized light l be the language of arithmetic and so star n this is our hyper Naturals model is is the is this ultra power so I'm going to write it with like a sort of superscript power notation so it's n to the power of n quotiented by the ultra filter which written in the ultra product notation is just the product over all natural eyes of just a bunch of copies of the natural numbers quotiented by Ultra filter so that's an Ultra power of n uh I was going to say ultra power of n copies of n but it's just an ultra power event consisting of n copies of N I guess question to um and that's for some Ultra filter Uh u and these are called the hyper Naturals so you can obviously criticize the usage of the there because it's sort of dependent on which Ultra filter you pick but again we can still say things about the we can still make co-finite sort of statements about hyper Naturals that are truth regardless of what Ultra filter you take and then so again for any valuation V which sends variables to things in to hyponaturals uh which splits into Vis for each of the copies of n um and any formula Phi uh you have truth in this model with a value with that valuation of Phi if and only if large set as in truth throughout and in most of the most of the uh models n um and so as in so because like you've got a different valuation VI in each particular pretending depending on I you might get a different uh a different you might be talking about a different uh natural number at each model even though they're just copies of the same model uh I said that that has to be a large set so um let's I want to now talk about some examples of hyper Naturals and we can sort of I'm sort of returning to what I was saying at the start and we're going to say um say some interesting things about them okay so uh I was gonna write so each numeral uh I won't write this each numeral in the language corresponds to uh like the secret the equivalence class of sequences uh and the representative is just you take just uh it's just the constant sequence of that normal number so maybe I should just write it uh when you interpret with any valuation uh the the numeral n so maybe I'll write it like this uh and there's n S's uh that thing is is this equivalence class of sequences which are equivalent to just this constant sequence [Music] um so that's your standard hyper Naturals um and then as I mentioned before there's k which is uh just the sequence of all the natural
numbers in order that thing is larger than any standard hyper natural um and so this is the point at which like you you can clearly see that like okay obviously when you have like sequences um just sequences of copies of n uh you get you get way more um elements than like just your accountable um standard model event but maybe when you take the quotient by the ultra filter like all of that weirdness goes away and you kind of just get an isomorphic thing back again um this is your sort of first indication that like there is genuinely way more stuffier and way more weird stuff that's not like the natural numbers um in a certain way uh so let me let me prove what I just said is it's just the observation that uh the set of indices such that uh I guess this is a bad notation but let's say k is sort of identified with that representation I wrote down well maybe I shouldn't say the representation I should just say such that I is greater than n which is the which is the the standard hyper natural projection here that's it is equal to say n plus 1 and plus two and so on um which is co-finite so I have a I have a object in my interpretation of uh yeah I have an um an object in my structure which is larger than all the things named by a numeral in the language of arithmetic um and so when you do things like when you do things with the compactness theorem where you uh you force there to be more objects uh I'm sorry I guess what I mean sort of more in the spirit of the lowenheim scholar theorem where you force there to be more objects by sort of introducing a constant a new constant and saying it's not equal to all the other uh to all the other like named natural numbers or uh yeah if you if you do that like this could be the sort of new object you have in mind this is a number that that's uh who's this is an object in the model who which is not equal to any of the named objects um and it's also larger than all of those things um any questions about that example I've got some more but same no matter what you interpret zero as there's no number of times you can apply the successor to it to get this object K that's the point right uh yeah yeah so it's yeah it's not equal to any of the because applying the successor just means like you just get one of these standard hyper Naturals but it's never equal to this because uh you always have a co-finite amount of uh places where it's not equal um is a dumb question let me continue on and do it some more can you just say from the beginning that
actually my axioms for the natural numbers are that everything is a successor of zero or a success yes sir so that's that's a key point is like kind of the whole like a large Point here is that it's uh not possible to do that uh as in okay maybe you decide that you can do that in your formalism um but because of what I don't need to talk about why can't I say that all right because then what you're doing is not a first order thing uh you will and you'll end up violating the compactness theorem and therefore it will not be possible to uh write down a logic and a semantic such that you have soundness and completeness hmm sorry it's it's all a game with respect to that so like we would really like to like okay uh maybe so of course maybe like whatever there's no rules you can you can say everything is the successor of some as a finite amount of successes of zero but yeah then you're not playing the first order game anymore so how rude yeah fun I said how rude yeah so so I I guess a a core point I I want to get to is is well with any of these seminars is kind of on first or logic is what what can you fit into versatile logic like plainly what are the sort of limits of first order logic uh in what you can get it to do and uh what you just said sort of uh making a making the the the first order theory of uh arithmetic is not categorical is that is the way to say that because it's exactly this in that there's no way to follow all the first order rules um such that you end up with only a one uh model of arithmetic up to isomorphism um actually I I I've got some cool more hyper Naturals but let me just State the the sort of theorem that kind of blew my mind and made me want to uh talk about this is uh I guess I I see this sort of next to the incompleteness theorem as sort of an implication of that something uh it sort of contradicts something that I thought was kind of implied in the in the incompleteness theorem um which is let me talk about all right let me just write a theorem and go back into into boring details mode uh one moment Okay so ethereum 2.0.6 for every first order formula Phi in the language of arithmetic uh starting Oh no just just n entails Phi if and only if star n entails Phi um this is this is just sort of uh it's the same as the previous theorem but I'm I'm just uh the premise in in both directions is that well this entails this symbol is just saying for any evaluation so pick any evaluation you like the premises that uh the formula is true with that valuation
in this model and then you apply the previous theorem to to get the same fact about truth in this model for any valuation so there's some details in figuring that out and then you can also do the same thing in the other direction um and so a corollary of this I'll come back to maybe make you not want to believe the corollary and then you can scrutinize the theorem um corollary of this uh this is the weird thing uh star n is a model of this set the theorems of n which is the set of all the Phi such that n entails Phi which is also called uh true arithmetic so let me rerun for a second uh I guess the What will what will uh was thinking about with the incompleteness theorem is okay let me remind myself that the point of it was okay we really want to axiomatize piano arithmetic and uh the point was we cannot get a uh we cannot make a complete theory of arithmetic if we restrict ourselves to computable axioms so if the axioms of the theory are computable then there is no way to make it such that we can prove all of the formulas in this set um does do you agree with that will does that make sense yeah yeah that's what I was saying so in that characterization sort of this set was kind of like the Holy Grail it's like okay we really want to be able to prove all the things that are true about n um because that's that's what n is all about right it's true arithmetic but we can also we can only ever achieve with computable axioms the sort of like a part of true arithmetic and um so in in my mind this made True arithmetic the the Holy Grail it's really the the like this is the thing to achieve but this theorem here is saying like well actually like even if you got there it's like you haven't categorized the models of arithmetic you've um you've got all the like it like the the whole thing about non-standard models is not about like like non-standard models don't go away if you even if you allow yourself to have uncomputable axioms you know all the axioms you want um it's first sort of logic itself as in the Restriction to only reasoning in a first order way where you can apply selected predicates to uh like quantifications over objects that is a limitation in itself in actually categorizing the models of arithmetic but why should I be offended by that what's what's wrong with star n being a model of the theorems event because well it has uncountably many objects um I don't know is that is that enough for you or not well not really because couldn't I make lots of unaccountable things
models of the natural numbers like it doesn't where's the search activity requirement right I mean like I can just interpret the I can interpret first order or something yeah so so when you say interpret the natural numbers as blah blah blah I think in your head you're already assuming it interpret the piano axioms um as in like like there's clearly they like say a limited number of things um so like and yeah so so they say a limited number of things about the actual natural numbers but like if we're talking about like okay the the real natural numbers that maybe like we can't ex we don't know like until this theorem whether or not you can sort of say all the things about it to identify it with first order formulas but the the real natural numbers themselves like that model um the point is there's no way to sort of to categorize just that with first order formulas yeah that's what you mean so what about induction so so would it be maybe I'm off track here but so is it true then in these in this non-standard model that if you can prove a proposition four zero and that p n implies p n plus 1 then it's true for everything even though not everything is of the form successor of successor of successor of zero that's crazy crazy yeah so that's insane yeah I don't believe it okay so the it's because there's no uh like um yeah like you to imply like you want to imply the induction Axiom to like like the formula which says like is the successor of something um but uh we'll uh you know yeah the point is you uh you don't get anywhere I think like maybe a different non-standard model is like I think I don't know it super well but you kind of just have all the natural numbers and so on and then way out at Infinity you just like Chuck in another object a and I think you can set it up so that you have a and b where B is the successor of a and a is the successor of B and so everything is the successor of something because all you can like you want to say for all X uh ex like you want to say like x equals zero or uh exists y Y is equal to the accessor of zero wait not zero of x sorry um yeah foreign [Music] because like a is is the successor of B and B is the successor of a like there's no like you you're the only thing you can hope to do is sort of quantify over like a number of successes applied to like X uh or or to zero or something and you can't express that in first order logic and this is exactly where um where the incompleteness theorem where like uh let me remind myself it's it's
there's a you've got this formula which says of itself that it isn't provable um and you can't prove that because there is a model in which there supposedly is a proof that it's not provable um and what actually happens in that model is uh when you're you're talking about coding encodings like girdling encodings of formulas uh and you have some kind of Axiom or some formula that sort of asserts like the sort of like this is a proof tree that has these uh these formulas as the premises and this formula is the conclusion and all the deductions are like valid along the way um that formula is expressed by like talking about numbers but there's nowhere to say hey I when I say for all X or when I say exists x i I just want to be talking about the numerals the sort of named numbers um there's no way to do that like you're every time that like in the the non-standard model case there's always the the non-standard model case where like you could interpret your numbers in a weird way so that like you have these infinite numbers that I don't know maybe so they're sort of sort of correspond to some infinite proof or something or like it's just the the weird numbers that have this weird successor relationship between them they they mess up the game and they allow you to have uh they they allow it to be true that this formula says that there's a proof that there's no proof of G um and there actually is no proof because uh the actual proofs are sort of correspond to like the actual encodings the murals yeah wow that's fascinating so yeah so why do people do first order logic why not go into second order logic where you can make these statements of like and that's all that there is yeah and then goodbye completeness um that's that's basically why um why why do we need completeness so badly um if we don't have completeness we're not we're not touching the transcendental man we're like puny beings pushing symbols around no but like we have soundness so every time we do that it still reflects truth it's just we don't have the other statement everything of that internal models is provable so it might be like things that are true out there that we can't prove well but at least we can only restrict ourselves to the yeah so what what completeness gives you that if you have a consistent Theory then there is a model that's the sort of statement of completeness that sort of is in line with what Dan was just saying and like like completeness is the sort of is that is the sort of formalist is the is the only hope for
like formalism where like like formalism is just like oh I'm just pushing symbols around but like and and the sort of justification for doing your formalism is well I know this set of sentences is consistent or at least I think it is and completeness then sort of justifies that by saying well if you have full if you have syntactic consistency then there is a is a model that actually uh yeah but I still feel like that you don't have to be so attached to that I mean that kind of sounds like Hilbert's definition of existence or whatever where if it is described in some consistent system then it exists right you could abandon that and you can say consistency isn't sufficient for me to believe that we're talking about something so I need to have a consistent system and I need a proof that the model exists of that consistent system and then what have I do inside that formal system reflects truth inside at least one model yes I'll say that like the study of second order logic exists so people still do it and talk about it so that like I can't just just say that like okay this uh it's not complete so like there's nothing to say there um but like losing the the different like it's just that you lose completeness and if you're interested in having completeness then you don't get to uh you don't get to leave first order logic um in this yeah you see because what I'm taking from this like the non-standard models the Googles and completeness they're um it feels like genuine flaws in first order logic which can be described as you can't say the word only right it's like there's a the only things in the model that I want to talk about like I have the numerals and I have the numerals only kind of thing um and that seems like this recurring the thing that kind of blows our mind like we sit there and go like wow that's so mind-blowing that's crazy like blah blah but it's because we don't realize that we have that restriction and we don't like swallow that so once you've swallowed that and you're like okay we really want to be able to formalize that sentence but we can't do that so we need a system that can and what that means is that we need to give up completeness so that puts completeness onto the spotlight but then you realize oh maybe the Deep realization is that consistency is not sufficient for realness existence a consistent system does not necessarily talk about any real phenomena and that one seem to me like a reasonable philosophical take from this and girls and completeness
that's my immediate reaction at least but yeah I'll think about it it's really interesting this is this was a good choice of topic thanks for doing this no worries foreign like one or two more weird hyper Naturals so you have unfortunately I need to go I think I need to run after something but I can check the notes or the recording or something okay see what thanks for coming all right see you guys okay so you've got the the sequence of all the Primes that's also Prime uh it satisfies is prime they're sort of yeah the the first order statement is prime uh and also because of what I just read that that the N star is a model of true arithmetic uh and sorry star n also satisfies for Rolex exists y is greater than x and uh is I see sorry this is a hyper natural Prime but there's also infinitely many uh hyper Naturals which are larger than that Prime and also Prime that's crazy uh then there's also this one there's two two times three two times three times five so this is uh Euclid's proof of the if infinitude of primes in an object um it's this hyper natural is divisible by every uh hyper natural Prime this is like uh it's the product of all standard hyper Naturals I guess you could say it like that um this is pretty but yeah yeah this I mean this has a very geometric flavor to me so what is the Spectra of this thing I mean I can do Zed right uh all right so I don't actually know what what's what a spectrum I say all right so you just take you just take all the prime ideals of a ring so if I take if I take this the hyper Naturals and uh change it from a semi monoid from a monoid into it into a ring by adding inverses so I'll add negations um then that's going to be a commutative ring and the spectrum of a commutative ring is the set of all prime ideals I don't know if this correspondence are in Prime ideals and primes still works uh um so if I say I mean if you can go via the this theorem here then and do it all with the first order statement then it will carry through yeah but I mean is there an is there an operation of multiplication on the hyper Naturals yeah yeah it's just as usual pairwise yeah I I guess I'll you can um yeah so this actually let me just erase uh I'll go to the next one um I don't know if this helps but I I said this was Prime um and it's because take any product of two hyper Naturals that it's that is equal to this um so this is equal to this times something uh so I'm saying it's Prime so I'm saying one of these things is is p is all is is this hyper natural
Prime and one of them is the number one it's like equal to the numeral one and what you do is okay well maybe like it's either two one or one two there and maybe three's there one's there maybe one's their fives there and then one's there five's there um um and so on you make all those choices depending how you do it um well you've got two complementary indexing sets where like so either right either the uh this one of these is equal to this numeral and the other one is equal to two three five seven so on yeah um so that's why that thing is prime um so I I guess actually this didn't that didn't really lead into the point I wanted to make but uh yeah so the the weirdness is so uh I I stated a lot of this sort of out loud and even in the notes I kind of just said it informally because the whole point is you can't sort of describe this relationship as in like this hyper natural is larger than all the standard hyper Naturals I can't say that like by talking about a first order formula because I can't sort of with first of all ordered with a festival formula say hyper natural or say standard hyper natural that's a really iterating a point already made and so like uh the the number K or or the or this this weird Prime hyper natural is larger than every standard hyper natural but it's not larger than all of them so all all the logic about the normal Naturals about like there's no there's no number there's no natural which is larger than all other Naturals the the that corresponds to there's no hyper natural which is larger than all hyper Naturals um and so the weird like you can only identify the weirdness from The Meta Theory where you can sort of see these are the naturals these are the like these are the standard hyponaturals these are the weird hypernaturals yeah this is a very instructive example I mean the hyper Naturals themselves not this particular Prime thing yeah so yeah there's the hyper reels too just do the same thing sequences of reels or something like some Ultra filter um and yeah that so so the the second the second reference in the notes is where I sort of what it really inspired me because it's a really good overview there's a sort of series of blog posts about it and they they go further and sort of talking about um like they have a couple more examples I still these sort of prime examples Prime hyper natural examples from there um but yeah you can also go on to prove the compactness theorem in a direct way like normally compactness theorem is like all