WEBVTT 00:00:00.000 --> 00:00:02.839 Welcome to the deep dive. Today, we're stepping 00:00:02.839 --> 00:00:04.960 away from the kind of mathematics that deals 00:00:04.960 --> 00:00:07.120 with calculating everyday answers. Yeah, we're 00:00:07.120 --> 00:00:08.900 getting into the architecture. Exactly. We're 00:00:08.900 --> 00:00:10.220 stepping into the architecture of mathematics 00:00:10.220 --> 00:00:13.060 itself. We're going to strip logical systems 00:00:13.060 --> 00:00:15.919 down to their bare studs to examine the absolute 00:00:15.919 --> 00:00:18.899 limits of what can exist. It's a heavy topic. 00:00:19.079 --> 00:00:21.379 It really is. If you've been looking to understand 00:00:21.379 --> 00:00:24.199 the Loewenheim number, abstract logic, infinity, 00:00:24.359 --> 00:00:28.339 or non -constructive mathematical proofs, consider 00:00:28.339 --> 00:00:32.020 this deep dive your shortcut. Our mission today, 00:00:32.200 --> 00:00:34.960 based on a fascinating Wikipedia article, is 00:00:34.960 --> 00:00:38.159 to explore the boundaries of model theory. Right, 00:00:38.240 --> 00:00:40.740 and the foundational rules that dictate exactly 00:00:40.740 --> 00:00:43.280 how large a mathematical universe needs to be 00:00:43.280 --> 00:00:46.820 to satisfy a given logic. So, to start us off, 00:00:46.939 --> 00:00:49.259 how should we visualize this? Well, imagine a 00:00:49.259 --> 00:00:51.979 visual backdrop for our conversation. Picture 00:00:51.979 --> 00:00:54.600 a sprawling illuminated star chart, but instead 00:00:54.600 --> 00:00:57.969 of constellations, the void is... It's layered 00:00:57.969 --> 00:01:00.649 with complex mathematical symbols. Expanding 00:01:00.649 --> 00:01:03.909 endlessly. Endlessly into the dark. Yeah. That 00:01:03.909 --> 00:01:06.209 is the vast abstract scale we're dealing with 00:01:06.209 --> 00:01:08.010 today. We're not just talking about infinity. 00:01:08.370 --> 00:01:11.930 We are talking about finding the ceiling of infinity 00:01:11.930 --> 00:01:15.510 within higher order infinitary logic. Okay, let's 00:01:15.510 --> 00:01:17.930 unpack this. Because before we even get to Leopold 00:01:17.930 --> 00:01:19.689 Loewenheim and the number that bears his name, 00:01:19.810 --> 00:01:22.469 we have to define the environment this concept 00:01:22.469 --> 00:01:25.000 actually lives in. The abstract logic. Right. 00:01:25.060 --> 00:01:27.640 It operates within an abstract logic. Yeah. And 00:01:27.640 --> 00:01:31.040 according to the text, an abstract logic is built 00:01:31.040 --> 00:01:33.480 on three core components. I like to think of 00:01:33.480 --> 00:01:35.680 it like building a miniature world or maybe setting 00:01:35.680 --> 00:01:38.159 up a really complex board game. I like that analogy. 00:01:38.400 --> 00:01:41.200 Yeah. So you have your source code, essentially. 00:01:41.459 --> 00:01:43.659 The first component is a collection of sentences. 00:01:44.319 --> 00:01:46.659 Which, in your board game analogy, those are 00:01:46.659 --> 00:01:48.659 the rules, the axiomatic statements that govern 00:01:48.659 --> 00:01:51.439 the play. Exactly. The rules of the game. Then 00:01:51.439 --> 00:01:53.319 the second component is a collection of models. 00:01:53.560 --> 00:01:56.200 Each model is assigned a specific cardinality 00:01:56.200 --> 00:01:59.340 or size. Right. These models are the actual worlds, 00:01:59.359 --> 00:02:01.200 the different shaped boards where these rules 00:02:01.200 --> 00:02:03.540 are tested. The physical spaces where the game 00:02:03.540 --> 00:02:06.340 happens. Yeah. And the third component is the 00:02:06.340 --> 00:02:08.639 bridge between them, the satisfaction relation. 00:02:08.939 --> 00:02:12.719 Which determines whether a specific model successfully 00:02:12.719 --> 00:02:15.620 satisfies a specific sentence. Right, like checking 00:02:15.620 --> 00:02:17.939 if a move on the board actually follows the rules 00:02:17.939 --> 00:02:20.659 in the rulebook. If the environment you're analyzing 00:02:20.659 --> 00:02:23.780 has those three things, sentences, models with 00:02:23.780 --> 00:02:26.419 cardinalities, and a satisfaction relationate, 00:02:26.460 --> 00:02:29.360 you are officially working within an abstract 00:02:29.360 --> 00:02:32.669 logic. What's fascinating here is the sheer flexibility 00:02:32.669 --> 00:02:35.430 of this universal framework. It's totally bare 00:02:35.430 --> 00:02:38.229 bones. Exactly. It doesn't demand any rigid, 00:02:38.409 --> 00:02:40.550 predefined properties from the sentences of the 00:02:40.550 --> 00:02:43.349 models. It's just a generalized skeleton for 00:02:43.349 --> 00:02:46.030 logic itself. So it can apply to almost anything. 00:02:46.370 --> 00:02:48.789 Right. Because of that, it applies to a massive 00:02:48.789 --> 00:02:51.569 spectrum of systems. It works for standard first 00:02:51.569 --> 00:02:53.949 -order logic. but it scales up perfectly into 00:02:53.949 --> 00:02:56.729 higher -order logics and even infinitary logics. 00:02:56.849 --> 00:02:58.650 Where the sentences can be, what, infinitely 00:02:58.650 --> 00:03:01.569 long? Yes, infinitely long strings of conjunctions 00:03:01.569 --> 00:03:03.889 or quantifiers. It's mind -bending. That flexibility 00:03:03.889 --> 00:03:05.969 is what makes the Loewenheim number so powerful. 00:03:06.169 --> 00:03:08.789 So let's define it. Let's do it. The Loewenheim 00:03:08.789 --> 00:03:11.469 number of a given logic, let's call that logic 00:03:11.469 --> 00:03:14.530 L, is defined as the smallest cardinal number. 00:03:14.770 --> 00:03:16.729 We'll call that cardinal kappa. Okay, kappa. 00:03:16.889 --> 00:03:19.849 The rule is this. If an arbitrary sentence in 00:03:19.849 --> 00:03:23.120 that logic has any model at all, It is mathematically 00:03:23.120 --> 00:03:25.719 guaranteed to have a model of a size no larger 00:03:25.719 --> 00:03:27.919 than kappa. To put that into perspective for 00:03:27.919 --> 00:03:30.400 you, you might write a logical sentence that 00:03:30.400 --> 00:03:34.060 seems to demand a universe of unmeasurable, unfathomable 00:03:34.060 --> 00:03:37.379 size just to satisfy it. Just massive. Right. 00:03:37.460 --> 00:03:40.319 But if that sentence is satisfiable at all, the 00:03:40.319 --> 00:03:42.520 Loewenheim number is the absolute ceiling on 00:03:42.520 --> 00:03:45.000 how big a model ever needs to be. There is always 00:03:45.000 --> 00:03:47.680 a perfectly working model out there with a cardinality 00:03:47.680 --> 00:03:50.659 of kappa or smaller. Always. It functions as 00:03:50.659 --> 00:03:52.780 this ironclad guarantee of structural efficiency. 00:03:53.080 --> 00:03:55.680 But establishing that a universal ceiling exists 00:03:55.680 --> 00:03:58.379 for every possible logic requires a bulletproof 00:03:58.379 --> 00:04:01.439 proof. It does. This is where Leopold Loewenheim's 00:04:01.439 --> 00:04:03.879 work, which was originally published around 1915, 00:04:04.199 --> 00:04:06.639 really laid the groundwork for modern model theory. 00:04:06.860 --> 00:04:09.639 How does the proof actually establish this boundary, 00:04:09.800 --> 00:04:12.000 assuming the collection of sentences forms a 00:04:12.000 --> 00:04:15.139 set? The mechanism is conceptually quite elegant. 00:04:15.419 --> 00:04:18.220 Okay. For every individual sentence, let's call 00:04:18.220 --> 00:04:22.339 it psi, You identify the smallest possible cardinality 00:04:22.339 --> 00:04:25.300 of a working model that satisfies it. So finding 00:04:25.300 --> 00:04:28.259 the absolute most efficient board for that specific 00:04:28.259 --> 00:04:31.480 role. Exactly. We designate that specific size 00:04:31.480 --> 00:04:35.269 as kappa sub phi. Now, if the sentence is logically 00:04:35.269 --> 00:04:38.110 contradictory and simply has no model, it just 00:04:38.110 --> 00:04:40.569 gets assigned to zero. So the process involves 00:04:40.569 --> 00:04:43.910 sweeping through the entire set of possible sentences, 00:04:44.290 --> 00:04:46.990 isolating the most efficient model for each one, 00:04:47.110 --> 00:04:49.949 and recording that model's cardinality. Precisely. 00:04:50.240 --> 00:04:51.899 This generates a vast collection of cardinal 00:04:51.899 --> 00:04:54.779 numbers. Right. A huge list of sizes. But mathematically, 00:04:55.180 --> 00:04:57.379 you cannot just sweep infinities into a pile 00:04:57.379 --> 00:04:59.180 and declare it a bounded set. Because that breaks 00:04:59.180 --> 00:05:02.240 math. Right. Yeah. Doing so haphazardly is what 00:05:02.240 --> 00:05:05.500 leads to paradoxes in early set theory. To rigorously 00:05:05.500 --> 00:05:07.860 group all these isolated cardinals into an official 00:05:07.860 --> 00:05:10.540 mathematical set, the proof relies heavily on 00:05:10.540 --> 00:05:12.870 the axiom of replacement. I want to pause on 00:05:12.870 --> 00:05:14.529 the axiom of replacement for a second, because 00:05:14.529 --> 00:05:16.689 that is a massive pillar of the Melo -Frankel 00:05:16.689 --> 00:05:19.029 set theory. It absolutely is. Without getting 00:05:19.029 --> 00:05:21.709 barked down in the dense notation, how does it 00:05:21.709 --> 00:05:24.470 practically allow us to group these models? Well, 00:05:24.490 --> 00:05:26.970 the axiom of replacement essentially states that 00:05:26.970 --> 00:05:29.769 if you take a valid set, in this case, our initial 00:05:29.769 --> 00:05:33.009 set of sentences, and you map every element in 00:05:33.009 --> 00:05:35.509 that set to another mathematical object, which 00:05:35.509 --> 00:05:37.970 would be our minimal cardinalities, the resulting 00:05:37.970 --> 00:05:41.019 image of that mapping is also a valid set. So 00:05:41.019 --> 00:05:43.759 it's a safe transfer. Exactly. It ensures that 00:05:43.759 --> 00:05:45.899 replacing the elements of a set with other well 00:05:45.899 --> 00:05:48.680 -defined elements doesn't somehow break the boundaries 00:05:48.680 --> 00:05:51.360 of set theory and create a class that's too large 00:05:51.360 --> 00:05:53.720 to measure. It validates the collection as a 00:05:53.720 --> 00:05:56.699 true set. Yes. So once the axiom of replacement 00:05:56.699 --> 00:05:59.139 secures this collection of cardinal numbers, 00:05:59.420 --> 00:06:02.399 what's the final step to finding the Lohenheim 00:06:02.399 --> 00:06:04.779 number? You simply find the supremum of that 00:06:04.779 --> 00:06:07.500 set. The supremum? Yes, the least upper bound. 00:06:07.899 --> 00:06:10.220 It is the lowest possible ceiling that still 00:06:10.220 --> 00:06:12.639 manages to cover every single cardinality in 00:06:12.639 --> 00:06:15.420 that collection. By definition, that supremum 00:06:15.420 --> 00:06:17.800 is the Lohenheim number for your logic. Here's 00:06:17.800 --> 00:06:20.279 where it gets really interesting. The logical 00:06:20.279 --> 00:06:22.500 flow of that proof, gathering the sentences, 00:06:22.959 --> 00:06:25.899 finding the minimal models, using the axiom of 00:06:25.899 --> 00:06:28.399 replacement to form a set, and finding the supreme 00:06:28.399 --> 00:06:31.639 motive, it sounds like a very clear step -by 00:06:31.639 --> 00:06:34.100 -step algorithmic recipe. It does sound like 00:06:34.100 --> 00:06:36.490 one, doesn't it? But it's not. This proof carries 00:06:36.490 --> 00:06:39.689 a massive mind -bending caveat. It is strictly 00:06:39.689 --> 00:06:42.649 non -constructive. This raises an important question 00:06:42.649 --> 00:06:45.269 about the philosophy of mathematics itself. How 00:06:45.269 --> 00:06:48.910 so? A non -constructive proof provides absolute 00:06:48.910 --> 00:06:53.009 ironclad certainty that a specific mathematical 00:06:53.009 --> 00:06:56.350 object exists. In this case, the Lowenheim number. 00:06:56.509 --> 00:06:58.970 We know for a fact it's there. Right. But it 00:06:58.970 --> 00:07:01.709 offers absolutely no method, no algorithm, and 00:07:01.709 --> 00:07:04.250 no pathway to actually calculate what that number 00:07:04.250 --> 00:07:06.930 is. It's a fascinating intellectual space to 00:07:06.930 --> 00:07:09.449 occupy. You have a mathematical guarantee that 00:07:09.449 --> 00:07:11.829 the ceiling is there. You know exactly what properties 00:07:11.829 --> 00:07:14.329 the ceiling has. But the proof doesn't give you 00:07:14.329 --> 00:07:16.410 the coordinates to actually find it. Not at all. 00:07:16.490 --> 00:07:18.879 Why does mathematical logic... tolerate, and 00:07:18.879 --> 00:07:21.079 even celebrate non -constructive proofs like 00:07:21.079 --> 00:07:24.199 this. Because abstract logic often deals with 00:07:24.199 --> 00:07:26.860 scales of infinity that fundamentally transcend 00:07:26.860 --> 00:07:30.439 human or even theoretical computation. It's just 00:07:30.439 --> 00:07:33.920 too big for us to crunch. Exactly. In model theory, 00:07:34.139 --> 00:07:37.060 demanding that every proof be constructive would 00:07:37.060 --> 00:07:39.459 severely and artificially limit our understanding 00:07:39.459 --> 00:07:41.800 of mathematical architecture. So we'd be flying 00:07:41.800 --> 00:07:45.199 blind to the bigger picture. Right. Non -constructive 00:07:45.199 --> 00:07:47.720 proofs, which often rely on principles like the 00:07:47.720 --> 00:07:50.199 axiom of choice or the law of excluded middle, 00:07:50.339 --> 00:07:53.199 allow us to perceive the structural topography 00:07:53.199 --> 00:07:56.420 of a logical universe, even when we are entirely 00:07:56.420 --> 00:08:01.500 incapable of running the numbers. That is a brilliant 00:08:01.500 --> 00:08:03.240 way to frame it. And of course, in mathematics, 00:08:03.500 --> 00:08:05.720 once a structural topography is identified, the 00:08:05.720 --> 00:08:08.040 immediate instinct is to scale it up. Naturally. 00:08:08.139 --> 00:08:10.819 The Loewenheim number establishes a ceiling for 00:08:10.819 --> 00:08:13.779 a single sentence. But what happens when a system 00:08:13.779 --> 00:08:16.240 requires an entire network of rules to function 00:08:16.240 --> 00:08:18.759 simultaneously? That scales us up to the first 00:08:18.759 --> 00:08:21.660 major extension, the Loewenheim -Skolem number. 00:08:21.860 --> 00:08:25.300 Skolem, S -K -O -L -E -M. Correct. Instead of 00:08:25.300 --> 00:08:27.139 analyzing a single sentence, we are now looking 00:08:27.139 --> 00:08:29.860 at an entire set of sentences, which we'll designate 00:08:29.860 --> 00:08:32.679 as T. So this is upgrading from a single line 00:08:32.679 --> 00:08:35.480 of code to an entire software library. Exactly. 00:08:35.620 --> 00:08:38.480 The Loewenheim -Skolem number is the smallest 00:08:38.480 --> 00:08:41.279 cardinal, kappa, where if your entire set of 00:08:41.279 --> 00:08:44.519 sentences t has a model, it's guaranteed to have 00:08:44.519 --> 00:08:47.379 a model of a size no larger than the maximum 00:08:47.379 --> 00:08:50.919 of either the size of the set t itself or kappa. 00:08:51.039 --> 00:08:54.019 That introduces a variable constraint. You have 00:08:54.019 --> 00:08:56.759 to factor in the sheer volume of rules you are 00:08:56.759 --> 00:08:59.240 imposing on the system. You do. If your set of 00:08:59.240 --> 00:09:02.659 sentences is uncountably massive, larger than 00:09:02.659 --> 00:09:05.200 kappa, then the ceiling for your model has to 00:09:05.200 --> 00:09:07.399 scale up to accommodate the size of the rulebook. 00:09:07.519 --> 00:09:09.539 Right. But if your rulebook is relatively small, 00:09:09.720 --> 00:09:11.919 kappa remains the absolute ceiling. It accounts 00:09:11.919 --> 00:09:14.620 for the complexity of the logical demands, but 00:09:14.620 --> 00:09:17.059 the architecture goes one layer deeper. arriving 00:09:17.059 --> 00:09:19.259 at the Loewenheim -Skolem -Tarski number. Adding 00:09:19.259 --> 00:09:22.440 Tarski to the mix. Yes. This applies when a logic 00:09:22.440 --> 00:09:24.679 incorporates a specific structural feature known 00:09:24.679 --> 00:09:27.200 as an elementary substructure. How should we 00:09:27.200 --> 00:09:29.399 visualize an elementary substructure in this 00:09:29.399 --> 00:09:31.779 context? Think of an elementary substructure 00:09:31.779 --> 00:09:34.840 as a perfectly representative microcosm. Okay. 00:09:35.000 --> 00:09:38.720 If you have a massive, complex mathematical structure, 00:09:38.840 --> 00:09:41.700 let's call it structure A, an elementary substructure 00:09:41.700 --> 00:09:44.759 is a smaller subset of A that preserves all the 00:09:44.759 --> 00:09:47.250 logical truths of the original. So it's identical 00:09:47.250 --> 00:09:51.110 in its internal logic. Yes. Any sentence that 00:09:51.110 --> 00:09:53.850 evaluates as true in the massive structure is 00:09:53.850 --> 00:09:56.549 also definitively true in the substructure. It's 00:09:56.549 --> 00:09:59.049 like a holographic plate. Even if you shatter 00:09:59.049 --> 00:10:01.269 a hologram and take a tiny shard of the glass, 00:10:01.509 --> 00:10:04.570 that single shard still contains the entire three 00:10:04.570 --> 00:10:07.169 -dimensional image of the original object, just 00:10:07.169 --> 00:10:10.379 at a smaller scale. A very apt analogy. The Loewenheim 00:10:10.379 --> 00:10:12.820 -Skolem -Tarski number is the smallest cardinal 00:10:12.820 --> 00:10:15.500 such that for any given structure A within that 00:10:15.500 --> 00:10:17.899 logic, there is guaranteed to be an elementary 00:10:17.899 --> 00:10:20.720 substructure, a perfect holographic shard of 00:10:20.720 --> 00:10:23.519 a size no more than kappa. That progression reveals 00:10:23.519 --> 00:10:25.940 a very strict hierarchy among these ceilings. 00:10:25.940 --> 00:10:28.220 It does. Assuming all three numbers exist for 00:10:28.220 --> 00:10:31.240 a specific abstract logic, the standard Loewenheim 00:10:31.240 --> 00:10:34.000 number will always be less than or equal to the 00:10:34.000 --> 00:10:36.559 Loewenheim -Skolem number. And the Loewenheim 00:10:36.559 --> 00:10:39.259 -Skolem number will always be less than or equal 00:10:39.259 --> 00:10:42.720 to the Loewenheim -Skolem -Tarski number. The 00:10:42.720 --> 00:10:45.100 hierarchy reflects the increasing strictness 00:10:45.100 --> 00:10:48.220 of the preservation. Moving from preserving the 00:10:48.220 --> 00:10:51.009 truth of a single sentence... to an entire set 00:10:51.009 --> 00:10:53.590 of sentences to preserving the total internal 00:10:53.590 --> 00:10:57.029 reality of an entire structure via a substructure 00:10:57.029 --> 00:10:59.990 inherently demands a higher potential ceiling 00:10:59.990 --> 00:11:02.490 that makes perfect sense it's also worth noting 00:11:02.490 --> 00:11:05.629 that in deep model theory research mathematicians 00:11:05.629 --> 00:11:08.009 will occasionally use the phrasing has a model 00:11:08.009 --> 00:11:11.080 strictly smaller than rather than no larger than? 00:11:11.220 --> 00:11:14.100 It's a subtle semantic shift. It is, but it provides 00:11:14.100 --> 00:11:16.299 a more granular classification when dealing with 00:11:16.299 --> 00:11:18.840 the extreme upper limits of set theory. And the 00:11:18.840 --> 00:11:20.899 extreme upper limits of set theory is exactly 00:11:20.899 --> 00:11:23.220 where this gets visceral. Visceral is a good 00:11:23.220 --> 00:11:25.460 word for it. The abstract framework is beautiful, 00:11:25.679 --> 00:11:28.700 but the specific real -world mathematical examples 00:11:28.700 --> 00:11:31.100 of these numbers in action demonstrate just how 00:11:31.100 --> 00:11:33.620 volatile these infinities can be. The friction 00:11:33.620 --> 00:11:35.980 between different logical systems is incredible. 00:11:36.659 --> 00:11:39.169 Let's look at first -order logic. First order 00:11:39.169 --> 00:11:41.809 logic is the bedrock for the vast majority of 00:11:41.809 --> 00:11:44.570 modern mathematics. It allows for quantification 00:11:44.570 --> 00:11:47.909 over individual objects using modifiers like 00:11:47.909 --> 00:11:51.629 for all or there exists. The classics. Right. 00:11:52.009 --> 00:11:53.889 If we look at first -order logic, specifically 00:11:53.889 --> 00:11:56.909 utilizing countable signatures, meaning its alphabet 00:11:56.909 --> 00:11:59.590 of symbols, functions, and relations is countable, 00:11:59.669 --> 00:12:02.830 the Loewenheim -Skolem -Tarski number is Alephnaut. 00:12:03.129 --> 00:12:05.950 Alephnaut, the absolute smallest infinity. Yes. 00:12:05.990 --> 00:12:07.929 It is the cardinality of the natural numbers, 00:12:08.110 --> 00:12:11.129 one, two, three, onward, forever. So what the 00:12:11.129 --> 00:12:13.870 text says this means is, if a sentence in first 00:12:13.870 --> 00:12:16.669 -order logic is satisfiable at all, it can always 00:12:16.669 --> 00:12:19.769 be satisfied in a countably infinite space. Which 00:12:19.769 --> 00:12:21.669 leads to some deeply countable... counterintuitive 00:12:21.669 --> 00:12:24.450 phenomena such as Skolem's paradox. I love Skolem's 00:12:24.450 --> 00:12:27.169 paradox. It's fascinating. You can write a set 00:12:27.169 --> 00:12:29.289 of rules in first -order logic that explicitly 00:12:29.289 --> 00:12:32.409 defines an uncountably infinite concept like 00:12:32.409 --> 00:12:34.629 the continuous spectrum of all real numbers. 00:12:34.830 --> 00:12:36.889 But because the Loewenheim -Skolem -Tarski number 00:12:36.889 --> 00:12:39.889 is aleph -naught, first -order logic is structurally 00:12:39.889 --> 00:12:42.850 incapable of distinguishing between that uncountable 00:12:42.850 --> 00:12:45.730 reality and accountable model that simply mimics 00:12:45.730 --> 00:12:48.789 it perfectly from the inside. Exactly. It is 00:12:48.789 --> 00:12:51.090 structurally blind to the massive scale it's 00:12:51.090 --> 00:12:53.889 trying to describe. It gets tricked by a smaller, 00:12:54.070 --> 00:12:57.269 aleph -not -sized universe. It does. But the 00:12:57.269 --> 00:12:59.389 moment you transition from first -order logic 00:12:59.389 --> 00:13:02.330 to second -order logic, that blindness vanishes. 00:13:02.529 --> 00:13:05.070 And the required scale just explodes. Completely 00:13:05.070 --> 00:13:08.429 explodes. Second -order logic is vastly more 00:13:08.429 --> 00:13:11.149 expressive. It doesn't just quantify over individual 00:13:11.149 --> 00:13:14.330 objects. It quantifies over entire subsets of 00:13:14.330 --> 00:13:16.730 objects and complex relations. So it's much... 00:13:16.779 --> 00:13:19.860 Yes. Because it has a much richer vocabulary, 00:13:20.279 --> 00:13:23.000 it can enforce much stricter structural demands 00:13:23.000 --> 00:13:25.620 on its models. And the resulting Loewenheim -Skolem 00:13:25.620 --> 00:13:28.740 numbers are staggering. In second -order logic, 00:13:29.039 --> 00:13:31.799 the ceiling shoots past Alephnaud entirely. It 00:13:31.799 --> 00:13:33.820 leaves it in the dust. It becomes larger than 00:13:33.820 --> 00:13:36.460 the first measurable cardinal, assuming a measurable 00:13:36.460 --> 00:13:39.120 cardinal is even capable of existing within the 00:13:39.120 --> 00:13:41.860 universe of set theory. To understand the gravity 00:13:41.860 --> 00:13:43.600 of that jump, we have to touch on the concept 00:13:43.600 --> 00:13:46.039 of large cardinals. A measurable cardinal isn't 00:13:46.039 --> 00:13:48.860 just a bigger infinity. It is an infinity so 00:13:48.860 --> 00:13:52.480 incomprehensibly massive that its existence cannot 00:13:52.480 --> 00:13:55.659 be proven using the standard Zermelo -Fraenkel 00:13:55.659 --> 00:13:58.279 axioms. So standard math can't even reach it. 00:13:58.500 --> 00:14:01.340 Right. It requires mathematicians to introduce 00:14:01.340 --> 00:14:04.820 entirely new, stronger axioms just to permit 00:14:04.820 --> 00:14:08.440 it to exist. A measurable cardinal requires the 00:14:08.440 --> 00:14:10.539 existence of something called a non -trivial, 00:14:10.679 --> 00:14:14.549 two -valued countably complete ultrafilter. That 00:14:14.549 --> 00:14:17.789 is quite the mouthful. It is. In simpler terms, 00:14:17.870 --> 00:14:19.889 it requires a mathematical structure that can 00:14:19.889 --> 00:14:22.590 measure the size of subsets in a way that standard 00:14:22.590 --> 00:14:25.190 infinity simply cannot support. And the escalation 00:14:25.190 --> 00:14:27.690 doesn't stop there, does it? Not at all. When 00:14:27.690 --> 00:14:29.269 you look at the universal fragment of second 00:14:29.269 --> 00:14:31.649 -order logic, the Loewenheim number sits just 00:14:31.649 --> 00:14:34.570 beneath the first supercompact cardinal. Supercompact 00:14:34.570 --> 00:14:36.950 cardinals dwarf measurable cardinals. They do. 00:14:37.090 --> 00:14:39.850 They are a deeply theoretical class of large 00:14:39.850 --> 00:14:42.029 cardinals that reflect properties of the entire 00:14:42.029 --> 00:14:44.950 absolute universe of sets down into smaller bounded 00:14:44.950 --> 00:14:47.870 universes. We are operating at the absolute bleeding 00:14:47.870 --> 00:14:50.389 edge of mathematical philosophy here. We really 00:14:50.389 --> 00:14:53.559 are. Finally, we hit the Lohenheim -Skolem -Tarski 00:14:53.559 --> 00:14:56.279 number for second -order logic, which the text 00:14:56.279 --> 00:14:59.460 defines as the supremum of all ordinals definable 00:14:59.460 --> 00:15:02.759 by a pi sub 2 formula. Let's clarify what a pi 00:15:02.759 --> 00:15:05.340 sub 2 formula is, because it highlights exactly 00:15:05.340 --> 00:15:08.460 why this ceiling is so unreachably high. A pi 00:15:08.460 --> 00:15:11.429 sub 2 formula. refers to a specific level of 00:15:11.429 --> 00:15:14.090 logical complexity in the arithmetical or analytical 00:15:14.090 --> 00:15:17.509 hierarchy. Structurally, it's a statement that 00:15:17.509 --> 00:15:20.009 begins with two alternating blocks of quantifiers, 00:15:20.210 --> 00:15:23.549 specifically a universal quantifier followed 00:15:23.549 --> 00:15:26.899 by an existential quantifier. For all x, there 00:15:26.899 --> 00:15:29.259 exists a y. So it's essentially a statement saying 00:15:29.259 --> 00:15:31.159 that no matter what condition you impose, there 00:15:31.159 --> 00:15:33.480 will always be a subsequent element that satisfies 00:15:33.480 --> 00:15:35.919 a following condition. Correct. When you gather 00:15:35.919 --> 00:15:38.019 every single ordinal number that can possibly 00:15:38.019 --> 00:15:40.419 be defined by that level of alternating infinite 00:15:40.419 --> 00:15:43.039 complexity, and then you find the supremum of 00:15:43.039 --> 00:15:45.360 all of them. That incomprehensible boundary is 00:15:45.360 --> 00:15:47.500 the Lohmenheim -Skolem -Tarski number for second 00:15:47.500 --> 00:15:50.379 order logic. Yes. It's staggering. If we connect 00:15:50.379 --> 00:15:52.980 this to the bigger picture, the contrast is what 00:15:52.980 --> 00:15:56.190 makes this so profound. In first -order logic, 00:15:56.509 --> 00:15:58.970 your universal ceiling is aleph -naught, the 00:15:58.970 --> 00:16:01.389 baseline of infinity. Right. But the second you 00:16:01.389 --> 00:16:03.669 upgrade the expressiveness of your language to 00:16:03.669 --> 00:16:07.129 second -order logic, your ceiling violently skyrockets 00:16:07.129 --> 00:16:10.289 into the realm of measurable cardinals, supercompact 00:16:10.289 --> 00:16:13.389 cardinals, and the extreme limits of pi sub 2 00:16:13.389 --> 00:16:16.309 definability. It demonstrates a fundamental law 00:16:16.309 --> 00:16:18.889 of mathematical reality. Yeah, it is. The language 00:16:18.889 --> 00:16:21.610 you use to describe a universe directly dictates 00:16:21.610 --> 00:16:24.070 the maximum size required to manifest that universe. 00:16:24.330 --> 00:16:28.049 Wow. There is an inescapable proportional relationship 00:16:28.049 --> 00:16:31.070 between the syntax of our rules and the semantics 00:16:31.070 --> 00:16:33.389 of the worlds those rules create. So what does 00:16:33.389 --> 00:16:35.769 this all mean for you listening? Why does understanding 00:16:35.769 --> 00:16:38.210 the Loewenheim number matter when these scales 00:16:38.210 --> 00:16:40.610 of infinity have absolutely no bearing on our 00:16:40.610 --> 00:16:43.769 dirty physical lives? A fair question. It matters 00:16:43.769 --> 00:16:45.750 because it proves that even within the abstract 00:16:45.750 --> 00:16:48.110 formless void of infinity, infinite possibilities, 00:16:48.549 --> 00:16:51.230 there is undeniable structure. We tend to view 00:16:51.230 --> 00:16:54.110 infinity as a messy, boundless ocean where everything 00:16:54.110 --> 00:16:56.889 simply goes on forever. Right? Unchecked. But 00:16:56.889 --> 00:16:59.850 model theory shows us that infinity has a rigid 00:16:59.850 --> 00:17:03.399 geometry. There are absolute, calculable... or 00:17:03.399 --> 00:17:05.839 at least provable ceilings, dictating exactly 00:17:05.839 --> 00:17:09.359 how large a reality needs to be to satisfy a 00:17:09.359 --> 00:17:12.539 specific set of rules. It is about finding hard, 00:17:12.619 --> 00:17:15.119 unforgiving boundaries in the middle of the infinite. 00:17:15.319 --> 00:17:17.779 It brings a sense of architectural order to concepts 00:17:17.779 --> 00:17:20.960 that defy human intuition. It reassures us that 00:17:20.960 --> 00:17:23.200 logic does not break down at massive scales. 00:17:23.500 --> 00:17:27.059 It simply adheres to deeper, more complex invariants. 00:17:27.259 --> 00:17:28.859 I want to leave you with a thought regarding 00:17:28.859 --> 00:17:31.039 that non -constructive proof we dissected earlier. 00:17:31.599 --> 00:17:34.099 Mathematics is the most rigorous demanding discipline 00:17:34.099 --> 00:17:36.930 in human history. Yet, at its highest levels, 00:17:37.170 --> 00:17:39.430 it fully accepts proofs that guarantee a ceiling 00:17:39.430 --> 00:17:41.430 exists without ever giving us the mechanism to 00:17:41.430 --> 00:17:43.750 actually find it. It maps the shadows. Exactly. 00:17:43.890 --> 00:17:46.329 If a fundamental bedrock of logic allows for 00:17:46.329 --> 00:17:48.529 absolute truths that we can prove exist but can 00:17:48.529 --> 00:17:51.230 never actually touch, how many other absolute 00:17:51.230 --> 00:17:53.569 truths in our physical universe exist completely 00:17:53.569 --> 00:17:56.170 out of our human reach to ever calculate or comprehend? 00:17:56.549 --> 00:17:58.690 That is the enduring mystery of mathematics. 00:17:59.190 --> 00:18:01.849 We can map the shadows of these structures, even 00:18:01.849 --> 00:18:03.920 if we can never hold them. Something to mull 00:18:03.920 --> 00:18:07.660 over until next time. Keep questioning the boundaries 00:18:07.660 --> 00:18:09.680 and we'll see you on the next deep dive.