Joel David Hamkins
Joel David Hamkins
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Joel David Hamkins: Set-theoretic forcing as a computational process
This was a talk by Joel David Hamkins for the Midwest Computability Seminar at the University of Chicago, May 2, 2023, taking place in "The Barn" at the University of Chicago.
Abstract. I shall explore several senses in which set-theoretic forcing can be seen as a computational process on the models of set theory. Given an oracle for the atomic or elementary diagram of a model (M,∈^M) of set theory, for example, there are senses in which one may compute M-generic filters G⊂ℙ∈M over that model and compute the diagrams of the corresponding forcing extensions M[G]. Meanwhile, no such computational process is functorial, for there must always be isomorphic alternative presentations of the same model of set theory that lead by the computational process to non-isomorphic forcing extensions. Indeed, there is no Borel function providing generic filters that is functorial in this sense. This is joint work with myself, Russell Miller and Kameryn Williams.
The paper is available on the arxiv at arxiv.org/abs/2007.00418.
Further commentary available at the speaker's blog jdh.hamkins.org/forcing-as-a-computational-process-mcs-chicago-may-2023.
The seminar web page is available at www.math.uchicago.edu/~drh/mcs/Jointseminar.html.
มุมมอง: 2 460

วีดีโอ

How to Count to Infinity and Beyond
มุมมอง 9K2 ปีที่แล้ว
Anyone can learn to count in the ordinals, even a child. Let us learn together how to count to ω², the first ordinal that is a limit of limit ordinals. Joel David Hamkins of Oxford University explains the transinite ordinals, a fascinating and beautiful number system discovered by Georg Cantor in the 19th century. The poster used in this presentation is available on his blog at jdh.hamkins.org/...
Joel David Hamkins-Computable quotient presentations of models of arithmetic and set theory
มุมมอง 1.3K3 ปีที่แล้ว
Talk at Fudan University in Shanghai, at the Conference on Model Theory and the Philosophy of Mathematics, 23 August 2021. "The Tennenbaum phenomenon for computable quotient presentations of models of arithmetic and set theory" Abstract. Tennenbaum famously proved that there is no computable presentation of a nonstandard model of arithmetic or indeed of any model of set theory. In this talk, I ...
The Banach-Tarski paradox
มุมมอง 26K3 ปีที่แล้ว
Joel David Hamkins, with tongue in cheek, illustrates the Banach-Tarski paradox by forming two unit cubes from one, using only rigid motion. In a second follow-up video, to be posted, the professor will apply the same process again to those two cubes, making four, and then again, making eight cubes, and so on. There will be a plethora of colored cubes!
Joel David Hamkins: The Math Tea argument-must there be numbers we cannot describe or define?
มุมมอง 3.1K3 ปีที่แล้ว
A talk for the seminar in mathematical logic at the University of Warsaw, jdh.hamkins.org/definability-and-the-math-tea-argument-warsaw-22-january-2021. Abstract. According to the math tea argument, perhaps heard at a good afternoon tea, there must be some real numbers that we can neither describe nor define, since there are uncountably many real numbers, but only countably many definitions. Is...
Joel David Hamkins: Are there natural instances of nonlinearity in consistency strength?
มุมมอง 1.3K3 ปีที่แล้ว
This was a talk for the University of Wisconsin Madison Logic Seminar, January 25, 2021. jdh.hamkins.org/natural-instances-of-nonlinearity-in-the-hierarchy-of-consistency-strength-uwm-logic-seminar-january-2021/ Abstract. It is a mystery often mentioned in the foundations of mathematics that our best and strongest mathematical theories seem to be linearly ordered and indeed well-ordered by cons...
Set Theory and the Philosophy of Set Theory
มุมมอง 22K3 ปีที่แล้ว
Joel David Hamkins, Professor of Logic, Oxford University This lecture is based on chapter 8 of my book, Lectures on the Philosophy of Mathematics, published with MIT Press, mitpress.mit.edu/books/lectures-philosophy-mathematics. Chapter 8. Set Theory We shall discuss the emergence of set theory as a foundation of mathematics. Cantor founded the subject with key set-theoretic insights, but Freg...
Joel David Hamkins - Set-theoretic and arithmetic potentialism: the state of current developments
มุมมอง 1.2K3 ปีที่แล้ว
This was a plenary talk for the Chinese Annual Conference on Mathematical Logic (CACML 2020), Nankai University, 13-15 November 2020. jdh.hamkins.org/set-theoretic-and-arithmetic-potentialism-the-state-of-current-developments-cacml-2020/ Conference web page: logic.nankai.edu.cn/2020/0930/c22018a304507/page.htm Abstract. Recent years have seen a flurry of mathematical activity in set-theoretic a...
The Gödel incompleteness phenomenon
มุมมอง 17K3 ปีที่แล้ว
Joel David Hamkins, Professor of Logic, Oxford University This lecture is based on chapter 7 of my book, Lectures on the Philosophy of Mathematics, published with MIT Press, mitpress.mit.edu/books/lectures-philosophy-mathematics. Chapter 7. Incompleteness David Hilbert sought to secure the consistency of higher mathematics by finitary reasoning about the formalism underlying it, but his program...
What is Computability?
มุมมอง 9K3 ปีที่แล้ว
Joel David Hamkins, Professor of Logic, Oxford University This lecture is based on chapter 6 of my book, Lectures on the Philosophy of Mathematics, published with MIT Press, mitpress.mit.edu/books/lectures-philosophy-mathematics. Lecture 6. Computability What is computability? Kurt Gödel defined a robust class of computable functions, the primitive recursive functions, and yet he gave reasons t...
What is Proof?
มุมมอง 11K3 ปีที่แล้ว
Joel David Hamkins, Professor of Logic, Oxford University This lecture is based on chapter 5 of my book, Lectures on the Philosophy of Mathematics, published with MIT Press, mitpress.mit.edu/books/lectures-philosophy-mathematics Lecture 5. Proof What is proof? What is the relation between proof and truth? Is every mathematical truth true for a reason? After clarifying the distinction between sy...
Geometry - a paragon of mathematical deduction?
มุมมอง 7K3 ปีที่แล้ว
Joel David Hamkins, Professor of Logic, Oxford University This lecture is based on chapter 4 of my book, Lectures on the Philosophy of Mathematics, published with MIT Press, mitpress.mit.edu/books/lectures-philosophy-mathematics. Lecture 4. Geometry Classical Euclidean geometry is the archetype of a mathematical deductive process. Yet the impossibility of certain constructions by straightedge a...
Climb to Infinity!
มุมมอง 7K3 ปีที่แล้ว
Joel David Hamkins, Professor of Logic, Oxford University This lecture is based on chapter 3 of my book, Lectures on the Philosophy of Mathematics, published with MIT Press, mitpress.mit.edu/books/lectures-philosophy-mathematics. Lecture 3. Infinity We shall follow the allegory of Hilbert’s hotel and the paradox of Galileo to the equinumerosity relation and the notion of countability. Cantor’s ...
The Rise of Rigor in the Calculus
มุมมอง 12K3 ปีที่แล้ว
Joel David Hamkins, Professor of Logic, Oxford University This lecture is based on chapter 2 of my book, Lectures on the Philosophy of Mathematics, published with MIT Press, mitpress.mit.edu/books/lectures-philosophy-mathematics. Lecture 2. The Rise of Rigor in the Calculus Let us consider the problem of mathematical rigor in the development of the calculus. Informal continuity concepts and the...
What is a number?
มุมมอง 56K3 ปีที่แล้ว
Joel David Hamkins, Professor of Logic, Oxford University This lecture is based on chapter 1 of my book, Lectures on the Philosophy of Mathematics, published with MIT Press, mitpress.mit.edu/books/lectures-philosophy-mathematics. Lecture 1. Numbers Numbers are perhaps the essential mathematical idea, but what are numbers? There are many kinds of numbers-natural numbers, integers, rational numbe...
Joel David Hamkins: Modal model theory as mathematical potentialism
มุมมอง 1.8K3 ปีที่แล้ว
Joel David Hamkins: Modal model theory as mathematical potentialism
Joel David Hamkins: Categorical Cardinals
มุมมอง 1.8K4 ปีที่แล้ว
Joel David Hamkins: Categorical Cardinals
In memory of John H Conway, who could do impossible things
มุมมอง 11K4 ปีที่แล้ว
In memory of John H Conway, who could do impossible things
Arguing with the Yes-No box
มุมมอง 1.4K4 ปีที่แล้ว
Arguing with the Yes-No box
Introducing the Triare!
มุมมอง 8575 ปีที่แล้ว
Introducing the Triare!
Introducing the Squircle!
มุมมอง 2.9K5 ปีที่แล้ว
Introducing the Squircle!

ความคิดเห็น

  • @topherthe11th23
    @topherthe11th23 3 วันที่ผ่านมา

    This is really good. The reason I find it SO good is that I expected it to be the old trick of folding your arms (a.k.a. "crossing your arms") as if tired, on your chest, and then leaning over the table so that your hands can pick up the ends of the rope. (Your right hand, which is now closer to your left shoulder, picks up the left end of the rope. And vice versa). Then all you have to do is unfold/uncross your arms. The tangle that had existed in your arms is transferred to the rope because it has nowhere else to go. So, this video was a great surprise to me. I probably wouldn't have bothered, since I THOUGH I knew what it was, but I saw that it was only 14 seconds, so I gave it a go. I have to wonder if John Horton Conway thought of this by transferring the folding of the arms in time to folding BEFORE you pick up the rope to a net-effects same thing AFTER you pick up the rope. It can't be EXACTLY like that, of course, but I wonder if he evolved it from that.

  • @hilaleristi5513
    @hilaleristi5513 14 วันที่ผ่านมา

    Özet Joel David Hamkins’in Oxford’daki matematik felsefesi dersleri, sayıların doğasını ve farklı sistemlerde nasıl temsil edilebileceğini inceliyor. Konular 📚 Hamkins’in dersleri, öğrenciler için geniş bir perspektif sunuyor. 🔢 Sayıların farklı sayı sistemlerinde nasıl temsil edildiği tartışılıyor. 📜 Platonizm, Mantıksallık ve Yapısalcılık gibi felsefi bakış açıları irdeleniyor. 🌌 Cantor-Hume ilkesi ve Dedekind aksiyomları ele alınıyor. 🎲 Sayılar, oyunlar olarak kavramsallaştırılıyor. 🔄 Yapısalcılık, matematiksel nesnelerin sistem içindeki yapısal rolünü vurguluyor. 🧠 Psikoloji ve bilişsel bilimlerin sayı kavramını anlama üzerindeki etkileri tartışılıyor. Açıklamalar 🧩 Felsefi Perspektifler: Platonizm, mantıksallık ve yapısalcılık gibi farklı felsefi akımlar, sayıların doğasını çeşitli şekillerde yorumluyor. Bu, matematiksel düşüncenin derinleşmesine katkı sağlıyor. 🔍 Çeşitli Tanımlar: Sayıların ne olduğu konusunda farklı görüşler ve tanımlar var. Bu çeşitlilik, matematiğin dinamik yapısını gösteriyor ve yeni tartışmalara yol açıyor. ⚖ Cantor-Hume İlkesi: Bu ilke, sayıların varlığını ve doğasını sorgularken, matematik felsefesinde önemli bir temel oluşturuyor. 🧠 Bilişsel Bilimler: Psikoloji ve bilişsel bilimler, sayıların nasıl anlaşıldığını etkileyerek, matematiksel düşüncenin doğasına dair yeni perspektifler sunuyor. 🎲 Sayılar ve Oyunlar: Sayıların oyunlar olarak düşünülmesi, matematiksel yapının eğlenceli ve etkileşimli bir yönünü ortaya koyuyor. 🔄 Yapısalcılığın Önemi: Yapısalcılık, matematiksel nesnelerin bireysel kimliklerinden ziyade, sistem içindeki rollerine odaklanarak matematiksel anlayışı derinleştiriyor. ❓ Soruların Açık Ucu: Sonuç olarak, sayıların ne olduğu sorusu hala açık ve süregelen bir keşif ve tartışma alanı olarak kalıyor.

  • @paschalcharles6097
    @paschalcharles6097 18 วันที่ผ่านมา

    I think mathematicians do not know math at all

  • @WmTyndale
    @WmTyndale หลายเดือนก่อน

    Wonderfully illuminating presentation even for this nonspecialist mathematician!

  • @DrThalesAlexandre
    @DrThalesAlexandre หลายเดือนก่อน

    Thank you for all the enlightening lectures in this series!

  • @oldsachem
    @oldsachem หลายเดือนก่อน

    Fascinating overview

  • @oldsachem
    @oldsachem หลายเดือนก่อน

    Professor Hamkins, Is an irrational number a number if we cannot identify or write it with a specific and precise notation? Can we even think of it, if we have no way to pinpoint it relative to other numbers which are known? Does the fact that such an irrational number cannot be specified, except as a member of a general abstract class of such "numbers," detract from its fundamental character as a number?

  • @DrThalesAlexandre
    @DrThalesAlexandre หลายเดือนก่อน

    Thank you for opening up these lectures to the general public. It is a privilege being able to watch them.

  • @pmcate2
    @pmcate2 หลายเดือนก่อน

    @1:03:30 How about the definition of a number is the intersection of all possible definitions of a number 😂

  • @DanielPerezMunoz-mf3iz
    @DanielPerezMunoz-mf3iz 2 หลายเดือนก่อน

    The only raminder question here it's. Where may I buy that cube? Please, let me know.

  • @elcapitan6126
    @elcapitan6126 2 หลายเดือนก่อน

    do you explore any type theory w.r.t. foundations of mathematics (hott as they call it)?

  • @clickaccept
    @clickaccept 2 หลายเดือนก่อน

    This is the shit you have to go through to become an oxford graduate?

  • @davidwright8432
    @davidwright8432 2 หลายเดือนก่อน

    Many thanks! I understood about 'adding 1 to 'omega' (tho how you know this is possible I don't know - unless you simply stipulate that any ordinal can have one added, as an axiom. What I was missing was the notion of a supremum as a number without immediate predecessor. What if so do you get by subtracting 1 from 'omega'??! Unless neither the operation of subtraction, or negative numbers are defined in this system.

    • @tomkerruish2982
      @tomkerruish2982 หลายเดือนก่อน

      Subtraction is not in general defined in this system. We can define a-b in case there is some ordinal c such that c+b equals a, but there's no guarantee such a c exists. Subtraction can be defined in the surreal numbers, which include the ordinals as a 'subset' (actually a subclass since there are 'too many' ordinals to be a set).

  • @cessromer7078
    @cessromer7078 2 หลายเดือนก่อน

    I only know in the Euclidean plane, two figures that are equidecomposable with respect to the group of Euclidean motions are necessarily of the same area, and therefore, a paradoxical decomposition of a square or disk of Banach-Tarski type that uses only Euclidean congruences is impossible.

  • @Ruktiet
    @Ruktiet 2 หลายเดือนก่อน

    ….WHAT?!

  • @Achrononmaster
    @Achrononmaster 2 หลายเดือนก่อน

    @1:33:00 isn't a shorter answer that Peano Arithmetic does not admit power sets. So you do not have ℝ .

  • @Achrononmaster
    @Achrononmaster 2 หลายเดือนก่อน

    @1;23:00 this was lovely, had not heard my affinity with platonism expressed so well before. I am definitely a platonist, but I like this plurality idea of different conceptions of "set", and it follows there could be different conceptions of ℝ. Just like your analogy to non-Euclidean geometry. But it is a fascinating analogy. Non-Euclidean geometries have such clear concrete manifestations (any curved surface or hypersurface). What I've never heard from a set theorist is a similar obvious concrete alternative concept of "set", only different but _very abstract _*_in_*_ their difference_ axiom schemas.

  • @Achrononmaster
    @Achrononmaster 2 หลายเดือนก่อน

    @1:14:00 no? Not necessarily? Woodin's V=Ultimate-L conjecture would render this conception immune to Cohen forcing. It seemed pretty simple. It is _only_ Cohen forcing that gives rise to models where CH is false. I think this is a terrific possible resolution for CH. It does not imply there is only one unique set theoretic platonic universe, just that there is this nice rigid one!

  • @Achrononmaster
    @Achrononmaster 2 หลายเดือนก่อน

    Just a quick note, which I got from Hugh Woodin. Given two doors you can choose one to go through, forbidding forever entry to the other. One leads to an Oracle for Set Theory, the other to an Oracle for Number Theory (and only NT). No one in their right mind is ever going to choose the NT door, since the ST Oracle can already tell you if Number Theory is consistent or not. The number theory Oracle cannot tell you the converse. If the ST Oracle tells you ST is inconsistent, then you can continue on wherever you go developing NT with no worries. But your stupid cousin who took the NT door will never know whether the greater richness of ST could have been a thing.

  • @Achrononmaster
    @Achrononmaster 2 หลายเดือนก่อน

    @1:18:00 you got to my last question, but stopped too short! The thing about CH is that it enriches the conception of ℝ. We could not claim to have realized a conception of ℝ at all before. (And philosophically certainly still no one knows what a true continuum means! You can always squeeze in hyperreals or surreals between the reals.) In some sense the metaphysical continuum is the core issue, and that's just way beyond all of axiomatizable mathematics. What am I trying to say? I think that it is fine to add CH to ZFC as a _provisional theory._ Many logicians might balk at that, but this is the way mathematics is inexorably going. If ZFC+CH cannot force any inconsistency not already in ZFC of course, but more mildly could allow some very obviously stupidly wrong statement to be provable, then so much the worse for CH, but the mathematical exploration is the thing worth pursuing, not the final rejection/acceptance of CH. The thing is, what on earth could be "very obviously stupidly wrong" but not disprovable in ZFC? (Do not tell me Banach-Tarski dissections, since they are beautiful "true" results imho. Showing the richness of sets like ℝ .) To find some such statement surely would require a helluva intuition at the ω-Ramanujan level, and so it'd be only one geniuses intuition, to the rest of us it'd be unfathomable, most likely, unless she could explain it to us children.

  • @Achrononmaster
    @Achrononmaster 2 หลายเดือนก่อน

    @1:17:00 one thing I've never been totally clear about is whether "Set Theory" (or any other axiomatization of the notion of arithmetic or the like) is a lot like Euclidean geometry: If you cannot prove a nice conjecture like the CH, and you really want to prove it, not take it as an axiom, then you are looking at maybe an analogue of non-Euclidean geometries which turned out to "be a _thing!"_ (Who knew?!! We were all flat-earthers before Gauss, Lobachevsky and Bolyai, haha.) So the deeper question is what are non-standard set theories? And can we realize them concretely in a similar way to the way a sphere gives elliptical geometry or Poincaré disk gives hyperbolic geometry?, and the real mind-bender --- the set theoretic or number theoretic analogue of Riemann geometry! To my knowledge we are still "Flat Setters" in this respect. I note that Woodin's V=Ultimate- L conjecture, if true, would establish the CH, and the way he speaks makes it more a set theoretic Universe rather than Multiverse. But surely that is only gauged to CH. I imagine there really should be other varieties of set theory that would not destroy the CH, and so "secure" the finite part of Peano arithmetic relative to ZFC+CH, but still allow for different conceptions of the transfinite, analogous to how Riemann geometry warps Euclid. Thoughts?

  • @Achrononmaster
    @Achrononmaster 2 หลายเดือนก่อน

    @1:16:00 and @51:20 --- is it fair (correct?) to say Tarski gave a criterion for a truth predicate, and showed the predicate is not definable in any formal system? The criterion for a predicate is not a definition _of the predicate_ in a formal system, right? So it's not as paradoxical as you make out?

  • @Achrononmaster
    @Achrononmaster 2 หลายเดือนก่อน

    Related to resiliency, @1:23:30 that issue of a root finding function when there is a tangency is critical. The idea to "solve" it is topological. You deform the function to within floating point tolerance, if you get two roots close by in tolerance the machine can either report them both or consider them "the same". Also related: I used to employ a library that defined an UncertainNumber data type. For work in microwave measurement metrology. I no longer work in metrology, but the idea is of fundamental importance in computing, more people should be computing with uncertain numbers, and someone aught to write an efficient library for them, they should be a fundamental data type, as should data types for dimensioned quantities. These low hanging fruit could go a long way towards establishing some minimal resiliency standards. Still further related is the idea of stochastic computation or analogue or noisy computing, which for a lot of everyday tasks is "good enough" and yet saves tremendous fractions of your real electricity bill! (Learned this from Tim Palmer.) Still even further related: fintec industry is a massive waste of human lives and parasitic on the real economy. As are loads of neural net programs. Really, it should be nearly a crime these days to run an inefficient program if it is sucking megawatts of power from the grid for no good public purpose (internet advertising --- needs to be utterly eliminated, which governments can do. Web companies relying on ad revenue need not exist, if they serve a public purpose the government can subsidize them, without currency inflation risk, to do so the public purpose has to be known and those companies books open, so their real resource and wages are transparent, if the wages can always be paid then there is no need for a profit motive.).

  • @Achrononmaster
    @Achrononmaster 2 หลายเดือนก่อน

    @1:28:00 what insanity is that? We certainly do have a word for "tell the truth", _honestly._ If some grammar nerd tells me, no that's an "adverb," I'll honestly slap them in the face, and can _testify_ to that. I'll also _avow_ to it my dude.

  • @Achrononmaster
    @Achrononmaster 2 หลายเดือนก่อน

    @1:25:00 another answer is that historically people did not understand "=" does not mean "the same" or "identically equal". So _x_ + _x_ = 2 _x_ = _x_ 2, is fine for finite numbers "up to isomorphism". But Cantor, yeah, he's the GOAT.

  • @Achrononmaster
    @Achrononmaster 2 หลายเดือนก่อน

    @1:17:00 not having a clear concept of ℕ, and thinking Dedekind-Peano is "murky" is fine. It gives mathematicians things to do. It does not force one to adopt pluralism. Pluralism is fine too, as a framework. But it is fragile, right? If you could get a non-murky definition of ℕ, then that kills pluralism for ℕ (a damn good thing in my opinion). However, Gödelman always flies to the rescue, no? There's always some structure you can validly desire to be pluralist about, like today the CH. Or "The Continuum" in general --- hyperreal continuua, Surreal continuum (the Surreals are non-murky?), or Cantor-Rucker "The Absolute" continuum (cardinality of Ω = the absolute infinite.) However, does not a lot of this boil down to words? With finer grain knowledge of mathematical structure we can name things that were once thought the same differently, provided we do not run out of words. Like 1960's particle physicists. 🤣 (that's a joke). I honestly think this game can go on forever, the question is whether mathematicians will last for _more than forever_ so the game terminates! 😜

  • @Achrononmaster
    @Achrononmaster 2 หลายเดือนก่อน

    @1:14:00 I am pretty sure, gut instinct, that what Ahmed meant was whether there is a problem with mathematical induction when we allow transfinite numbers, like ω +1, or ω·ω. Maybe? The answer is no (I think). The issue he might have had in mind was that if we can "go to" ω+1 then how come the induction over ℕ is valid? The answer is that it is valid because the proof using induction is induction only over ℕ. But the same idea extends to validity of transfinite induction. For these things I find the category theoretic style better than the indexing style. The CT folks tend to write an "index set" and try to never use an index subscript. It is just conceptually clearer that then one is staying in ℕ (or whatever index set one is using). Although I think I am not expressing this well. I got it off Misha Gromov ("I hate subscripts." He is also known to have said, "I don't know what a matrix is, a matrix is stupid." --- which being a Clifford algebra user I fully agree with.... not just for the laughs! Although mostly for the laughs.).

  • @Achrononmaster
    @Achrononmaster 2 หลายเดือนก่อน

    @59:40 yeah, but you need to be very careful about educating economists. They can be a bit nerdy and autistic in a dangerous way. In economics the Pareto principle is often grossly violated. It goes under various fallacies of composition. Such as the paradox of thrift: everyone is at least slightly better off if they net save. But if everyone net saves then everyone starves and dies (in a monetary economy). In a monetary economy with no counterfeit (so strict accounting laws) some agency has to net dissave. That is typically government, and government can _always_ net spend, since they are the monopoly currency issuer. It is called "running a deficit". Most Europeans think this is a shockingly appalling bad thing to not have people begging for the state currency (hence they introduce insane "fiscal rules" in the EMU which cause needless austerity and mass unemployment), but they've got basic accounting 101 all wrong, it is a *_good thing_* for government to run a deficit --- no ones "tax dollars" are paying for this --- although the government deficit need not be targeted, the supply can float in order to accommodate a stable price level at full employment --- by which I mean zero involuntary unemployment, not the NAIRU myth, the reality is a NAIBER not a nairu, or "non accelerating inflation buffer employment ratio". Employment and price level are all about buffer stocks, which mainstream macroeconomists never seem to consider when theorizing about inflation, being the idiots they are, they are pretty evil really. (Bet you didn't think a few words about reflexive and partial orderings would lead to life & death morality in the real world. But this comment is just basic math for state money accounting systems.) I think it can also be viewed as a comment about directed acyclic graphs? So from DAG's you get an understanding about the source of the price level, which no mainstream economist understands (I am willing to bet). Government deficits accommodate non-government savings desires (to the penny). (Inflation has nothing to do with this.) But try telling this to a Neoclassical or New Keynesian "professional" and they'll wet their pants.

  • @Achrononmaster
    @Achrononmaster 2 หลายเดือนก่อน

    @38:50 what I like is that you "construct" all algebraic numbers by "listing" all the polynomials over rationals --- surely countable, and they're bijective with the strictly finite sub-lists of rationals or integers that are the coefficients. But you never _actually_ construct this list. You only "know you could" if you were a god. This presumption of humble god-like powers is what finitists often object to, but the humility is beautiful. It is "well, if _only_ I had infinite paper and infinite time, or could do the proverbial ω many tasks in 2 seconds by working twice as fast each task step. While I am against tying mathematics to "physical tasks" (or even ω −Turing machine tasks) like this, since it leads to seeming pathologies like Banach-Tarski, it does cry out for a better verb than "construct". It is "mental construction". But what's the shorthand for that? It is "conceivability" or "computability" is it not? But _strong conceivability_ --- one must have a means of computation available _as if_ one were a humble god who merely has vast resources (infinite time or time steps). Which brings me to the question: has anyone got some of these proofs (involving the reals that use such god-like constructive arguments) in the form of computations involving tasks that could be completed in finite time and finite time steps? (i.e., non god-like.) Not that I care, I am fine with the nonconstructive proofs, and infinite tasks. But I'd like to have the occasion now and then to throw old Norman Wildberger some scraps.

  • @thiagof414
    @thiagof414 2 หลายเดือนก่อน

    Thank you

  • @Achrononmaster
    @Achrononmaster 2 หลายเดือนก่อน

    @1:21:00 who is your colleague in New York? Carl ????

  • @svarupa
    @svarupa 2 หลายเดือนก่อน

    excellent. you can also demonstrate the concept of anti-time by putting the cubes back into one. bravo. 👏

  • @Achrononmaster
    @Achrononmaster 2 หลายเดือนก่อน

    @41:15 The Asgaardians did one better than Yuen Ren Chao, they figured out the any amount of beer *_or_* mead is allowable theorem.

    • @Achrononmaster
      @Achrononmaster 2 หลายเดือนก่อน

      And Abraham Robinson died of beer thirst because he only allowed himself a hyperreal ball amount of beer.

  • @jurgenblick5491
    @jurgenblick5491 2 หลายเดือนก่อน

    So you are saying that math is mechanical?

  • @atlas4074
    @atlas4074 2 หลายเดือนก่อน

    On continuous induction at 37:20, this can be extended to connected topological spaces, and linear orders are connected only if they are dense orders with the least upper bound property. The more general versions states that if A is non-empty, whenever x is in A there's a neighbourhood of x contained in A (i.e. A is open), and whenever V is an open subset of A then A contains the boundary of V, A is the whole, connected space. The last condition is like saying 'if points arbitrarily close to x are in A, then x is in A.' The second and third properties tell you A is both open and closed, which in connected spaces means A is empty or the whole space.

  • @nikitasmarkantes5046
    @nikitasmarkantes5046 3 หลายเดือนก่อน

    I am a lawyer too. Its amazing how this lecture reiterates Socrates' parmenidis. A real and genuine platonist view.

  • @Tysm_for_1k_subs
    @Tysm_for_1k_subs 3 หลายเดือนก่อน

    0 1 2 Omega Omega +1 Omega +2 Omega x2 Omega x2+1 Omega x2+2 Omega ^2 Omega ^2 +1 Omega ^2 +2 Omega ^2 +Omega Omega ^2 +Omega +1 Omega ^2 +Omega +2 Omega ^2 +Omega x2 Omega ^2 +Omega x2+1 Omega ^2 x2 Omega ^2 x2+1 Omega ^2 x2+2 Omega ^2 x2+Omega Omega ^2 x2+Omega +1 Omega ^2 x2+Omega +2 Omega ^2 x2+Omega x2 Omega ^2 x2+Omega x2+1 Omega ^2 x2+Omega x2+2 Omega ^Omega

  • @nyworker
    @nyworker 3 หลายเดือนก่อน

    Just like politics...depends on pov.

  • @dimitardimitrakov2841
    @dimitardimitrakov2841 3 หลายเดือนก่อน

    I am listening on my laptop full volume and somehow struggle to hear. May be it's a good idea if you can edit the video and raise the volume up a bit

  • @draconyster
    @draconyster 3 หลายเดือนก่อน

    Man I am drunk and I have no idea what this is about, but I feel so smart listening to this and kind of picking up on stuff. I don't know why TH-cam suggested me this but I am listening for like an hour now, "general comprehension principle" man, you rock!

  • @user-mn6bb6gi6v
    @user-mn6bb6gi6v 3 หลายเดือนก่อน

    Many highly educated people can't see the forest for the trees. Too "strong/rigorous" demonstrations destroy any useful applications. They are a bit like ants that spin around. The hard part is not to be “rigorous”-that can be done by robots-but to set from the beginning the context/axioms and to see where you can go with the results and how to use them. The real is so vast and dense that we can start from anywhere, construct logically some model/structure/theory, that it is self-consistent 99% or more. Nevertheless, it could be completely not fitted for practical use, because it is not compatible with our reality. You can't trust maths as a dogma; this is intellectual fanaticism. Or, as Feynman put it ..., i let u find the citation yourself. Mathematics is tools we have being invented and continues to be invented, not a gift from GOD :)

  • @nothingposted9056
    @nothingposted9056 3 หลายเดือนก่อน

    I fell asleep listening to this, woke up and, for the first time in my life, picked up pen and paper and made math ecuations for fun

  • @user-nb3mq3cg8k
    @user-nb3mq3cg8k 3 หลายเดือนก่อน

    ❤❤❤❤❤

  • @flux81
    @flux81 3 หลายเดือนก่อน

    @joeldavidhamkins5484 Are you aware of any research into synesthesia along the exact lines that you mentioned? It seems to be a quite promising line of research.

  • @flux81
    @flux81 3 หลายเดือนก่อน

    The jokes from the book are quite funny. Good stuff!

  • @pinkraven4402
    @pinkraven4402 3 หลายเดือนก่อน

    3

  • @peterwaksman9179
    @peterwaksman9179 3 หลายเดือนก่อน

    Infinity in no way depends on a set theory analysis. Un-boundedness was probably understood by the Greeks.

    • @AlainGenestier-tj8fn
      @AlainGenestier-tj8fn 3 หลายเดือนก่อน

      When you comment on something, you should have an idea of what you are talking about. Clearly, you don't. The infinities of set theory are not a mere "unboundedness". Precisely, Aristotle considered that an "actual" infinity (as opposed to a mere potential infinity, to which "unboundedness" refers) was contradictory. With Bolzano and Cantor, the actual infinities got rehabilitated --but of course they introduced a new ingredient, which was not present in the Greek tradition

    • @peterwaksman9179
      @peterwaksman9179 3 หลายเดือนก่อน

      @@AlainGenestier-tj8fn Clearly you cannot read English. Aristotle did not know set theory. My comment was about the lack of necessity of set theory - not about how the Greeks understood infinity.

    • @AlainGenestier-tj8fn
      @AlainGenestier-tj8fn 3 หลายเดือนก่อน

      Aristotle didn't know set theory": of course, and that's obviously not what I claimed (if it is what you choose to understand, that's you problem!). "Lack of necessity": for what ? Modern math needs set theory (or more or less equivalent alternatives: type theory, category theory, which also deal with actual infinities). If you didn't notice, math has made progress since the Greeks. And so has our understanding of the concept of infinity. You are not supposed to accept those theories, but at least if you want to criticize them, don't make a superficial and uninformed criticism.@@peterwaksman9179

    • @AlainGenestier-tj8fn
      @AlainGenestier-tj8fn 3 หลายเดือนก่อน

      @@peterwaksman9179 Where did I claim that Aristotle knew about set theory? It's that's what you understood, your reading skills are questionable. On the contrary, Aristotle thought that it was inconsistant to consider infinite totalities. A set theory with only finite sets would be a quite rudimentary theory, and set theory is expressely built against this Aristotelian taboo of infinite totalities. "The lack of necessity of set theory": well, set theory is necessary for modern math. Or, you at least need o theory (like type theory, ...) admitting actual infinities, against Aristotle's prescription. In case you didn't notice, math has made progress since the Greeks. You are not committed to "accept" set theories (or even actual infinities). But if you want to criticize them, make at least an informed criticism!

  • @CriticalDispatches
    @CriticalDispatches 3 หลายเดือนก่อน

    I just bought your book, so I hope you don't give away the ending in this lecture.

  • @atol71
    @atol71 4 หลายเดือนก่อน

    Instead of going into higher levels of mathematics one should ponder: If you have nothing on cut board (0) and you try to divide ( / ) the nothing (0) with your one knife (1). How many peaces of bread you have in cut board? AKA 0/1==? Before going forward in academic curriculum of mathematics.

    • @atol71
      @atol71 4 หลายเดือนก่อน

      I do hope you do not declare empiric example (empiric example of cut board and knife) void like the last one I talked about it. If you do all your Academic mathematics is VOID.

  • @martin2ostra
    @martin2ostra 4 หลายเดือนก่อน

    Thanks