I know Talagrand because some of his work comes up in topological dynamics (work around Rosenthal's l¹-dichotomy culminating in the Bourgain-Fremlin-Talagrand dicothomy for compact sets of Baire class 1 functions), but I had no idea he has such accomplishments in other fields! Impressive
"An oscillator generates a 12-volt, 200-kilohertz (kHz) pulsed electrical signal, which is applied to a small plate on one side of the blade. The signal is transferred to the blade by capacitive coupling. A plate on the other side of the blade picks up the signal and sends it to a threshold detector. If a human contacts the blade, the signal will fall below the threshold. After signal loss for 25 micro seconds (µs), the detector will fire. A tooth on a 10-inch circular blade rotating at 4000 RPM will stay in contact with the approximate width of a fingertip for 100 µs. The 200-kHz signal will have up to 10 pulses during that time, and should be able to detect contact with just one tooth.[4] When the brake activates, a spring pushes an aluminum block into the blade. The block is normally held away from the blade by a wire, but during braking an electric current instantly melts the wire, similar to a fuse blowing."
It is a theorem of ZFC that uncountable sets exist and every model of ZFC will have a set that the model believes to be uncountable. It doesn't matter than the metatheory might believe that model to be countable (why should the metatheory have the correct notion of what it means to be countable anyway?).
> It is a theorem of ZFC that uncountable sets exist
This is simply false, as I already explained.
> and every model of ZFC will have a set that the model believes to be uncountable.
That is something else. (And I wouldn't use the nebulous term "believes" here, it's just that the model lacks an object which maps A to P.)
> It doesn't matter than the metatheory might believe that model to be countable (why should the metatheory have the correct notion of what it means to be countable anyway?).
"The meta theory" here is simply sentences expressed in natural language, or beliefs held by people expressing those sentences. It is the language in terms of which everything formal is ultimately defined. It's the only thing that ultimately matters.
It is absolutely not false! This is taught in every undergraduate set theory course. Please point to me the step in the above proof where there is an error.
Certainly not in every undergraduate class, though I don't doubt that the subtleties around these issues may often be taught wrong. I already did point you to the errors, and I included the reference to Shapiro's book.
You keep making vague references to concepts without showing how they even remotely contradict Cantors theorem. You explained to me the power set axiom, which, thanks, I guess? But I don’t understand what your point is. Are you claiming X is not in the power set of the natural numbers? X is, unambiguously, a set. And it is clearly a subset of A. Therefore, it is in the power set. If you don’t understand that, I think you need to review the power set axiom in ZFC. Then you said “Cantor's theorem stating that there is no mapping f from A onto P merely means that the mapping f itself can't exist inside a model of ZFC”. Which is literally identical to saying “under ZFC, there are uncountable sets”. You just don’t like it because that statement isn’t wrapped in eight layers of indirection with model theory.
I don't exactly understand your construction of X. But note that you are relying on the existence of ZFC's so-called "powerset", which, as we already know, can be countable. ZFC has no ability to talk about infinite powersets, since it can't and doesn't state that all subsets exist, and thus it doesn't imply the existence of a powerset. Accordingly, both f and X may not be what you want.
> Then you said “Cantor's theorem stating that there is no mapping f from A onto P merely means that the mapping f itself can't exist inside a model of ZFC”. Which is literally identical to saying “under ZFC, there are uncountable sets”.
No, it only means that ZFC can't contain a function f from A to P in its model, which doesn't make P uncountable. (Things can be true even if the theory itself can't express them. E.g. Gödel's second incompleteness theorem says that a theory can't prove its own consistency, but that doesn't mean that the theory is inconsistent.)
For any f and A, we can define X as { a in A | a is not in f(a) } . That set exists in ZFC by the axiom of separation, also known as the axiom of subsets or the axiom of comprehension.
I recommend you pick up a book on ZFC if you are interested in understanding set theory. I found Enderton’s “Elements of Set Theory” to be a really good introductory text.
I'm sorry but this is just wrong. Since you seem to like Shapiro's book more than traditional set theory books let me quote from page 144 that the existence of an uncountable set is a theorem of ZFC: "Let C be the statement of Cantor's theorem. It entails that the powerset
of the collection of finite ordinals is not countable. Since C is a theorem of
first-order ZFC..."
Also this is not how the metatheory is understood in mathematics, not even in Shapiro's book, who dedicates two whole chapters to the metatheory
> Since you seem to like Shapiro's book more than traditional set theory books let me quote from page 144 that the existence of an uncountable set is a theorem of ZFC: "Let C be the statement of Cantor's theorem. It entails that the powerset of the collection of finite ordinals is not countable. Since C is a theorem of first-order ZFC..."
You didn't finish reading the quote. It continues:
> "... Since C is a theorem of first-order ZFC, m ⊨ C, but, as just stated, m is itself countable and so are its elements. This, again, is the so-called Skolem paradox."
That is, he was making an (informal) contradictory statement in order to illustrate the paradox. But there is actually no contradiction (otherwise ZFC would have been proven inconsistent), so we already know the statement of the paradox must have been inaccurate. He then goes on to explain where the inaccuracy was. It turns out that it was mainly in the mistaken, but common, assumption that the "powerset axiom" implies the existence of a powerset:
> [The powerset axiom] is supposed[!] to assert the existence of the set of all subsets of each set. But the variables (like all first-order variables) range over the elements of the model. So the powerset axiom only guarantees the existence of a set of all subsets of (say) ω that are in the model. The subsets of ω that are 'guaranteed by the axioms' to exist in a given model m are those that are first-order m-definable, and only those. In some cases there are only countably many of them.
As I explained in a previous comment, you can't say in first-order logic "every possible combination of elements of this infinite set forms a set" (more precise expression of "all possible subsets exist"). ZFC's powerset axiom only states that all existing subsets are element of some set P, but it doesn't imply they exist in the first place. So it doesn't imply the existence of infinite powersets (finite powersets wouldn't require the axiom anyway). Indeed, only those subsets are implied to exist that are implied by the other axioms, which isn't very many. (See the Löwenheim-Skolem theorem.)
And Cantors theorem only states that inside the model no function f (which would be just another set) exists that maps A to its supposed powerset P, even if both A and P are countable. So there no contradiction between Cantor's theorem and the countability of P in ZFC.
> Also this is not how the metatheory is understood in mathematics, not even in Shapiro's book, who dedicates two whole chapters to the metatheory
See this (well-known) quote on page 254 where he talks about the metatheory of second-order logic:
> The language of set theory is employed, without apology, and no anti-realist interpretation or reduction its envisaged. Indeed, no explicit interpretation is envisaged at all. There is no perspective outside this language from which to discuss its interpretations, or its models, or at least none is contemplated. The set-theoretic universal quantifier reads 'for all sets' and the existential quantifier reads 'there is a set'. Thus sets are in the ontology of the background theory. If asked 'which sets?' or 'how many?', there is only one answer: 'all of them'. This is what it is to take the language literally.
Alternatively, one could already take the axioms of second-order logic as "rock bottom", insofar they are grounded in natural language (which he also offers arguments for). In any case, using other formal theories as a tower of meta- and meta-meta-theories only leads to an infinite regress. Everything bottoms out in natural language. Normal mathematicians don't bother with formal languages in the first place, it's only logicians which make this excursion, but even they resort to natural language on the (meta) meta level.
Löwenheim-Skolem gives you a countable elementarily equivalent submodel (assuming you're working in a theory in a countable language, otherwise it gives you an elementary substructure of the same cardinality of the language at best), but plenty of interesting properties of familiar mathematical objects cannot be captured by a first-order theory and are not preserved by elementary equivalence, completeness of the reals being the standard example
Yet the very notion of countability in ZFC, which is itself a first-order theory, is rendered completely relative by Löwenheim-Skolem. ZFC itself has a countable model.
If "plenty of interesting properties of familiar mathematical objects cannot be captured by a first-order theory" then that also undermines ZFC, which is a first-order theory.
ZFC was specifically designed to be immune to the classic paradoxes of naive set theory: Russell's paradox, the Burali-Forti paradox, and Cantor's paradox.
You are arguing that the ground moves to perfectly fit the shape of a puddle.
Zermelo was one of the first to reference "Cantor's theorem" in his papers.
These paradoxes do not occur in higher-order logic. You don't need ZFC or any first-order set theory for that. (Also, your comment doesn't address the sentence I quoted.)
I don't understand what you mean. Higher-order logic does have classical negation. First-order PA tries to approximate the (second-order) induction axiom by replacing it with an infinite axiom schema, but that doesn't rule out non-standard numbers.
Cantor's proofs showing that Z and Q are countable and R is uncountable.
Cantor's "diagonalization proof" showed that.
Turing extended to the computable numbers K, which can be conceptualized as a number where you can write a f(n) that returns the nth digit in a number.
The reals numbers are un-computable almost everywhere, this property holds for all real numbers in a set except a subset of measure zero, the computable reals K which is Aleph Zero, a countable infinity.
The set of computable reals is only as big as N, and can be mapped to N.
It is not 'non-standard numbers' that are inaccessible, it is most of the real line is inaccessible to any algorithm.
Beginners do that exactly until the first time they have a render crash halfway through, then they learn about rendering each frame as an image (yes I learned this lesson the hard way)
If it's not acyclic then you haven't broken down the knowledge graph enough. But that's probably a waste of time, trying to come up with a perfectly ordered plan of study for all of mathematics. When you find an apparent cycle, it means the two domains are strongly interrelated and you'll be studying part of one, then the other, then the first again, repeat until you're done (whatever that means to you). No need to try and break every subject down into one-week or one-day chunks and finding a perfect ordering, just figure out the roughly course-lengthed chunks of study and start working through them, concurrently if needed as described.
Right when you read something you don't need to understand it all to understand something which may be needed to understand something else elsewhere.
But still I think it would motivate me to keep on learning if somebody could show me an accurate acyclic pre-requisites graph and tell me: "These are the thing you need to understand before you should go to the next topic. If someone could come up with the time to come up with an accurate acyclic "knowledge-graph" it would help millions of students of mathematics.
If you try hard and long enough you will understand what you're trying to understand, you will. The question is what would make that more fun and less tedious. It is about precision and not needing to learn something you don't need to learn, to understand something that you need to learn. Spend your time on learning stuff you need to learn to understand what you want to learn.
Frequently these ISP level blocks also redirect the normal port 53 traffic to their own servers so these settings are meaningless. What does work is DNS-over-HTTPS which can be enabled quickly and easily in both Chrome and Firefox and will side-step this block even against adversarial ISPs.
"Frequently these ISP level blocks also redirect the normal port 53 traffic to their own servers"
No ISP is going to do nonsense like that for something like copywrite related compliance unless lots of money is forthcoming. An ISP can NXDOMAIN on their own DNS servers at a minimal cost. Doing 53/udp fiddling for all customers means CAM or similar expensive resource usage.
If you do find that your DNS is being redirected as TrueDuality describes, then yes DoH is an option. However if this is an issue then you have far bigger problems than not being able to access Anna's Archive.
Weird to write an article about Liz Truss as a math snob (a description that seems rather stretched to be honest) without mentioning that her father, John Truss, is a fairly well known model theorist! (Even though they're not really very close from what I've been told by students of the latter)
Oh I see, I was confused because I didn't mention PPE in my comment, maybe you meant to answer to the other top level comment which does use this abbreviation!
The pdf includes famous altered images from the 2006 Lebanon war but not other well known staged ones from the same conflict, so maybe the author does consider staging a different category?
Enver Hoxha, the Albanian dictator from 1944 to 1985, also had the habit of having photos altered to remove former allies that fell out of favour and just generally making himself look better, but he's not included in this pdf. To be fair there's probably examples of historical altered photos from any dictator or sufficiently authoritative government after the invention of photography.
