North Korea Envoy Executed Over Trump-Kim Summit, Chosun Reports — Maw

LOL. Already debunked. The "Philosophy forum" is a hotbed of fake news when you are around. Just how gullible are you, anyway?

North Korea execution reports - why we should be cautious

But there is a reason we treat reports about North Korean officials being executed with extreme caution. The claims are incredibly difficult to verify and they are very often wrong.

Both the South Korean media and the government in Seoul have reported on purges in the past - only for the "executed" officials to turn up a few weeks later looking alive and well next to the North Korean leader Kim Jong-un.

On this occasion, a single anonymous source has told a newspaper in Seoul that Kim Hyok-chol, the former North Korean envoy to the US and a key figure in talks ahead of the summit between Kim Jong-un and Donald Trump in Hanoi, was executed at an airport in Pyongyang.

https://www.bbc.com/news/world-asia-48477248

The US had to make a decision between throwing everything and the kitchen sink at him so that there's a higher probability something will stick when he's in a US court and a better chance for a successful extradition. They chose the former, which suggests to me the likelihood of conviction on all counts is very low. This was necessary then to request extradition for everything because it is not allowed to request extradition for something and then charge that person for additional crimes once extradited.

The death penalty is a no go in any case but I'm sure they have given assurances they won't pursue it or the extradition request would be stupid. That leaves the sheer amount of years and the extraordinary grounds that might suggest it's politically motivated or disproportionate. Unfortunately, disproportionately is a specific ground for the European Arrest Warrant so you can reason a contrario it doesn't apply to an extradition to the US. So I don't think the chances of UK courts refusing extradition are very high, just that there's a possibility. — Benkei

A very reasoned and reasonable analysis. I can't disagree with anything, nor could I frame a response at that level of stylish erudition.

And yet ...

I'm disappointed others aren't as outraged as I am. I react viscerally to this case and others seem to impute the US government with good intentions and cleverness or strategic thinking. Of rationality, even of human decency.

I don't share anyone's high regard of the US government in this matter. In other posts I've expressed my intense feelings so I'll just state them here without going in to detail. I'm collecting mainstream opinion about this case, The NYT, WSJ, and even Rachel Maddow are expressing their dismay at this criminalization of standard journalistic practice. A common theme is, "Even if you hate Assange you have to be very concerned about these latest charges, which go right at the heart of the First Amendment."

So I'll just say for the record that I'm impressed by the clarity and insight of all the responses so far; but terribly disappointed at the lack of passionate concern for freedom of expression, the rights of journalists (whether you think Assange is one); and frankly, for Julian Assange. He revealed the US doing truly awful, immoral things as we "brought Democracy" to the world. If you're outraged about Assange's alleged "spying" but unaware of the war crimes he revealed, you should educate yourself about the particulars. Your outrage is misplaced.

If I used the phrase "good Nazis" that would be awfully inflammatory. I don't mean to inflame. What's a more measured phrase that would communicate the idea?. People who don't want to rock the boat. A few weeks ago Rachel Maddow was attacking Assange as part of her Russia Russia Russia schtick. Now that Mueller says no collusion and Maddow's ratings have tanked, she recently gave an impassioned defense of Assange and attack on this awful indictment.

In other words ... this latest indictment has snapped a lot of people to their senses. And Rachel Maddow, welcome back to the world of peace and civil liberties. Too many liberals have gone to the other side the last two years.

It's not like this forum is so intellectually dispassionate. Would the people recently telling me that I should be uniquely outraged because "Trump put kids in cages" and "Trump called Mexicans rapists" and "Trump separated families" please join me in a truly outraged chorus of:

Trump is trying to kill Julian Assange and criminalize journalism.

That's an outrage in opposition to which I'll gladly get out my torch and pitchfork.

But no. On the subject of Assange, everyone is suddenly very measured and rational. As if people want to salvage something from their former state of denial about the government's bad intentions and bad faith in this case.

The fact a 102 years old law that has never been employed for this purpose is used and the possibly disproportionate sentencing period might lead to extraneous considerations to refuse extradition. — Benkei

It's interesting that at least two people responded by saying this might make his extradition less likely. There are glass-half-full types around here! Interesting point though. Did the US act too early and thereby make it harder to get their hands on him? We shall see.

Maybe you quoted someone above. — fresco

My cat did it.

I started having trouble being certain about anything I can't perceive through my senses. — AnonThinker25

Don't worry. Read some Berkeley then you won't even believe your senses. Your senses are the only evidence you have that there's anything out there in "external reality." But that's no evidence at all! What if your senses are all that there is? And that there is nothing at all "out there?"

Nothing would be changed. Your senses would still be exactly the same as before.The existence of an external reality is a superfluous assumption. It's not needed to produce your senses.

Berkeley, being a bishop in the Catholic church, thought the cause of our senses was God. From a modern perspective, this is essentially the point of view of simulation theory. It's the great computer in the sky that creates our the illusions of our senses. Nothing is real. We only have our experience of the simulation.

This theory denies the existence of material substance and instead contends that familiar objects like tables and chairs are only ideas in the minds of perceivers and, as a result, cannot exist without being perceived.

Is that not exactly what simulation theory says?

https://en.wikipedia.org/wiki/George_Berkeley

By the way it's not entirely clear from your post whether this is an intellectual concern or whether it's bothering you personally in some way. If the latter, probably best to get "out" more. LOL. Jump into the simulation, the water's fine!

Assange has been formally charged in the US with espionage, which can carry the death penalty.

Does this change anyone's opinion? Do you think journalism should be punishable by death? Why didn't the owner of the New York Times face the death penalty for publishing the Pentagon papers?

Julian Assange Indicted Under Espionage Act, Raising First Amendment Issues

https://www.nytimes.com/2019/05/23/us/politics/assange-indictment.html

“Whatever a patron desires to get published is advertising; whatever he wants to keep out of the paper is news,”

https://quoteinvestigator.com/2013/01/20/news-suppress/

I am glad that you replied to my post, so I can explain in more detail my point of view — Mephist

Oh no now I am falling even further behind in my replies!

I don't believe that physics is founded on ZF Set theory, if that's what you mean by "classical math".
Physics is based on calculus (for the most part), that was created starting from XVII century (Newton's "principia" was published in 1687), at a time when logic wasn't even considered as part of mathematics. — Mephist

Yes but by the 19th century it became clear that the logical problems of calculus were becoming a problem and needed to be addressed. We needed to know exactly what was a real number and a continuous function in order to resolve the delicate convergence issues that were arising.

I agree of course that the physicists don't care and I didn't mean to imply that they do. But physics is based on the real numbers and mathematical analysis in general; and those things were discovered to need a foundation; and that foundation is currently set theory.

In fact, I believe that most of physicists don't even know the exact formulation of the axioms of ZF set theory, and that's because these axioms are never used in physics directly. — Mephist

Of course. In fact most working mathematicians couldn't state the axioms and don't use set theory directly! That's a fact. My point still stands. Once you get picky about which things converge and which things don't, a problem that arose in the infinite trigonometric series that arose out of 19th century physics, you need a foundation and set theory is it. Currently of course.

The thing that really matters is that you can use logic to prove that your results follow from the algebraic rules of calculus, differential geometry, group theory, etc... — Mephist

Sure, not disagreeing at all.

If this was true, the distinction between the various axiomatizations of real numbers would be irrelevant for physics. ZF set theory only happens to be the first axiomatization to be discovered. — Mephist

Perfectly well agreed. But we are not talking about whether Dedekind cuts or equivalence classes of Cauchy sequences are a better way to represent the real numbers. You are advocating nonconstructive foundations. It's in that context that my remarks make sense. We can NOT currently (as far as I understand it) get modern physics off the ground using nonconstructive math. The 't' in the Schrödinger equation is taken to be a variable that ranges over the standard mathematical real numbers, noncomputables at all.

I am aware of only one book that attempts to get physics working on nonconstructive math. That's a tiny drop of water in an ocean that needs to be filled.

I will agree that synthetic differential geometry is some kind of counterexample to my claim but I don't know enough about it. But SDG is not the same as nonconstructive approaches, is it? I thought SDG is just a categorical approach to infinitesimals so it's more like a complementary idea to nonstandard analysis. Not something directly related to homotopy type theory or neo-intuitionism or whatever. But I could be all wrong about all this, I have very little to no actual knowledge.

But this is in fact not completely true. There are some physical results that are not the same in all axiomatizations. For example, the Banach-Tarski theorem. — Mephist

I would never call B-T a "physical" result. On the contrary it's only a technical result regarding the isometry group of Euclidean 3-space, which happens to contain a copy of the free group on two letters, which has a paradoxical decomposition. It has nothing to do with the real world. As far as we know, of course.

But if earlier you said that physics doesn't depend on set theory, then surely we can't apply the axiom of choice to regions of physical space!

If you believe that the physical space can't be split in peaces and then reassembled with double volume, you have to conclude that the real numbers' axiomatization of ZFC is not the right model for physical space. — Mephist

I've often raised this point myself. If anyone seriously believed it was, then we'd see physics postdocs applying for grants to count the number of points in the unit interval and thereby discover the truth of the continuum hypothesis. The fact that the ides is absurd shows how far ZF, let alone C, is from the real world.

I'm not claiming the world's based on set theory. I apologize if my earlier wording made it sound like I did.

I claim that our best physical theories are based on the real numbers and the theory of limiting processes that go under the name of mathematical analysis; and that our current formulation of analysis is in terms of set theory. I don't state this as a metaphysical point, only as a historically accurate one. I don't care if you want to replace one definition of the real numbers with another. I do care that there is as yet no fully worked out constructive theory of the real numbers that will support modern physics. That is my belief. Certainly I could be wrong.

A more "practical" example are Feynman's path integrals: they can be used to calculate physical quantities with extreme precision, if you don't take for real the result that derives from ZFC's axioms. In fact, from ZFC axioms you can derive that the result is "infinite". But if you take only some terms of your infinite (divergent) series, the result is experimentally correct with a very high accuracy! — Mephist

I've heard about renormalization and I am not aware of whether it's been mathematically formalized or not. I found a stackexchange thread that some that "some" version of QFT are mathematically sound and others not! This is all inside baseball in the physics biz.

https://physics.stackexchange.com/questions/86597/qft-as-a-rigorous-mathematical-theory

Regardless, I take renormalization as a modern example of Newton's great use of calculus to work out gravity, two hundred years before we had a logically rigorous foundation for calculus. Physics often leads math that way. That doesn't mean the math of renormalization won't be fully patched up someday in terms of ZFC or some future foundation. Maybe tomorrow morning someone will derive renormalization from homotopy type theory. If that happens I'll have to shut up about constructivism.

I believe that the most popular explanation of this fact among physicists today is that physical space is in fact discrete. — Mephist

Yes we are now back to the ancient question of atomism. Zeno. All that stuff. We are not going to solve it today. The Planck scale doesn't say the world's a discrete grid. It only says that our contemporary theories don't allow us to speculate below that scale. Maybe there are little thingies in there.

Personally I do not think there are Euclidean points. I don't regard the real numbers as a good model of continuity. I'm not making any metaphysical claims for math at all. But I am making historical claims that our best continent theories DO need an underlying Euclidean model.

[By Euclidean I don't mean the geometry, I mean the Cauchy completeness of n-space],

But there is no experimental evidence of this until now.
( The same thing happens with energy: if energy is continuous you get infinite results for black body radiation. With Plank's quantization you get the correct results ) — Mephist

I hope my Planck point was clear. I didn't deny the Planck scale. It's that my own personal understanding is that the Planck scale is that scale of space and time below which we can't sensibly measure or calculate or speculate about. I'm agnostic on whether there's anything actually there.

But with physics you can assume nothing until you find experimental evidence, because nature has surprised us so many times! However, if I had to guess, discrete space (like pixels of a video game) is too "ugly" to be true: nature has shown to prefer "beautiful" mathematical structures. — Mephist

I think we're meeting in the middle. I don't think mathematical continuous space is quite the right model for the world, and you admit that neither is a discrete grid. Most likely it's something even stranger, yet to be discovered by a genius not yet born.

I think an interesting question is: is there an axiomatization of a continuous space (where you can take measures represented as real numbers) where Feynman integrals make sense and are convergent? I don't know, but I don't see a reason why this shouldn't be possible. — Mephist

Don't know enough about the subject, more stuff to read. But quantum physics is probably not something I'll be tackling in this lifetime. I do know a little functional analysis, enough to know what Hilbert space is. So if you tell me that an observable is a linear operator on some complex Hilbert space, I know what that means mathematically. Just not physically. And I'll probably never know.

I'll try to give you a simpler explanation: In logic you have propositions, terms and relations. Martin-Löf type theory ( and HOTT too ) reduces all these three concepts to only one: functions. That's basically the reason why it has a simple categorical model.

The functions that you consider in the theory are, exactly as for morphisms in category theory, not limited in any way. Now the question is: how is it possible to use and reason about functions that are not computable? Obviously, you can't "execute" the function on a given argument to check what's the result. Well, the "trick" is: lambda calculus! With lambda calculus you can "execute" a function by what's called "beta reduction". That means that you can transform and compose functions from one form to another (equivalent to the previous one) using very simple rules that are supposed to be respected by everything that can be called "function".
The whole point about constructivism is that you can assume existing for sure only the functions that you can effectively compute. All the others can enter in the theory only as premises of your deductions (no axioms!). And that is enough to prove all mathematical theorems that can be proved with classical logic, if you add the appropriate axioms corresponding to the "non computable" parts of classical logic. In this way you make clear the distinction between the computable and not computable parts of logic! — Mephist

I understand the words but not the ideas. I suspect it's the kind of thing where mere Wiki reading wouldn't help. I've heard of lambda calculus but don't know anything about it except that everyone thinks it's terribly important. I gave the Wiki a quick scan and they pointed out in their section on beta reduction that "The lambda calculus may be seen as an idealised version of a functional programming language ..."

Ok. But a functional programming language can't do anything a Turing machine can't, and in fact Turing proved that TMs are equivalent to the lambda calculus, and Turing machines can't crank out the digits of noncomputable numbers. So once again I'm up to my original brick wall of understanding. Or misunderstanding. I'm sure that if I grokked this subject I'd suddenly realize that I've been ignorant all these years, and that you can in fact do everything you need to do with mere computability.

I just don't see it yet.

[ that's enough for today.. :-) ]
a day ago — Mephist

Thank you much!

ps --

And that is enough to prove all mathematical theorems that can be proved with classical logic, if you add the appropriate axioms corresponding to the "non computable" parts of classical logic. — Mephist

If I could only understand this. You are saying that if I can prove a theorem at all, I can prove it using constructive methods, by adding certain axioms. This could be within my grasp. What are the axioms that make this work?

pps --

Ah this is that Curry-Howard business I bet. Programs are proofs. Fine, I believe that. A proof has to be computable, but the thing it's talking about need not be. But doesn't that mean something? Proofs aren't sufficient to know mathematical reality, ‎Gödel showed that. Same with physics. Restricting our epistemology to what we can prove with axiomatic systems or computers is not sufficient to understand reality. That perhaps would be my working thesis.

The original quote about 'definite truth value' was yours not mine. — fresco

You have a cat? Mine sometimes walks on my keyboard and writes half the stuff I post here.

"CH has a definite truth value. It's either true or false. — fresco

So you didn't write that? Ok.

r u ok? — Maw

Never better, thanks. @Wayfarer wrote a post that models the direct opposite of the "Orange man bad" school of political discourse. He didn't say, "Oooooh Trump said a bad thing about Mexicans," or "Oooooooh Trump separated families," as if turning children over to their traffickers represents a more humane policy. He wrote something intelligent. I'm incredibly gratified that someone can discuss Trump's policies without resorting to childish emotionalism. Made my day.

My views are rooted in the failure to pass the Immigration Bill of 2013. It wasn't perfect, but it was a good start. — Relativist

I agree with everything you said and I really appreciate your post. Yes you are right, there's a hard core of GOPs that simply will not allow any immigration reform at all. Hillary was right when she said that HALF of Trump's supporters were a basket of deplorables. Trump's rhetoric on Mexico panders to that base and I'm very unhappy about that. But the Democrats pander to much the same base. You may have seen recent news stories that Biden once called for a fence to keep out drugs. Now that Trump's for it, the Dems are against it.

https://www.cnn.com/2019/05/10/politics/kfile-biden-drugs-fence-2006/index.html?no-st=1558569322

So no, it's not just about family separations - but it IS about the intractable position of Trump and his ardent supporters - — Relativist

I just disagree that it's only Trump and the deplorables. You can look up the immigration rhetoric of every one of the prominent Dems over the past twenty years and they're all for border enforcement and all for fences and deportations and employment verification and the militarization of the border. And when there is a humanitarian crisis consisting of a flood of central Americans, Democrats ignore it.

Just the other day Kirsten Gillibrand, a Dem candidate for president, said that if she were president she would let all families into the country without reservation, and she would trust them all to show up for their court proceedings. Statistics show that about 2% of all asylum seekers released in the country show up for their hearings. Gillibrand's rhetoric is no more serious or useful than Trump's. Does anyone want to defend importing tens or hundreds of millions of the world's illiterate peasants into the US with no restrictions at all? Many of them not families but traffickers with their victims? What kind of thoughtful policy is that?

I see both sides as actively impeding any kind of meaningful immigration reform. I just can't see it as all Trump's fault. But yes now that you mention it I do recall the 2013 bill and its scuttling by the Tea party deplorables As Trump would. say ... Sad!

First of all, let's look for a definition of "constructive mathematics", because I have the impression that we are not speaking about the same thing. — Mephist

Wow you wrote a lot and I'm a little overwhelmed. I'll try to respond where I have some value to add. FWIW I have heard about HOTT. I do in fact know what a homotopy is from topology. I don't know the details of how homotopy is used to do logic or foundations. I've heard of the univalence axiom and my understanding is that it says, "Equivalence is equality" or some such. I'm sure I'm missing 100% of the philosophical and mathematical depth of the statement. I know enough category theory to vaguely imagine what a topos might be. I know who Grothendieck but not quite exactly what a sheaf is. I've heard of constructive math and I know about Brouwer. As I've said I regard HOTT and neo-intuitionism in general as Brouwer's revenge. So I know enough to make a little joke but not enough to discuss any of the technical aspects. I know of Bishop's book and I even know a book where a physicist tried to do constructive physics. That's a project in its infancy but you constructivists will have to face the problem someday. Physics is founded on classical math, ie continuity and the real numbers as defined in set theory.

This is all by way of indicating some of the boundaries of my knowledge so that you'll get a better understanding of the deep pool your posts just threw me into!

https://en.wikipedia.org/wiki/Constructivism_(philosophy_of_mathematics) — Mephist

Yes I've perused that article.

There are many forms of constructivism.[1] These include the program of intuitionism founded by Brouwer, the finitism of Hilbert and Bernays, the constructive recursive mathematics of Shanin and Markov, and Bishop's program of constructive analysis. Constructivism also includes the study of constructive set theories such as CZF and the study of topos theory." — Mephist

I know a little about Brouwer and once attempted to try to grok the idea of a free choice sequence. That's as far as I ever got with classical intuitionism. I find it murky in the extreme.

I wouldn't call Hilbert any kind of finitist, but you might know more about that than I do. Bernays I only know from ‎Gödel-Bernays set theory but beyond that I know nothing.

Shanin and Markov I've never heard of, I'll Google. Bishop's book I've heard of. Constructive set theory I know nothing about. As I said I know enough category theory to have a vague notion of what a topos is. I'd love to get to that point someday.

Well, the constructivist theory that I know quite well is the "Calculus of Inductive Constructions", or CIC ("https://en.wikipedia.org/wiki/Calculus_of_constructions"), implemented in the Coq proof assistant (https://coq.inria.fr/).

This is a particular version of "Martin-Lof Type Theory" (https://en.wikipedia.org/wiki/Intuitionistic_type_theory) — Mephist

I'll check out the Wiki link. Martin-Löf type theory I've heard of about a million times and have no idea what it is. I confess I give priority to trying to understand more of the classical math I wish I'd studied harder in my youth, and not so much with all the neo-intuitionist stuff. I like Steve Awodey's category theory book and Youtube videos and he's a big name in HOTT so I'm sure I'm missing a lot of good stuff. I'm just buried under an avalanche of stuff that I know a little "about" but that I'll never know. It's tragic, really. All this is part of that avalanche.

Here's a summary of the main properties of Coq's logic: https://github.com/coq/coq/wiki/The-Logic-of-Coq — Mephist

I know a little about Coq as a proof assistant but no details. I am grateful for your summary and I will try to work through it at some point. I have to confess I did not dive into this exposition but I'm glad you wrote it.

Basically, Coq is a programming language (let's cal this the "internal language" of the logic) based on dependently typed lambda calculus (https://en.wikipedia.org/wiki/Dependent_type) that allows only to build total functions: using this language it's impossible to define a function that does not terminate (then it's less powerful than the full untyped lambda calculus, or a Turing machine).

However, using the internal language you can operate on inductive (and co-inductive) data structures and on functions (the models about which you can speak about and on which you can operate using the internal language) that ARE NOT REQUIRED TO BE COMPUTABLE.

For example, you can say "let R be a field and let F be a function such that forall x:R, F(x)^2 = x; Let x be F(2)". You can represent for example real numbers as functions from natural numbers to the finite set {0...9 and comma} to make things simpler (the infinite decimal expansion of the number), and then you can prove that the algorithm that produces the digits of the square roots of two IS a real number that solves your equation. Of course "x" is not a recursive function (of course, it does not terminate if you execute it on all natural numbers! ), but to build it you only need a finitely defined algorithm.

In a similar way, you can say "Axiom excluded_middle: forall A:Prop, A \/ ~A." meaning: excluded_middle is a function that takes as input the prove of a proposition A and produces as output the prove of the proposition A \/ ~A. This function is not implementable in the internal language of Coq, but you can treat it as a pre-defined "external" function that you can call and use the result without knowing how it works (something like a function defined in an external library of a programming language).

The logic doesn't make sure that the function exists (and even it doesn't guarantee that i's consistent: the proof of consistency is related to the meta-theory of the internal language), but only says that if you have this function, you can use it. And the resulting logic is equivalent to classical higher order logic (not set-theory). — Mephist

Note to self, read and understand all of the above someday.

But if I'm skimming reasonably, you are saying that someone's figured out how to do a satisfactory theory of the real numbers using constructive math? Well ok if you say so but where are all the noncomputable numbers? Maybe you can explain that more simply. Also what about the various constructions that require the axiom of choice? The nonmeasurable set, the Hahn-Banach theorem used in functional analysis which is used in physics? Zermelo's well-ordering theorem. The fact that every vector space has a basis, and that every surjection has a right inverse? That every unital commutative ring has a maximal ideal. That every Dedekind-infinite set is infinite. That every field has an algebraic closure. Mathematicians are not going to give all those things up and they all depend on the axiom of choice hence LEM.

The correspondence between set theory and the various versions of dependent type theory is rather complex:
See for example (https://mathoverflow.net/questions/69229/proof-strength-of-calculus-of-inductive-constructions):
"IIRC, the calculus of inductive constructions is equi-interpretable with ZFC plus countably many "inaccessible cardinals" -- see Benjamin Werner's "Sets in Types, Types in Sets". (This is because of the presence of a universe hierarchy in the CIC." — Mephist

Ok I'll add this to my reading list. You know I could spend the summer just working through your post! But ... are you telling me you can believe in countably many inaccessible cardinals but not a noncomputable real number? I didn't even think constructivists believed in uncountable sets.

And this is a very good reference for intuitionistic type theory: https://plato.stanford.edu/entries/type-theory-intuitionistic/ — Mephist

More for my reading list.

I read that you can't define Cauchy or Dedekind real numbers in intuitionistic logic (without additional axioms), but you can use other definitions of "continuity": — Mephist

Ok right. This is perhaps a sticking point for me because I am so steeped in the traditional set-theoretic real numbers that I have a very hard time taking any other approach seriously. If you remove the noncomputable reals then there are Cauchy sequences that don't converge. Your "continuum" has more holes than points. I personally find that a serious problem. However I'm certain that all the smart people working on HOTT are perfectly aware of the problem and are ok with it somehow.

http://www.alternatievewiskunde.nl/QED/brouwer.htm
"Any function which is defined everywhere at an interval of real numbers is also continuous at the same interval. With other words: For real valued functions, being defined is very much the same as being continuous." — Mephist

Yes, I've heard that. If you restrict to continuous functions you can make all this work. Will add the link to my reading list. Seriously, this is an awesome research program you've outlined for an aspiring neo-intuitionist.

and MY FAVOURITE ONE:
https://plato.stanford.edu/entries/continuity/#9 "Smooth Infinitesimal Analysis" — Mephist

Oh yes another one I've heard of. Well differential geometers are closet physicists so they believe in infinitesimals and I gather you can take a category-theoretic approach that makes all this work. More stuff I know a little about without knowing any of the details. It's like my brain is filled with a skeleton but not enough is filled in. But yes I'll perfectly well agree that smarter people than me can make infinitesimals work in a categorical framework. I'm not saying they can't, LOL!!

What is your objection to this axiomatization of real numbers? — Mephist

I'm sure I have no objection. Lawvere I know from his elementary theory of the category of sets, ie you can do set theory without ever talking about elements. I'm sure in a hundred years nobody will remember set theory anymore, it's clearly doomed. I have no objection. I can only plead ignorance of even more stuff I wish I knew.

One thing I do know is that the axiom of choice implies LEM. To reject the axiom of choice involves throwing out quite a lot of modern math.
— fishfry

Intuitionistic type theory implies a somewhat "weaker" version of the axiom of choice (the "good" part of it :-) ) — Mephist

I'm gratified that we are in agreement on the areas where our knowledge overlaps. Weak forms of choice are often sufficient. However the facts that every vector space has a basis and every surjection has a right inverse (ie section) are both equivalent to full choice. I suppose the constructivists have a fix.

[ ..going to continue next time! ] — Mephist

I'm out of breath just typing. Let me tackle the other posts later.

"CH has a definite truth value. It's either true or false. — fresco

That is a Platonic claim. It can be strongly argued against. I"m not taking a position one way or another but only pointing out that your claim is arguable.

Consider a variant of the game of chess in which pawns may be promoted to queens or rooks but not knights or bishops. That is not a very radical change in the rules. There are in fact many variants of chess.

Now we come upon two expert chess players arguing over which version is true. But we can see that there is no truth of the matter at all. Chess and variant-chess are formal games. We make up the rules arbitrarily. The only requirement is that the rules are sensible enough so that the game is playable; and that enough people find it fun and enjoyable to play. There is no requirement with truth.

To a formalist, math is the same. It's a meaningless game played with marks on paper according to rules.

https://en.wikipedia.org/wiki/Formalism_(philosophy_of_mathematics)

To a formalist, CH has no definite truth value. We can play the game with CH or with its negation. And when it comes to CH it's a very interesting situation. All of the new axioms which set theorists have studied in order to get a handle on CH are CH-agnostic. You throw in a new large cardinal axiom, for example, and there's a CH and a not-CH version.

Now if someday some physicist determines that ZFC is instantiated in the physical world, then CH would become a research project and would have a definite truth value.

Till that day, if it ever comes, we can only ask if CH is true in the "correct true model of set theory out there somewhere." And the very existence of such a world is a Platonist dream. Gödel himself, as I've mentioned, was a Platonist. His incompleteness theorems to him mean that there is a realm of mathematical truth that's not accessible to the axiomatic method of symbol manipulation.

All that is by way of saying that when you say there is a definite truth value to CH, you might as well ask how pawns may "really" be promoted. The question is a category error. There is no truth in formal games.

That's semantics. But syntactically, we have no proof".
I'm not clear what you mean by 'syntax' here. The 'semantic point' is that the phrase 'definite truth value' automatically invokes the semantic context of classical binary logic. — fresco

Our syntax consists of:

* An alphabet of symbols;
* The usual rules by which we can form well-formed logical and mathematical formulas;
* The inference rules of first-order predicate logic, by which we can start from a set of wffs called the axioms, and derive other wffs called the theorems. Note by the way that an axiom is a theorem, since every axiom A has a one-line proof, namely A.
* The axioms of ZFC.

I should mention that the rules for wffs and the rules of inference are computable. You could write a program (ie Turing machine) to recognize a valid wff and a valid inference.

It is a fact that there is no proof in ZFC of CH nor its negation. That's syntax.

Semantics is an interpretation. Some universe of set theory, called a model, in which CH or not-CH are a matter of observable fact. The question is whether there is a Platonic "correct" model of set theory that settles the issue of CH. A lot of smart people haven't found one yet.

....on further consideration, I assume you mean 'rules governing what constitutes a valid form of answer'. On that assumption we are touching on 'Zen Koan' territory which forces the pupil to consider the assumptions regarding the structure of 'the question'.. In that case my identification the inapplicability of the rules behind the assumptions of classical logic could be regarded as a 'syntactic' point — fresco

No, nothing so woo-woo. A simple matter that syntax, the formal rules of deriving theorems from axioms, does not settle the question of CH when starting from ZFC. One can then find interpretations of the symbols in which CH is objectively true; and interpretations in which it's objectively false. And nobody knows an interpretation of set theory that's so obviously "the right one" that we're willing to call it the official model and thereby determine the truth value of CH.

I hope this wasn't too wordy and addressed some of your concerns. Syntax = derivations, semantics = interpretations.

No one is denying that the Democrats have been bad on immigration for nearly 30 years at least, but the reason we are focusing on Trump is because he has been president for over two years, and immigration has been his primary clarion call. All fishfry is doing is pure whataboutism. — Maw

I disagree that this is mere whataboutism. When Trump haters tweet out a photo of "Trump's child cages" that actually turn out to have been Obama's, I am entitled to call out the hypocrisy. When you fixate on Trump's awful rhetoric on Mexico and compare it to Obama's actual record on Mexico, you find that on balance, if you're a man from Mars, you would conclude that Obama did far more damage to the US-Mexican border than Trump has. You don't like Trump's style. Well yes Obama had great style. And did a lot of damaging things. Obama's border policy was awful. Obama's malfeasance on the border has led to the humanitarian and political disaster we have now. And yes Trump's rhetoric's made it worse. But that doesn't mean you can say that and then stop thinking. Try to have TWO thoughts. Orange Man Bad, ok. Now try to have ANOTHER thought as well.

Here's another example from only two days ago. Trump gave a speech and said something sensible -- or at least arguably sensible -- about immigration. He said we should prioritize merit instead of family ties.

This is a perfectly sensible statement, even if you don't agree with it. One can make a case that a country should screen immigrants based on their potential ability to thrive or at least survive on their own in our society.

Nancy Pelosi, Democratic Speaker of the US House of Representatives, spoke out against merit.

“It is really a condescending word. They’re saying family is without merit?," Pelosi said at her weekly press conference.

...

"Are they saying most of the people who have ever come to the United States in the history of our country are without merit because they don’t have an engineering degree?" Pelosi said, drilling into the administration's argument.

https://thehill.com/homenews/house/444047-pelosi-says-merit-based-immigration-is-a-condescending-word

Have any of you worked in the tech industry? The tech industry is full of H1b immigrants from India who have technical degrees in computer science and engineering. There are in fact about half a million H1b's in the country at any given time.

India is a country of 800 million people. I'm sure they could find 50 or 100 million illiterate peasants to send to the US. And why not? Does Nancy Pelosi think we should take in India's illiterate peasants? Don't their families have merit?

We import illiterate peasants and laborers from Mexico; and college educated professionals from India. Why? Because the government is helping out the farmers with farm labor, and the tech companies with tech labor.

And by the way why don't we import India's teachers? Because the teachers have a better union than the programmers.

But really, why not engineers from Mexico? Mexico has bridges, power plants, roads. Mexico has excellent engineers. But Silicon Valley isn't lobbying Congress to increase the cap on Mexican H1b engineers.

Why is this, anyway? Our immigration system makes no sense. But here is Nancy Pelosi literally denying the reality of Indian immigration of highly skilled professional workers. Why? Because]she knows that her listeners don't know shit about our actual immigration system hence don't even know about the H1b's from India; and two, she doesn't care. Pelosi knows about the H1b's because it's Congress who authorizes their presence.

Pelosi damn well knows immigration's based on merit. She and Congress agree on that fact. She just denies it in public because Trump tried to say something sensible on immigration.

I object to this level of hypocrisy. Again, if ALL you see is that "Trump called Mexicans rapists" then you are missing the evil hypocrisy in the news every single day. Do you think Nancy Pelosi is really advocating for 100 million illiterate Indian peasants to come to the US?

Or do you think she's just saying the sky is green simply because Trump said it's blue?

It's a sick joke that Pelosi mocked the idea that we'd restrict immigration to people with engineering degrees. THAT IS EXACTLY WHAT WE DO and Pelosi knows it because she signs off on the legislation making it possible.

Please try to see past your dislike of Trump's style, to the bullshit emanating from literally everyone in Washington about literally everything.

Trump's zero-tolerance policy treated all border-crossers as criminals, which resulted in separating children from parents whose only crime was crossing the border. — Relativist

I'm not defending Trump's immigration policies, since in fact I oppose them.

I'm simply calling attention to, and expressing my deep frustration with, the bipartisan decades-long legacy of bad decision making and bad policy that's resulted in a terribly inhumane and indecent situation. And if ALL you can see is "Trump separated families," I can only repeat that I find that kind of thinking ignorant (if you simply don't know anything about US immigration policy), disingenuous (if you do, but pretend not to for partisan purposes); and in any event, childish. Yes Trump's border policy sucks. But both parties are to blame for how the situation got to this point. So ignorance doesn't help here. Nor does it convince me that you are trying to make a serious point about immigration.

ps --

https://www.washingtonexaminer.com/policy/defense-national-security/dna-tests-reveal-30-of-suspected-fraudulent-migrant-families-were-unrelated

They did a pilot program where they DNA-tested illegal border crossers with kids. 30% of the kids didn't belong to the parents. They also busted a ring of criminals recycling kids to act as family members.

So say you are in charge of US border policy. When adults bring kids across the border and say they're family, do you decree that "OK, come on in?" Or do you separate the families until you can determine who is a loving parent and who is a child trafficker?

Come on, please give me an honest answer. You're in charge of policy. What do you do?

Thakyou. I stand corrected on the technicalities of Cohen's work. But as an example of 'problems' with classical logic I still claim validity.
I have no idea where you are hoping to go with my alleged 'confusion' between syntax and semantics etc. As far as I'm concerned the contexts in which you want to differentiate between those terms is nothing to do with the context of my anti-classical logic position. — fresco

I only read your post and commented on your remarks regarding CH. I didn't take a position on logic. Sorry for any confusion.

The point about syntax and semantics is that in terms of syntax, we can neither prove nor disprove CH within ZFC. But we can exhibit a model, or interpretation of ZFC, in which CH is true (‎Gödel 1940) and another model in which it's false (Cohen 1963). In any given model of ZFC, CH has a definite truth value. It's either true or false. That's semantics. But syntactically, we have no proof.

As I say if you are making a larger point, I didn't address it.

A similar issue seems to be the suggested of Cohen's award of the Field's Medal, for proving both that there was and was not 'another infinite set of cardinality between Cantor's infinite sets. — fresco

Good God. Cohen did nothing of the sort. He showed (in conjunction with ‎Gödel) that CH was formally independent of ZFC. ‎Both ‎Gödel and Cohen believed that CH is false -- in other words, that it has a definite truth value. Just one that's not accessible via ZFC.

You are confusing syntax with semantics, formal systems with models.

By the way you even stated CH incorrectly. CH doesn't say that there's a set "between Cantor's infinite sets." Rather, the negation of CH is that the real numbers have a cardinality that's larger than Aleph-1. ‎Gödel believed the reals had cardinality Aleph-2. Cohen thought it might be much larger than that. But all the Alephs are Cantor's cardinals.

I get that you don't like Trump's style.
— fishfry

Do you like Trump's style, of inciting hatred for the purpose of political advantage? — Metaphysician Undercover

No I find some of the things Trump does appalling. As I've mentioned I'm what you might call a Mexicophile. I moved to California as a young adult and always had a great affinity for Mexican culture. I travelled through country years ago and recently lived there for a few years. All things being equal I am closer to an open-borders type. I regard Mexico as a friend and neighbor. I oppose Trump's policies on Mexico and I am sickened by some of his rhetoric.

So how the hell come I am here seeming to defend Trump?

It's because I can see what Trump is doing; butwhat the Democrats have done on border issues over the past couple of decades is much worse.

Democrats talk a great game on compassion. Which frankly I appreciate because I have tremendous compassion for the plight of the people whose best hope in life is to somehow get into the United States by any means necessary.

But in order to defend themselves against political charges of being "soft on immigration," the Dems have passed some of the most harmful bills and pursued some of the most inhumane and literally inhuman policies imaginable. They passed the Secure Fence Act of 2006. I've heard "liberals" say, "Oh that's a fence, not a wall." Spare me the sanctimony.

Google some of the immigration rhetoric of Hillary, Obama, Biden, Bill Clinton. Look at the laws they passed. Go back to the 1980's. Reagan signed a huge amnesty. The Bushes as you know have close ties with Mexico both in business and in their own family. They were always good on immigration. In fact Bush proposed a very sensible program of immigration reform. The right of course rejects any talk of immigration reform so they objected; and the left hated anything that came out of Bush's mouth (with very good reason of course) and so Bush's actually pretty good idea quickly disappeared.

Bill Clinton was tough on immigration. Obama deported records of Mexicans and hardened the border. All to placate the right so he could get his domestic programs through.

And the drug war. 100,000 Mexicans died between 2000 and 2010 in a bloody drug war down there. Financed by US Democrats like Hillary and my own California Senator Dianne Feinstein, who is called by the right a "liberal" but who is the most bloodthirsty warmonger and opponent of civil liberties in the Senate. She votes for the wars and her husband profits. You could look it up. Don't get me started on DiFi.

US government financial aid to Mexico was conditioned on the money being used to fight the drug war. As if Mexico "pushes" drugs on the US. On the contrary it's spiritually sick Americans who smoke, shoot, snort, and pop every mind-numbing substance known to man in order to cope.

You don't know about the American backing of the bloody drug war in Mexico run by powerful Democratic politicians because Rachel Maddow didn't tell you about it. You could Google it.

I can't give you chapter and verse on every dirty deed the Dems did in the past 30 years because this is a forum post and not a book that needs to be written. The Dems funded all the surveillance and interior checkpoints (awful violation of the Constitution) and the militarization of the border to buy off the Republicans who wanted tough action. So it got harder and harder and harder to cross the border. Migrants had to go farther out into the desert.

In the meantime the same Dems pass sanctuary city laws (which I happen to support). What is the net result?

We leave people to die of thirst in the desert. And if they make it through, we give them drivers licenses, job, legal protection.

What kind of fucked up immoral system is that?

So when someone says, Oh Trump said something awful; or that his policies are awful, you get no argument from me. It only seems that way.

It's that when you tell me that Trump personally injured your family because he "caged children." GIVE ME A FUCKING BREAK.. Obama caged children. The big joke is that the caged children meme got started because someone tweeted a photo of kids in cages and said they were Trump's cages. But the photo was from 2014 and was one of Obama's cages.

If you don't separate the families then you will be turning kids over to traffickers. Obama had documented cases of that and plenty more that were not documented. Better optics than separating the kids from the adult to find out who's a family member and who's a trafficker.

"Trump put kids in cages" is a slogan, not an actual thought.

Or when you tell me that "Oh he called Mexicans rapists."

Bill and Hillary Clinton and Obama and Biden and Pelosi and DiFi, Harry Reid, Chuck Schumer -- all the big "centrist" Dems of the past 20 years -- passed laws that damn near destroyed Mexico.

And now it's all Trump did a bad thing and Trump said a bad thing and that's all you want to know?

I have three words for that kind of thinking. Ignorant. Disingenuous. Childish.

Ok this has all been on my mind for a few days. This is how it came out tonight. I wish I could write the book. I can document everything. It's all well known. I'm not letting the GOPs off the hook but frankly only half the GOP hate the immigrants, the social cons. The business-oriented GOPs love the cheap labor. And of course when it's illegal the workers can't complain if you don't pay them.

So it's a sick, depraved, hypocritical, inhuman, inhuman, and evil system we've developed over the southern border. It's bipartisan but the Dems have been much worse because at least the Bush family regards Mexico as a friend. I for one would like to see some meaningful immigration reform in my lifetime.

But "Oooh Trump put kids in cages" and "Trump called Mexican rapists."

Yeah. Those things are true. And so is a lot more. So stop throwing out slogans as if this is the politics forum on Craigslist. Try to see beyond your angry emotions. I get you don't like Trump. Try to have another thought besides that.

Ok that's what I have to say about all this.

-- forall is a loop that stops returning false if it finds a false value, and it's value as proposition is false if it stops and true if it loops forever — Mephist

I'm not sure what that means. You can't iterate through a set unless it's well-ordered. And there are sets that are not well-ordered, unless you accept the axiom of choice, which implies LEM. How would you iterate through the real numbers, for example? Here's how I'd do it. First, well-order the reals. Second, use transfinite recursion. Those are two powerful principles of math not available to constructivists. I don't know how constructivists feel about transfinite recursion, but it applies to very large sets that constructivists can't possibly believe in.

But the "for all" operator can be applied to the reals without the need for such powerful machinery as the axiom of choice and transfinite recursion. You just say "for all reals" and you're good. That's much more powerful than iteration IMO.

I do not believe that the universal quantifier is the same thing as iteration, for this reason. There are sets that you can apply "for all" to but that you can't iterate through in any obvious manner.

This tranlsates to the english: "however you choose to match elements of A with elements of B, there will always be an element of B that is not matched by any element of A"
I think this is a quite intuitive definition of the fact that B has at least one element more than A. — Mephist

I can't agree that intuition is useful here. Intuitively, the even numbers adjoined with a single odd number are "larger" than the even numbers; and in fact this is confirmed by the proper subset relationship. But these two sets have the same cardinality. This is specifically a point of confusion for beginners. I reject naive intuition here and insist that "bigger" means injection but no surjection, end of story and no intuition allowed! Intuition is exactly what gets students into trouble when they first learn the math of infinity.

Moreover if you reject the axiom of choice (necessary if you reject LEM) there are "amorphous" sets that have no sensible cardinality at all. Such sets defy anyone's intuition.

https://en.wikipedia.org/wiki/Amorphous_set

I should add that constructivists have considered these issues regarding the axiom of choice.

https://en.wikipedia.org/wiki/Constructivism_(philosophy_of_mathematics)#Axiom_of_choice

Yes. But then you can't get the theory of the real numbers off the ground and the intermediate value theorem is false. The constructivists counter: The IVT becomes true again we only consider computable functions.
— fishfry

Sorry, I don't understand what you mean here: The constructivist (intuitionist) logic is only a "more general" logic than classical logic, since it has less axioms. As a result, it is valid for a bigger set of models: — Mephist

That's an interesting point. Here's why IVT is false in constructive math. IVT, you will recall, says that if a function from the reals to the reals is continuous, and if it takes a negative value at one point x and a positive value at another point y, then it must be zero at some point between x and y.

That is if we have f(x) < 0 and f(y) > 0 then there exists p with x < p < y and f(p) = 0. This is intuitively correct about a continuous function f.

If you take the standard real line and omit all the noncomputable points, you get the computable real line. The computable real line is full of holes where the noncomputable reals used to be. You can have a continuous function whose graph passes through the x axis at one of the holes. To a constructive mathematician that zero does not exist. You can drive a continuous function through a hole in the computable real line.

For example for a noncomputable p, there's a straight line that passes through the x-axis at p. But the point (p,0) does not exist for constructivists. We have a straight line that crosses the x-axis without ever taking on the value zero.

I personally find such a model of the continuum unsatisfactory in the extreme. It violates every intuition about what a continuum should be. I think that's both a mathematical and a philosophical problem for the constructivists.

You mentioned a lot of other interesting things that I'll try to get to later. One was about LEM and HOTT. I was not under the impression that HOTT is a constructivist theory. I'm sure Voevodsky believed in uncountable sets. They are a simple consequence of the first-order axioms of set theory. A machine could verify Cantor's theorem. So I am not sure that HOTT and constructive math are the same. On the contrary I'd assumed they're different. But I'm mostly ignorant regarding HOTT.

One thing I do know is that the axiom of choice implies LEM. To reject the axiom of choice involves throwing out quite a lot of modern math.

https://en.wikipedia.org/wiki/Diaconescu%27s_theorem

OK, probably this is not the conventional point of view in mathematics, but I'll try to explain my point of view: — Mephist

Thanks for this interesting post.

If I am reading you correctly you are espousing a computational or constructivist point of view. Perfectly valid, and even trendy lately as computer science begins to influence math. Homotopy type theory, etc. I like to think of it as the revenge of Brouwer!

In ZF Set theory you have the "axiom of infinity" that says ( in a simplified form ):
There exists a set A such that:
-- ∅∈A,
-- for every x∈A, the set (x∪{x}) ∈ A.

This, from my point of view, is a complicated definition for a recursively defined structure that is equivalent to Peano's natural numbers:
You can build ∅ (equivalent 0), ∅∪{∅} (equivalent to S0), ∅∪{∅}∪{∅∪{∅}} (equivalent to SS0), etc... — Mephist

Yes and no. The axiom of infinity does provide a model of PA inside ZF.

However there is a crucial distinction. PA says that if n is a number then S(n), the successor of n, is a number. That gives us (after the standard naming conventions) a handy collection 0, 1, 2, 3, 4, ... which we can use to do a fair amount of number theory.

What the axiom of infinity says is that {0, 1, 2, 3, ...} is a set, not just a collection. That's a much stronger statement. Without the axiom of infinity we still have PA and everything that we can do with it. But we don't have the powerset of the natural numbers nor do we have an easy way to get the theory of the real numbers off the ground.

The axiom of infinity is a strong statement that says that we not only have successors; but that the collection of all successors forms a set. In fact this gives us an easy way to visualize proper classes. In ZF minus infinity, the collection of all natural numbers is a proper class because it's too big to be a set.

Then, first order logic postulates the existence of functions.

I asked on math.stackexchange once what functions are in PA, since we can't be sure we have enough sets. I don't recall getting an answer that satisfied me. But if we naively regard functions as mappings, we can probably get away with it. I'm sure logicians have a good answer for this point.

Basically, Cantor's theorem proves that, for every set A, the function A -> (A -> Bool) is always bigger than A (in the usual sense of no one-to-one correspondence). — Mephist

Let me be picky here (as if I'm ever any other way!) First, it's not the function that's bigger, it's the powerset. But Cantor doesn't prove that P(A) is bigger than A. He proves that there is no surjection from A to P(A); then we DEFINE "bigger" to mean there's no surjection. This is a common point of confusion among those who complain that it's absurd to call one infinite set bigger than another. They are quite right! Rather, "bigger" and "smaller" in this context refers to the existence or nonexistence of injections and surjections. I see that you basically said that by mentioning 1-1 correspondence, but I wanted to emphasize this point. There is no surjection from the positive integers to the reals. Whether the reals are "bigger" in any meaningful sense, I have no idea.

Then, since (A -> Bool) is itself a set, you can build an infinite chain of sets of the form (((A -> Bool) -> Bool) -> Bool) -> ...
So, from the infinite set you can build an infinite hierarchy of infinites. — Mephist

This blew my mind when I first saw it. Still does. Cantor really put a zap to the world of math. This was quite a revolutionary discovery. Or fraudulent sophistry, depending on one's point of view.

The crucial point here to decide the cardinality of "A -> (A -> Bool)", is what you take as a model for functions. — Mephist

Well again, yes and no We do NOT need to know what is the cardinality of P(N). We only know that there is an injection from N -> P(N) and no surjection. So we say P(A) has a larger cardinality. In fact nobody has the foggiest idea what is the cardinality of P(N). That's the Continuum hypothesis.

If functions are only recursive functions (what I can evaluate using a decidable set of rules applied to the symbols of the language), I can never build anything that has non-numerable cardinality. — Mephist

Perfectly well agreed. I hope you don't think I'd be shocked or would object. The computable powerset of N is countable of course, since there are only countably many Turing machines. And now we're into constructivist philosophy. And since we live in the age of computation, there's growing support for this point of view. Can't fight the tide.

In this case, it is
true that for every set you can find a bigger set, but their cardinality will be numerable: at the same way as it is true that for every integer, you can always find a bigger integer. — Mephist

Yes. But then you can't get the theory of the real numbers off the ground and the intermediate value theorem is false. The constructivists counter: The IVT becomes true again we only consider computable functions. And so the argument goes. I'm sure we both understand each other's point of view here. There's no right or wrong to the matter. Just history and philosophy.

If instead you define functions in the usual way as "sets of couples", then there are 2^|A| possible functions of type A -> Bool, and there exists a non-numerable set of sets. But, in any way, inside your language you are able to describe only an enumerable set of them. — Mephist

Yes agreed. We agree on everything. I just think the world of the full powerset of N is richer and more interesting than the computable powerset. That's an aesthetic judgment. And it's more useful. That's a pragmatic judgment.

(https://en.wikipedia.org/wiki/Skolem%27s_paradox):
"From an interpretation of the model into our conventional notions of these sets, this means that although u maps to an uncountable set, there are many elements in our intuitive notion of u that don't have a corresponding element in the model. The model, however, is consistent, because the absence of these elements cannot be observed through first-order logic" — Mephist

Right. The model thinks it's uncountable but from "outside" we can see it's countable.

As I understand it, Skolem himself thought that his paradox showed that our notion of set is murky and not entirely coherent. This is a point of view I can't disagree with. Nobody knows what a set is. Sets as conceived in set theory are much more strange than the "collections of objects" that we teach in high school.

Since both models for functions are consistent, what's the reason to take the standard interpretation as the "true" one? — Mephist

Aha! Well, I don't think Cantorian set theory is true any more than I think chess is true. Does that knight "really" move that way? It's a silly question. Chess is a formal game. And when pressed into my Cantorian corner, I put on my formalist hat and say that we can adopt the axiom of infinity or reject it. Accept the full powerset of N or only the computable powerset. Accept the law of the excluded middle or reject it. Accept the axiom of choice or reject it.

Now the question is: Why adopt the full powerset? The answer is pragmatic. In the 20th century, full Cantorian set theory has proven supremely useful. It lets us develop a logically rigorous theory of the real numbers. The physicists use the real numbers to build their theories. Throw out Cantorian math and you lose a lot. Nobody has yet succeeded in developing a computable or constructive theory of physics. People are working on it. Perhaps we'll need another few decades to get more insight into this question.

No I don't believe so. This is confusing syntax with semantics. A formal theory consists of finite-length strings of symbols. But models of those theories are not strings of symbols. They're sets, often quite large ones.
— fishfry

See Herbrand interpretation: https://en.wikipedia.org/wiki/Herbrand_interpretation. — Mephist

I read that article and didn't fully understand it or the point you're making. But whether math refers to anything outside itself is another one of those philosophical questions. Chess doesn't. Why should formal math?

The demonstration of Godel's incompleteness theorem is using a similar technique to map natural numbers on the syntactic structures of the language and properties of arithmetic operations on syntactical rules of logic. — Mephist

Math is more than ‎Gödel numbering. But now I'm making a Platonist argument when a couple of paragraphs ago I was making a formalist one. I tend to use whichever argument is handy for my point. But ‎Gödel's incompleteness theorems are about syntax and not semantics. ‎Gödel himself was a Platonist. I found that surprising when I first learned it. He believed there is mathematical truth "out there" that's beyond the limitations of formal systems.

As usual, a citation from the old good wikipedia can come to the rescue :-)
"Set theory (which is expressed in a countable language), if it is consistent, has a countable model; this is known as Skolem's paradox, — Mephist

Right. A countable model. Not a finite one.

This of course doesn't mean that uncountable sets "do not exist", but only that you cannot use a finitary
first order logic theory to prove that they exist. — Mephist

I don't believe this is true. ZF is a first-order theory with a countably infinite language that easily proves uncountable sets exist. See Cantor's theorem, as simple a proof as one can imagine for a fact so profound.

So, you could even say that "every mathematical object is a string of characters", — Mephist

No I don't believe so. This is confusing syntax with semantics. A formal theory consists of finite-length strings of symbols. But models of those theories are not strings of symbols. They're sets, often quite large ones.

However unlike Trump, Obama never called Latin Americans "rapists", — Maw

Confusing rhetoric with policy. I get that you don't like Trump's style. Obama deported record numbers of undocumented immigrants. You could look it up. Perfect illustration of why I won't participate in these insipid political discussions. Obama's actual record on border issues was awful. He always had great rhetoric. And a jump shot.

Your statement that Trump called Latin Americans rapists is a lie, of course. You could look up the quote. Orange Man Bad. Not conducive to thought.

Here's a little light reading to bring interested readers up to speed on Obama's reality versus rhetoric on immigration.

https://abcnews.go.com/Politics/obamas-deportation-policy-numbers/story?id=41715661

https://www.migrationpolicy.org/article/obama-record-deportations-deporter-chief-or-not

https://www.politico.com/story/2017/08/08/trump-deportations-behind-obama-levels-241420

orry, I wanted to write "finitary", in the sense of "recursively enumerable" (of course not finite, if you can build natural numbers with sets) — Mephist

Oh yes ok, thanks for clarifying.

However, this idea is not mine: (https://www.youtube.com/watch?v=UvDeVqzcw4k) see at about min. 8:23 — Mephist

very non contradictory axiomatic theory based on first order logic has a finite (non-standard) model — Mephist

Oh HOTT. Very interesting. I don't know much about it beyond the basics. I only watched the vid starting a little before the 8:23 mark and didn't find what you were quoting. I saw where he was showing that every mathematical object is a tree once you expand its set-theoretic nature into the sets that contain sets and so forth. Very nice little insight. Maybe I'll watch more of this. Very tragic about Voevodsky's premature demise. If he said there are finite models I'm sure there are!

As far as separating families, Obama did the same.
— fishfry

Seriously, you cannot tell the difference between Donald Trump and Barack Obama? — tim wood

In terms of the crisis on the border? The main difference is the way the MSM ignored Obama's 2014 humanitarian disaster on the border and politicized Trump's. FWIW -- since someone earlier asked about my personal life -- I formerly lived in Mexico for several years and follow border issues with great interest. No, I do not see much substantive difference between Obama's clusterfuck on the border and Trump's.

I have a number of other mentions on political topics. I hope nobody minds if I don't reply to those. I find political conversations here futile. "Seriously, you cannot tell the difference between Donald Trump and Barack Obama?" That's disingenuous.

Political conversations are tedious when they are so unserious. Political philosophy is not political advocacy. People who viscerally hate Trump and who can't see beyond that are missing a lot. In this case the past several decades of bipartisan failed immigration policy leading directly to today's crisis. If all you know is Orange Man Bad you just can't even think. I see so much of this lately.

A whole is always quantitatively greater than a part, but a part is always proportionately greater than a whole. — Troodon Roar

What do you think about the even numbers within the set of positive integers?

These examples show that it is a general metaphysical principle that, whenever there is a whole, it always lacks the full nature of each of its parts, — Troodon Roar

Isn't that just what they call emergence? Hydrogen and oxygen aren't wet, but water is wet. Etc.

"Under past administrations ..., — Relativist

LOL. Fact-check from Trump-hating Wapo. https://www.cnn.com/2019/04/09/politics/fact-check-trump-claim-obama-separated-families/index.html?no-st=1557361075

Here's one of Obama's kid cages.



But again, completely off the point. Someone claimed that Trump did "unspeakable" thinks to their family. That's a lie. A politician implementing a policy you don't happen to like is not an "unspeakable" personal attack on your family. The person who made that claim has been unable to back it up and lacks the integrity to withdraw their hyperbole.

My heartfelt advice to people who viscerally hate Trump would be to get the DNC to pick a better candidate next time. Hillary was a corrupt, incompetent warmonger disliked by most Americans. Or as Obama said in 2008, "You're likable enough, Hillary." Ouch! Remember that Trump's 2016 campaign against Hillary was virtually the same as Obama's in 2008. Label her corrupt and unlikable, call out her support for the Iraq war. Obama wrote the playbook and won with it. Trump read Obama's playbook and won with it.

To win an election, run a better candidate. That's politics. Not every election you lose is a direct attack on you personally.

What I wanted to say is that Russel's paradox invalidates the use of "naive set theory", that is the kind of set theory used on Principia Mathematica — Mephist

Of course. Perfectly correct. Perhaps you're thinking of the kind of set theory used by Frege, not by Russell. But that's a historical point and I'm not familiar enough with the specifics of Russell.

Regarding type theory, sure. No problem there. Type theory's even making a modern comeback. But I don't think I wrote a word in opposition to your remarks on type theory. You're defending a point I didn't even disagree with.

Every non contradictory axiomatic theory based on first order logic has a finite (non-standard) model — Mephist

Oh my, no. Not at all. You should read the link you posted. There's no nonstandard finite model of ZF. Please reread the Wiki page you linked. It says nothing in support of the false claim you made. There is no nonstandard finite model of ZF. That's not what the compactness theorem says.

If you use first order logics on the domain of real numbers, the set of all subsets of real numbers is the same thing as "the set of all sets" — Mephist

No not at all. For example let us consider the set of all real numbers that are not members of themselves. That's a legal set formation according to the axiom schema of specification. That is, we start with a known set, the reals, and then reduce it by a predicate.

So, what is the set of all real numbers that are not members of themselves? Well, 14 is not a member of 14. Pi is not a member of pi. The cosine of 47 is not a member of the cosine of 47. In short, the set of all real numbers that are not members of themselves is ... drum roll ... the real numbers.

That's exactly how specification avoids Russell's paradox.

You have claimed that Russell's paradox invalidates the powerset axiom but I still don't follow your logic. In fact if the powerset axiom were false, I would have heard about it.

You will note that the formal expression of the powerset axiom is in fact first order. But perhaps there are some subtleties that you can elucidate in this regard.

https://en.wikipedia.org/wiki/Axiom_of_power_set

ps -- The Wiki article says:

The axiom of power set appears in most axiomatizations of set theory. It is generally considered uncontroversial, although constructive set theory prefers a weaker version to resolve concerns about predicativity.

Perhaps you are making a constructivist argument. It's true that the constructive powerset of the natural numbers is countable and Cantor's theorem doesn't apply. If you are making that type of argument it would be helpful to say so up front so as to not confuse the issue. If you are a constructivist and therefore don't believe in the full powerset as given by ZF, your remarks would make more sense.

pps -- Regarding my claim that the powerset axiom is first order, I refer you to the SEP article on set theory.

https://plato.stanford.edu/entries/set-theory/#AxiSetThe

ZFC is an axiom system formulated in first-order logic ...

The article then goes on to list the standard axioms, one of which of course is powerset.

If I may ask: What unspeakable things did Trump do to your family?
— fishfry

Why must it be personal and not what he has done, oh, say, to immigrant families by separating kids from parents? — Benkei

Because the person I'm challenging explicitly said that Trump had done "unspeakable" things to his own family. I'm challenging that assertion.

As far as separating families, Obama did the same. Obama also put kids in cages. You could look it up. Obama had a horrific humanitarian crisis on the southern border in the summer of 2014. He separated families, caged kids, and turned many kids over to traffickers. (Documented cases)

Think of it this way. An adult shows up with a kid. No paperwork. They could be family. Or they could be a trafficker and his victim. How do you know? You separate them till you can sort out the truth. Would you just take the trafficker's word for it? What kind of policy is that?

In one recent case, the same kid was used three times by three different people to pretend to be a "family." In another case, a kid turned out to have been taken by his mother against the wishes of his father, who had a good job and income in their home country.

One need not endorse Trump's sometimes awful rhetoric on immigration to call out liberal hypocrisy on the issue.

In many countries it's barely over fifty percent who vote, so saying that you're more politically active than the average person doesn't say much. — Metaphysician Undercover

There's little point in talking about one's personal life on an anonymous forum. I've done a lot more than vote. Out there in the world, in real life. But what is your point?

Likely this sex offender will be exposed later in history books. — ssu

Sex offender like Bubba, Teddy (who actually killed a girl) and JFK? Methinks your outrage is selective.

Trump did nothing to your family. Your kids are already seeing much worse on Pornhub. There's this thing called the Internet these days. You haven't made your point because you can't.

We're going to have to go beyond the Turing machine.
— fishfry

I support this idea — BrianW

Yay! I'll take all the support I can get around here.

If I may ask: What unspeakable things did Trump do to your family?
— fishfry
If you're an American you would not need to ask. — tim wood

I am an American and I do need to ask. You said Trump did unspeakable things to your family. I would like to hear what he did to your family. You made a claim. Back it up or retract it.

The demonstration is quite easy (https://en.wikipedia.org/wiki/Cantor%27s_theorem). But there is a problem with the statement of the theorem: Russel's paradox (https://en.wikipedia.org/wiki/Russell%27s_paradox). The concept of "set of all subsets" is contradictory. — Mephist

You have a detailed argument for that false claim? I'd like to understand your thinking here. The set of all subsets of a given set is in no way refuted by Russell's paradox.

So, in a sense, from the point of view of logic, all infinites are only "potential" — Mephist

Math goes beyond logic. Even Russell accepted the axiom of infinity, which posits an actual infinite set whose elements include all the natural numbers. If you deny the axiom of infinity you have Peano arithmetic, which is fine as far as it goes, but does not allow a satisfactory theory of the real numbers. So you have to throw out modern physics along with most of modern math.

The solution to Russell's paradox is the axiom schema of specification. And regardless, Russell's paradox does not contradict or invalidate the powerset axiom.

We all would recognize that the heroin junkie in the red-light district downtown is addicted to this substance. — darthbarracuda

I'll take the contrary view. It's my nature.

There was a great experiment performed in the 1960's and early 70's. It was called the Vietnam war. We took a large number of young men and sent them against their will to a jungle with people shooting at them in a war widely regarded, even by our political leaders, as pointless [The historical record shows that LBJ knew we couldn't win yet kept pouring troops in anyway].

Morale was so bad that fragging, throwing fragmentation grenades into the tents of officers, was a frequent occurrence. Heroin use was rampant. The military leaders were terrified that all these men were going to come home to the US and fuel a massive heroin epidemic.

Instead, when they got home, they never used heroin agin. Turns out that what we call substance addiction is, in many cases, nothing more than a rational response to awful external circumstances.

Lee Robins was one of the researchers in charge. In a finding that completely upended the accepted beliefs about addiction, Robins found that when soldiers who had been heroin users returned home, only 5 percent of them became re-addicted within a year, and just 12 percent relapsed within three years. In other words, approximately nine out of ten soldiers who used heroin in Vietnam eliminated their addiction nearly overnight.

https://jamesclear.com/heroin-habits

If you take those down-and-outers living on the street and give them a roof over their heads and a loving and nurturing environment, there's a good likelihood that they'll give up the heroin or the booze or the meth or whatever they use to get them through the night.

And if I took you out of your comfortable home life and put you on the street to live, in circumstances such that you had no hope of improving your lot in life, there's a good chance you'd use whatever substance was around to take the edge off.

It's not the substance. It's the circumstance.

ps -- This was a 3 year old thread? Didn't notice at first. But the story about the Vietnam vets is one that should be more widely known. It's wrong to blame the substances. You have to look at the circumstances surrounding the addiction.

For "Trump," substitute, "the guy who did unspeakable things to my family." — tim wood

I get it. Orange Man Bad. Not everyone feels that way, even those of us who clearly see Trump's many flaws. I wish the Dems had run a better candidate in 2016. I hope they do in 2020. That's the system we've got. I like Tulsi Gabbard for her pro-civil liberties and anti-war stance. She's polling at 0.8%. Not much of a constituency for peace these days.

If I may ask: What unspeakable things did Trump do to your family?

↪fishfry Yes. As I suspected. — frank

I can't begin to imagine what that remark means. I don't see anyone else here being asked for their personal life history. I'd say I've been more politically active in real life over the years than the average person. By a pretty good margin. I'll leave it at that. What difference does it make what someone says they've done? Where are you coming from here? There's a nasty streak on this forum sometimes.

Part of the implications is that AI which we seem to be in the process of perfecting, could turn out to be every bit as naturally intelligent as we (humans) are. — BrianW

It could be. I"m taking the opposite side of that debate. In my opinion, AI as currently conceptualized -- as (admittedly cleverly designed) software running on conventional digital computing hardware -- can never equal what humans do. We're going to have to go beyond the Turing machine.

Or, better yet, if our (human) intelligence is what operates machines and stuff, what is so artificial about it that we should conjure the term Artificial Intelligence? — BrianW

I wouldn't deny the analogies of memory, complex decision making, etc. Computers do seem to do things we think of as intelligent. AI's play chess and Go at master levels now. They drive cars. I agree that the word intelligence can be applied. That's why we need to be careful and try to think clearly when we impute souls to machines, or else decide that we ourselves must be Turing machines.

I believe that we may be machines. I'm not invoking mysticism. But we are not Turing machines. We need new physics and a new theory of computation before we can make progress on this mystery.

We refer to the organisation of the universe as intelligent; we refer to how components are organised into computer functionality as intelligent; we refer to a sports team as having an intelligent game when their organised activity yields positive results, etc, etc. Basically, intelligence for us is dependent on organisation and utility. I refer to computers and such as quasi-intelligent because their organised activity and utility is not inherently theirs even though they reflect/manifest it. — BrianW

Well sure, by that definition my chair is intelligent, being a highly organized configuration of atoms. I don't regard that as helpful in the debate about machine intelligence, since you just defined machines as intelligent. Well yeah ok then machines are intelligent. But what have we actually learned by this? Not much. If I define flying as standing on the ground eating peanuts, elephants can fly.