Whether or not they are as contagious once infected, they are infected at lesser rates. As continual testing of everyone is impractical, they therefore present less danger to the public than the unvaccinated. — hypericin

I have a small point and a large point to make. I'll start with the small. First, to recap. @Wayfarer posted:

Perhaps their freedom of movement may also be curtailed, though less so. Perhaps 'social distancing', the wearing of masks, and other hygeine measures, will henceforth remain as part of civil society. — Wayfarer

I pointed out that when the vaccinated acqire a breakththrough infection, they are just as infectious as the unvaxxed. Therefore their movement should be restricted too. And that's when you pointed out that they're infected at lesser rates.

This point is easily refuted. The fact that the average vaxxed person is statistically unlikely to infect you means nothing. After all, the average person is not a serial killer, but we endeavor to take serial killers out of society to protect the public. The argument that a random individual is unlikely to cause harm is no argument against separating that indvidual from society.

Likewise drunk drivers, which you mentioned.

The unvaccinated are making this choice to (in their mind) improve their well being, at the expense of the public well being. It is therefore rational public policy to restrict their freedom of movement, to both protect the public well being, and to discourage this selfish choice. — hypericin

In terms of protecting the public well being, you need to restrict the movement of the vaccinated as well, since they are just as contagious as the vaccinated, even if perhaps fewer in number.

So in the end, your point is purely punitive and unrelated to public health.

The situation is rather similar to driving. Everyone on the road presents some danger. But drunk drivers, as a result of their selfish decision to be drunk drivers, present a greater danger. Therefore their freedom of movement is restricted, to protect the public and to discourage drunk driving. — hypericin

But by your logic the contrary conclusion is forced on us. The average driver is statistically rare, even if all too common. Since contagious vaxxed people and drunk drivers alike are statistically rare, they should both be free to travel. After all, your likelihood of encountering either one is relatively low.

So your statistical argument is wrong, and all you have left is your feelings that the unvaxxed should be punished for their "selfishness," as you put it. How about people who don't get their flu shots? People who don't contribute enough to charity? Those with unpopular political opinions? If punishment is your only argument, you yourself wouldn't want to live in the world you wish for.

Now to the larger point. @Wayfarer suggests,"Perhaps their freedom of movement may also be curtailed ..."

Ok. Let's think that through. I can think of two extremes. One is what is done by the a grocery store near me. They have a sign out front that non-vaccinated people must wear masks. They don't check, and rely on the honor system. Then again I live in a relatively small, laid-back town with a relatively low infection rate.

The other alternative is full on police-enforced compliance. You're walking down the street, and the police may ask to see your papers. If you can't produce a vax card, you're arrested on the spot.

Those are the extremes. Perhaps you and @Wayfarer would like to say, specifically, how you think the restriction of free movement in the US (or your country, whatever it may be) should be implemented.

I well remember a few years back when the US state of Arizona wanted to implement a "show your papers" law to challenge brown-skinned people on their immigration status. Decent people across the country were rightfully outraged. Most people think of "show me your papers" as something said in a German accent in a late-night black and white movie from the 1940's. In the US, at least, we don't "show our papers" to the authorities without the police having probable cause or a damn good reason.

So perhaps you think this is a good reason, and that American citizens should be required to show their papers on demand. Can you see how this would quickly go south? Did you get your flu shot? Have any unapproved political opinions? Maybe you tweeted that "All lives matter," or that you believe in rationality and hard work. Those ideas are racist, according to the Smithsonian.

Can you look at history and give me an example of when "show your papers" ever came out well for a society and didn't quickly get abused?

How about when you're driving? Surely if freedom of movement is to be constrained, we need highway checkpoints. That's not so farfetched; there are already interior immigration checkpoints as far as 75 miles inside the US border, where travelers staying entirely within the US may be stopped, interrogated, and searched. Of course if they happen to find a joint or some other contraband, that's your bad luck. What, the Fourth Amendment to the Constitution forbids such an abomination? Sadly, courts have repeatedly allowed these interior checkpoints. It would be easy to set up a lot more of them to check people's vaccination status.

Do you think that's a good idea? Is that the country you want to live in?

Let me point out one more "inconvenient truth," as Al Gore once put it. Who in fact are the unvaxxed in the US? In the popular imagination they're white, MAGA hat-wearing deplorables with unapproved ideas.

In fact, the unvaxxed are blacks and Latinos. Don't believe me?

https://www.kff.org/coronavirus-covid-19/issue-brief/latest-data-on-covid-19-vaccinations-race-ethnicity/

https://thenewamerican.com/leftists-vaccine-passports-are-racist-under-the-lefts-own-thinking/

https://www.nytimes.com/2021/01/31/nyregion/nyc-covid-vaccine-race.html

So what are you going to do? Start pulling over or checkpointing black drivers, accosting blacks and Latinos on the streets and demanding their papers, refusing access to great numbers of blacks and Latinos to restaurants and movie theaters? Can't wait to see how that works out.

We have a real-life datapoint coming up. In New York City, restaurants and other indoor venues will soon require proof of vaccination for entry.

https://www.nytimes.com/2021/08/03/nyregion/nyc-vaccine-mandate.html

But it turns out that only a fraction, one third or so, of NYC blacks are vaccinated.

The policy takes effect this Monday, August 16, and enforcement begins in September. It will be administered by the health department and not the police. So can you imagine what it's going to be like when two thirds of the black people in New York City are banned from restaurants?

The WSJ has the summary. Most of the article is paywalled but the free part says plenty.

https://www.wsj.com/articles/bill-de-blasio-new-york-city-covid-vaccine-mandate-coercion-11628022693

The modern progressive speaks the language of high-minded purpose but always ends with coercion. Witness New York Mayor Bill de Blasio, the uber progressive, who announced Tuesday that New Yorkers will soon need proof of vaccination to do everything from dining out to working out at a gym. He’s proud that New York is the first U.S. city to impose such a mandate.

“It’s time for people to see vaccination as literally necessary to living a good and full and healthy life,” he said at his press conference. You gotta love Mr. de Blasio telling you what is necessary for a good and full life. According to the data, roughly 55% of the city’s residents are fully vaccinated, ranging from 46% in the Bronx to 67% for Manhattan.

His response is to exclude the unvaccinated from many of the functions of daily life. He doesn’t seem to care that this burden will fall heaviest on the city’s black population, which is only 31% fully vaccinated (versus 71% for Asian Americans, 42% for Hispanics and 46% for whites). — WSJ

@Wayfarer and @hypericin, is this what you want? 69% of black people in NYC excluded from public life? And if not, then what DO you mean when you talk about restricting people's movement?

Feedback appreciated. You disagree with my facts? My reasoning? Or are you you all in on "show me your papers" to every non-white face in New York City? You want to bring back stop-and-frisk but for vax cards instead of guns and knives?? And if you did implement nationwide walking and driving checkpoints, how long do you think it would be before the inevitable scope expansion and mission creep set in? Check for your vax card, check your wants and warrants. Behind on your child support? Carrying any unapproved contraband? Tweet any unapproved thoughts recently?

You serious? Anyone thinking this thing through? Or do you all want to live under the Chinese social credit system and can't wait till it's implemented here? I'm afraid that's exactly what some people want.

Some second hand books that I have leafed through, seemingly of the period when I was at school, averred that calculus was all about areas and speeds, though that had never had anything to do with the lessons I had "had". Is this a class thing? — Fine Doubter

Velocities and areas are applications of calculus, as in Newton's fluxions and fluents, corresponding to today's derivatives and definite integrals. In pure math, one only uses velocity and area as illustrations to help students understand, but they're irrelevant to the mathematical content.

Find Doubter, nice pun on "Find Outer."

reforms of the calculus in the 19th C? In terms of limits, as explained copiously hereabouts by fishfry — bongo fury

Thanks for remembering :-)

Well if you want a perfectly unhackable root, this idea might suffice. — Shawn

Do you know that in normal operations, the OS needs to constantly make changes to the kernel in privileged mode? How would you determine what's a legitimate change versus a malicious one? If you disallow all kernel changes the computer won't boot and won't run.

Well, as a direct outcome of having TimeShift running on the go, it would be a safer system by default. The root folder would restore itself once any alteration would be attempted on it by monitoring any attempted change to values on the kernel. — Shawn

How would necessary privileged mode (aka kernel mode or supervisor mode) operations be done? Such as adding a device, adding a new process to the process table, and so forth?

See Kernel.

The kernel is a computer program at the core of a computer's operating system and has complete control over everything in the system.[1] It is the "portion of the operating system code that is always resident in memory",[2] and facilitates interactions between hardware and software components. A full kernel controls all hardware resources (e.g. I/O, memory, Cryptography) via device drivers, arbitrates conflicts between processes concerning such resources, and optimizes the utilization of common resources e.g. CPU & cache usage, file systems, and network sockets. On most systems, the kernel is one of the first programs loaded on startup (after the bootloader). It handles the rest of startup as well as memory, peripherals, and input/output (I/O) requests from software, translating them into data-processing instructions for the central processing unit.

So the point is, how would the computer function if every time the OS needed to modify the kernel's data structures, you backed those changes out? How would you distinguish between legitimate and malicious alterations to the kernel?

What if a friendly actor needs to make a change to this root folder for legitimate reasons? — darthbarracuda

@Shawn, What he said.

I've had this idea of making computing faster — Shawn

For example, when a hacker tries to alter the system — Shawn

These two are in direct conflict.

After all, any intrusion detection scheme must necessarily slow down a computer. It takes extra cpu cycles to detect intrusions. It's like the extra time it takes you to get in your front door because you need to use a key. Any security measure always takes extra time.

When a piece of code attempts to alter memory, how does your system know when it's legitimate or not? After all doesn't a computer operate by way of software constantly making changes to memory?

Or (perhaps?) you are saying that when a piece of code attempts to alter the OS code it's automatically restored to its default state. The problem is that you often have to alter the state of the OS in memory. That's why computers operate in either privileged mode or user mode. In user mode you're not allowed to change the OS, in priv mode you are. I don't see how a computer could function if you disallowed priv access.

As a simple use case, suppose you buy a new printer and connect it to your computer. You have to install a driver and register the computer with your OS. In Windows there's a clunky user procedure and in Mac it's automatic, but the same things are being done either way. In your scheme you would reject all attempts to add new hardware.

The CPU would simple directly input integer values into the OS by having mounted on the RAM and bypassing slow hard drives through an application like TimeShift. — Shawn

I'm afraid I can't comment. But (for sake of discussion) how does the CPU input anything anywhere? It has to be instructed to do so by instructions stores in memory. And all instructions are in hardware at the time they're being executed. If the program (OS or application) is on disk, the page containing the currently executing instruction must be loaded into memory and into a CPU register for execution. So I am honestly not following your idea. But it could be me, my technical knowledge on operating systems and hardware is not current.

fishfry, may I ask for your opinion? — Shawn

I'm honored. I could not parse the following:

utilizing the ECC-RAM utilized nowadays in servers to be able to predetermine the state of a computer through directly interacting with the system OS itself. — Shawn

ECC Ram is just error-correcting memory. It wouldn't offer any functional difference from any other kind of RAM. So I didn't understand that part. How would it be able to "predetermine the state of a computer?" Are you talking about branch prediction? This is a 20 or 30 year old idea as far as I know.

What does it mean to directly interact with the OS? Of course the software directly interacts with the hardware, especially the privileged kernel. So I'm afraid I couldn't make sense of this line and kind of got stuck here.

Booting off RAM? Is this like a RAM disk?

I looked up TimeShift, it's a backup thingie, creates and restores snapshots.

https://wiki.debian.org/timeshift

I'm afraid I couldn't understand exactly what you're getting at. There's always Stackexchange or some of the Reddit groups for finding computer experts.

I get a lot if insights from your posts on maths. — Wayfarer

LOL. Can you comment on my point here? You said the unvaccinated should have their freedom of movement restricted because they may spread covid. I linked an article showing that the vaccinated spread covid at the same rate as the unvaccinated. In view of that, shouldn't the vaccinated be prohibited from free movement as well?

Have you a response?

But then, the person that might end up paying the price for that might be someone they infect. So perhaps the compromise is, anyone has a right to refuse to be vaccinated, but by so doing they forfeit the right to move freely in society. — Wayfarer

Excellent point. And by the same token, I assume you favor restrictions on the free movement of the vaccinated, since they too may infect others.

Vaccinated People May Spread the Virus, Though Rarely, C.D.C. Reports

You agree? If not, why not? If perhaps you're going to invoke the word "rarely," what's your standard for restricting free movement? Have you hard data on how many people are being infected by the unvaccinated? Is that rare, or common? What are the actual numbers? What is the science? I think people on all sides of these issues would like to see the data. Why is the Biden administration itself growing frustrated with the lack of CDC transparency?

From that latter article: "Public Health England published data collected through the end of July showing that vaccinated people are less likely than the unvaccinated to become infected with Delta, but once infected, they may be equally contagious."

I guess we SHOULD restrict the vaccinated individuals' freedom to move through society after all. If you have hard data on any of this I think we'd all be grateful, particularly the Biden administration. Of course you don't have data, because the CDC won't release it.

Smart men never get married lol. It's a bad deal. — hope

Newton either. LOL. This was an old thread!

not von Neumann but of neural networks — D2OTSSUMMERBUG

Neural nets run on conventional computers. They're a clever way of organizing data mining, but they are not a new paradigm of computation. They are physical implementations of Turing machines, in fact finite state machines. The hardware they run on is perfectly ordinary, off-the-shelf. There is no computational difference between a neural net and your laptop; or, for that matter, the 386 machines of the 1990's.

There is so much AI hype out there it needs to be countered. This subject is on my mind because just last night I heard this guy on the radio.

The Myth of Artificial Intelligence
Why Computers Can’t Think the Way We Do
Erik J. Larson.

Where does the creativity of AI come from? — D2OTSSUMMERBUG

An algorithm can never be creative. In particular, there is no hope whatsoever that the current big data and machine learning approach to AI can ever achieve creativity. The current approach is entirely based on datamining and statistical analysis of things that have already happened.

So … can we say that this rule is where all morality should stem from? — Deus

Masochist: Beat me.

Sadist: No!

I begin this thread in response to the backlash I received — Leghorn

Never helps.

In fine, courage used to be overcoming fear. Now it is succumbing to it. — Leghorn

Agree. I'm reminded of the blitz, when Hitler's Luftwaffe bombed London every night for eight months from 1940-41. If it happened today, the Brits would have surrendered to Germany "to prioritize their mental health." What they did instead was show incredible courage, huddling in the underground subway stations and stiffening their spines, till Hitler gave up and went off to attack Russia. In retrospect it was the bravery of those Londoners who turned the course of the war.

Is there anything wrong in stating that neurotransmitters are scientifically assumed to play a role in the regulation and experience of affective behavior? — Shawn

Yes. Here's one striking example. There's no proof that serotonin insufficiency causes depression and mental illness, yet millions are on SSRIs. These drugs have major side effects Literally every single 20-something mass murderer and school or movie theater shooter turns out to be on SSRIs. Are the drugs causing the bad behavior? We're not allowed to ask. The politicians blame the guns and nobody ever asks about the drugs. Of course the drug companies say that well, the kids were emotionally disturbed to start with which is why they were on psych drugs. Nobody ever looks into the high correlation between psycho killers and SSRIs.

A casual Google search on the phrase, "psycho killers and SSRIs' turned up a bunch of links.

https://www.bmj.com/content/358/bmj.j3697/rr-4

https://www.independent.co.uk/voices/antidepressants-side-effects-psychosis-nice-terror-attack-german-wings-pilot-extremism-terrorism-a7191566.html

https://nypost.com/2017/07/26/common-antidepressants-linked-to-at-least-28-murders/

https://www.nytimes.com/2005/02/16/us/boy-who-took-antidepressant-is-convicted-in-killings.html

And:

Do Antidepressants Increase Violent Behavior?

Antidepressants are supposed to make people feel happier and more at ease, but a study has linked several prescription antidepressants to an increased risk of violent behavior, including physical assault and homicide.

This is a major problem with the claim that in order to lead a happy life, all we need is a little pill to balance out our neurotransmitters. Not only is there no scientific evidence for the proposition; there is mounting evidence that these drugs cause harm.

The first video (I didn't watch the second video) is stupid nonsense and disinformation.

In this context, infinite summation is defined only for converging sequences. If the rules of definition are violated, then, of course, contradictions may be derived. There is no mystery or even problem about that.

The person at the blackboard says, "The problem with infinity is all sorts of weird things happen when you're dealing with infinity". First, that doesn't even mean anything. Second, instead of explaining that the fallacy is in using an undefined notion (infinite summation on a sequence that does not converge), the person at the blackboard doesn't even suggest how we may investigate further to see that there is not an actual conundrum.

The video is yet another example of Internet ignorance and disinformation. That person seems to be teaching a classroom. He should be told by the school administrators to clean up his act: If he wants to present mathematical challenges, then he should provide his students with the benefit of techniques and information for solving the challenges rather than obfuscate with "weird things happen". — TonesInDeepFreeze

:up: :up: :up: :up: :up:

Reminds me of Calvinball.

Contra that, in the item I cited, there are mentions of empiricism and the example of the applicability of the mathematics of elasticity to engineering a bridge; see (3) here. — Banno

Yes but your (3) is under the heading that says: "Gödel apparently characterises the syntactic view as consisting of three requirements:"

So he is characterizing the syntactic view. But we've already agreed that he's describing the syntactic view in order to disagree with it.

I would be extremely surprised to find that Gödel advocated Platonism because of the use of math in building bridges or whatever. On the contrary, Gödel's Platonism argued for the existence of large cardinals and other highly abstract and decidedly un-physical entities of set theory.

Again, your (3) is under the heading of the view that Gödel is trying to refute, not advocate for. Am I missing something?

so again:
It seems that for — Banno

, being consistent and being true are inseparable; that mathematical truth is displayed in mathematic's empirical applicability; and that hence the consistency of mathematics shows that it is empirically applicable and hence incompatible with the syntactic view.
— Banno

So if we trust this secondary source - I have no reason not to - Gödel held something like that we can speak of mathematical propositions being true because they are empirically applicable; — Banno

Two things may be true here:

1) It's perfectly possible that Gödel said that, or believed that; or at the very least, that some third-party interpreted his beliefs that way; and

2) I totally do not believe Gödel himself justified his Platonism on day-to-day physical grounds. Gödel believed in large cardinals, and that's one reason he did not believe we should adopt the axiom of constructability.

In other words: You may be right but I still don't believe it. Both those can be true. I believe many false things.

that truth is at a meta-level to the mathematics itself; and that together these show that maths is not just stuff we make up. — Banno

I just can not believe that Gödel paid much attention to the construction of bridges as a justification of large cardinal axioms. Like I say: even if I'm wrong, I'm sticking to my sense of the matter.

The exact argument remains opaque, but that is the implication. — Banno

But wasn't the use of the word empiricism listed under Gödel's description of the syntactic view? The very view that he's describing in order to refute?

I am feeling uneasy expressing strong opinions on things I know nothing about. I don't know what Gödel thought. I do know that he (later in life, at least) came to reject V = L (the axiom that says that the constructible universe is the true universe of sets) because L doesn't have enough sets. And the sets that it's missing are large cardinals, transfinite quantities so large they can't be proven to exist in ZFC and that are as far from empirical concerns as can possibly be.

That's really all I know about it; and I sincerely agree that I could well be completely wrong.

https://plato.stanford.edu/entries/set-theory/#GdeConUni — TonesInDeepFreeze

@TheMadFool This is also a good link if you're interested.

From what I could glean from Wikipedia, a constructible set is one which can be, well, constructed via set theoretic operations — TheMadFool

Wrong article. The Wiki page on "constructible sets" has something to do with topology. Different usage.

This is the relevant page.

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

The constructible sets are built out of first-order formulas in stages. I don't know enough about this to give a simplified explanation. Basically each stage is built from first-order statements with parameters and quantifiers that range only over the previous stage. It's very logic-y. Wish I could say more but I don't know too much about it. Only that Gödel cooked it up to prove the consistency of AC and CH.

Way above my paygrade! Thanks though. I hope to advance my knowledge in math ASAP. — TheMadFool

Above mine too actually. My point was that Gödel apparently believed in an expansive view of the set-theoretic universe, and that his Platonism was probably motived by that and not by practical considerations such as its use in physics.

FWIW Gödel cooked up a model of set theory called the constructible universe, in which the axiom of choice and the continuum hypothesis are true. That shows that they're consistent with ZF.

So why not just adopt the axiom that the constructible universe is the true universe of sets? If you did that, AC and CH would be theorems and we'd be done. The reason this assumption is not made is that most set theorists believe that the true universe of sets (if there even is such a thing) has way more sets in it than just the constructible ones. Gödel apparently first believed that the constructible universe was the true universe, and later came to not believe that.

It's trivial stuff, but may not have been around before — jgill

Or it has been waiting around since the big bang for you to come along and discover it.

Probably there are different versions of him to be consider of: Platonism, realism, relativism, etc... — javi2541997

Agreed, his thoughts were probably a lot more complex than the articles about him can capture.

One of the reasons why some, like Gödel I suppose, believe math is discovered is how math seems to,

1. Describe nature (math models e.g. Minkowski spacetime)

2. Describe nature accurately (we can make very precise predictions to, say, the 15th decimal place) — TheMadFool

My sense is that these mundane physical considerations were not on Gödel's mind. He believed in the Platonic existence of abstract sets including large cardinals, sets far too large to be of any conceivable interest to the real world. See
2.4.4 Gödel’s view of the Axiom of Constructibility
.

I really can't say what Gödel thought about or believed, since apparently he initially thought the axiom of constructability (the claim that the constructible universe includes all sets) was true, then came to doubt it. But my sense is that he was thinking of the Platonic reality of a very large universe of sets, and was not thinking about the utility of set theory in physics. On the other hand he did do some work in relativity, so who knows.

I so mean.


But we call them assumptions. — Banno

Ok, I think we're all clear and in agreement then.

We can start with the inference rules of ND, and, one-by-one, introduce as assumptions the standard axioms of ZF set theory. Of course there are infinitely many axioms because Specification and Replacement are actually axiom schemas, meaning that they each represent one axiom for each of infinitely many predicates. But presumably we can do that. [Specification can be derived from Replacement so technically we only need Replacement].

So far so good. Now as soon as you add the axiom of Infinity, you will have a model of the Peano axioms, and you'll introduce incompleteness. So in the end this is all exactly the same as standard set theory.

I did look this up in the Wiki article on first-order logic, which is the logic used for set theory. It says, in the section, "Hilbert-style systems and natural deduction,"

A deduction in a Hilbert-style deductive system is a list of formulas, each of which is a logical axiom, a hypothesis that has been assumed for the derivation at hand, or follows from previous formulas via a rule of inference. The logical axioms consist of several axiom schemas of logically valid formulas; these encompass a significant amount of propositional logic. The rules of inference enable the manipulation of quantifiers. Typical Hilbert-style systems have a small number of rules of inference, along with several infinite schemas of logical axioms. It is common to have only modus ponens and universal generalization as rules of inference.

Natural deduction systems resemble Hilbert-style systems in that a deduction is a finite list of formulas. However, natural deduction systems have no logical axioms; they compensate by adding additional rules of inference that can be used to manipulate the logical connectives in formulas in the proof.

So as far as I can understand, you get the same set theory either way.

Quick question: why can't we throw out self reference with regard to Gödel like they did with Russell's paradox? — Gregory

I found an answer on math.SE which you may or may not find satisfactory. See the checked answer by Asaf Karagila. I'm repeating it verbatim here:

Self-reference has a problem, if you want to think about it in terms of "I am not provable" sort of approach. A well-formed formula cannot refer to itself. Moreover, a formula cannot refer to the meta-theory (which is where proofs exist).

What Gödel did was two things:

1. Internalize the meta-theory into the natural numbers via coding, and show that this internalization is very robust.

2. Showed that there is a sentence with Gödel number n, whose content is exactly "the sentence coded by n is not provable".

The importance is in both points. They allow us both (limited) access to the meta-theory and the proofs; as well circumvent the problem of being a well-formed formula while still referring to itself. And while the importance of the incompleteness theorem is mainly in the fact that it shows there is no reasonable way to have a finitary proof-verification process to mathematics, and also prove or disprove every sentence; the proof itself is also important because it gives us the internalization of the meta-theory into the natural numbers.

https://math.stackexchange.com/questions/1962462/g%C3%B6dels-incompleteness-theorem-question-about-self-reference

FWIW we don't "throw out self reference" to fix Russell's paradox. Rather, we outlaw unrestricted comprehension and require restricted comprehension.

That is, if $P$ is a predicate, we outlaw set specifications of the form $\{x : P(x)\}$, which says we can form a set out of all the things that satisfy the predicate.

Instead we require that the predicate is used to cut down an existing set. So we have some set $X$ that already exists, we say that we can form a new set $\{x \in X: P(x)\}$. That makes all the difference.

For example if we form the set of everything that's not a member of itself, as in $R = \{x : x \notin x\}$, we get Russell's paradox.

But if we start with, say, the natural numbers, we may legally form the set $S = \{x \in \mathbb N: x \notin x\}$, we do NOT get any contradiction.

Let's walk through it Is 0 an element of itself? No, so 0 is in $S$. Is 1 an element of itself? No, so 1 is in $S$. Continuing like this, we see that $S$ is just the set of natural numbers. The paradox goes away.

No axioms. That's pivotal. Instead there is a rule of assumption: one can introduce any proposition on chooses at any time, and rules for deriving more theorems from those assumptions — Banno

Yes but there are no axioms in any deduction system. First you have the rules of deduction, then you add in some rules of set theory, say, and you crank out your theorems. It's like saying gasoline doesn't have a steering wheel. Gasoline is the stuff you put in your car, and the gas makes the car go. In order to apply the deduction system to something you have to write down some axioms. The axioms aren't part of the deduction system. So it's not a pivotal aspect of ND that there are no axioms. To do math, you write down some axioms and then use the rules of deduction to derive theorems. If you have no axioms you have rules of deduction but you can't prove anything.

Unless you mean that we can introduce our axioms in an ad hoc manner, for example, saying "if we assume the axiom of unions and we have sets X and Y then we can conclude that there a set X union Y. Is that what you mean by not having axioms?

SO the lesson might be that when the love of axioms tried to tighten up mathematics, it ended up toppling the axioms. — Banno

Indeed. It was Hilbert who said, "Wir müssen wissen – wir werden wissen ("We must know — we will know."). And as I recall. it was either a few days before or after that, that Gödel announced his incompleteness theorems. The search for mathematical certainty ended in the proof of uncertainty. Be careful what you wish for, or something like that.

I suspect that in mathematics any true formula can serve as an axiom form which to develop more cool mathematics. That's quite a different thing to an axiom in logic. — Banno

We don't know what the true formulas are. But it's true that math and logic are very different.

Can you tell me, from your knowledge of ND, is my Google-ish description that "To prove P implies Q we assume P and derive Q," is a fair summary of what it's about? I've seen natural deduction mentioned many times but never knew what it is.

Need to take a break so we don't post over each other. Yes, ND is just deduction. But in ND, any theorem can be taken as axiomatic, to be discharged as the deduction proceeds. As I say, they are functionally equivalent. — Banno

Ok I waited a bit. I did not understand ND from its Wiki page. But I did completely understand it from the Google description: To prove P implies Q, I assume P and derive Q. This is basic, everyday mathematical practice.

I should point something out. Mathematicians don't use formal logic. They use this kind of casual, everyday reasoning. In formal logic, the things people talk about are foreign to working mathematicians. By the standards of formal logic, no working mathematician has ever seen a proof, if you look at it that way.

So whatever ND is, if it's just "To prove P implies Q assume P and derive Q," then that's what I've been doing all my life and that's what everyone else does. Any distinction between that and some other kind of formal logic is "inside baseball" for logicians and apparently of little interest to mathematicians.

Bottom line, I don't see how ND can add anything to mathematical practice, nor relieve us of the need to start somewhere by writing down our axioms. If you add axioms one at a time making sure they are complete and consistent, you will never get to the arithmetic of the natural numbers, which is known to be incomplete.

Don't know if this was helpful to you, but it sure was to me. "Today I learned" that natural deduction is just the normal type of reasoning done by mathematicians. So in the end, thanks for mentioning it!

Too fast. Why? — Banno

Ok. I Googled "natural deduction as basis for math" and the following popped up right at the top:

What can you assume in natural deduction?
In natural deduction, to prove an implication of the form P ⇒ Q, we assume P, then reason under that assumption to try to derive Q. If we are successful, then we can conclude that P ⇒ Q. In a proof, we are always allowed to introduce a new assumption P, then reason under that assumption.

Now this is just what I call perfectly normal mathematical deduction. That's how mathematicians do proofs. To prove P implies Q we assume P and derive Q. If that's natural deduction, I've been using it all my life! Like the literary character who discovered he'd always been speaking prose.

So if that's all ND is, then it appears to be perfectly standard everyday mathematical reasoning.

I still don't understand the technical distinction being made by the Wiki article on ND, but based on this example, ND is just how everyone does math.

But I can see how we might have different pictures of mathematics, which are despite that functionally equivalent: one of maths as a series of axiomatic systems, the other as a language that becomes increasingly complex as new formation rules are added. — Banno

I don't think this is what's being said by ND. As the Google example shows, ND is basically normal everyday mathematical reasoning. To show P implies Q we assume P and derive Q.

Is there a reason to think that natural deduction is not as powerful as axiomatisation? — Banno

I don't know, I'm asking you. I don't know anything about it other than that it comes up when students are discussing logic. I glanced at the Wiki article on ND and perhaps you are making a valid point that I'm too ignorant to understand. I never studied formal logic, just picked up the basics from math classes.

Hmm. Your puzzlement has me puzzled, — Banno

I no longer have any idea where you are going with this. I am sure the fault is all on my side. I can't respond intelligibly because I just don't know what you are saying relative to what the thread is about.

Or we might treat it as a series of rules for derivation, never complete but always consistent, to which we add new rules of derivation in a similarly asymptote fashion. — Banno

Are you saying you can do modern math like that? I'm sure somewhere somebody's making the effort. I can't relate to what you are saying. My apologies.

Did you use an axiomatic system? — Banno

Math uses axiomatic systems, period. I didn't study much formal logic.

I studied logic forty years ago. — Banno

Yes but the subject of the OP is axiomatic systems in math. Not that threads don't wander, but I'm no longer understanding your point. Is ND somehow not subject to incompleteness? I tend to doubt that. Underlying set theory is first-order predicate logic. If there's some other basis for doing set theory I am unfamiliar with it. I've heard of ND used in logic, but I can't imagine it gives you different math.

https://iep.utm.edu/nat-ded/ — Banno

I thought we were talking about math, not logic. At least the OP was about math FWIW. Is ND offering an alternative foundation of math? Is it somehow not subject to incompleteness? I'm afraid you've lost me again but I don't know anything about ND.

Of course. I'm saying don't bother with axioms. — Banno

Now you really have me confused. What does that mean? Can you give me an example? You honestly have me at a loss. The entire conversation is about axiomatic systems and their relation to mathematical truth.

I intentionally left axioms out of my post. I've always been struck by the fact that what we select as our axioms is more or less conventional; we might have chosen different axioms. So in that way if one supposes that "all mathematical truth is derived from axioms" all one is doing is insisting on coherence and completeness. — Banno

Yes but no nontrivial collection of axioms (strong enough to found the usual arithmetic of the natural numbers) can be complete. The claim that all mathematical truth is derived from axioms was decisively falsified by Gödel. I'm not sure what you are saying here. You can't insist on completeness, it's been proven impossible to do that with any nontrivial system of axioms.

SO Gödel is clarifying the syntactic view while rejecting it. — Banno

Yes that's my interpretation of all this. I'm basing it on my prior knowledge that Gödel was a Platonist and held that the continuum hypothesis has a definite truth value, even if it's out of reach of our present axioms. Once you know Gödel was a Platonist, it's easier to understand the complicated arguments in the Ravitch paper and the SEP article you linked.

This is a vast topic. The more I look in to it the more it grows. And it is difficult. — Banno

Most definitely. Very deep waters. I know more about the math side and much less, basically nothing, of Gödel's philosophical thought.

I'll come out straight up and say that my own prejudice is that maths is made up as we go along, — Banno

I hold that view myself from time to time. Then again, is the proposition "5 is prime" made up as we go? I oscillate among Platonism, fictionalism, formalism, and "what difference does it make?" as it suits me in any particular argument. My only strongly felt philosophical stance is that the constructivists are missing something essential; and the Ravitch article showed me that Gödel is on my side.

In the SEP article I cited above, an account is given of an argument from Gödel against intuitionist and constructivist views, which Gödel grouped as the "syntactic view", such that the truth of mathematical propositions is determined by their relation to each other, by their syntax, and not by their dependence on facts. — Banno

I sort of agree but I'm unclear on one point. It is not my understanding that constructivists argue that mathematical truth is based on the axioms. Rather, constructivists merely say that in order to exist, a mathematical object must be constructed as from an algorithm or some kind of describable procedure. But I'm not sure whether that goes along with saying that all mathematical truth is derived from axioms.

It seems that for Gödel, being consistent and being true are inseparable; that mathematical truth is displayed in mathematic's empirical applicability; and that hence the consistency of mathematics shows that it is empirically applicable and hence incompatible with the syntactic view. — Banno

Might be but I'm unclear on that too. Gödel believed in mathematical truth, even though he knew no set of axioms can prove itself consistent (second incompleteness theorem). More murkiness.

The argument is unclear — Banno

I very much agree with that. The Ravitch article and SEP link are very complicated.

places Gödel in the position of having to choose between vicious circles and "classical" mathematics. Gödel rejects vicious circles. — Banno

Right. A lot of the basic math I was taught turns out to be "impredicative," defining things in terms of themselves. In math nobody cares or talks about it, but it's a bigger deal in philosophy. Gödel seems to be coming down on the side of accepting impredicative definitions and allowing math to just be math.

The classic example of an impredicative definition in math that's so common that math majors never even notice the circularity is the greatest upper bound of a set of real numbers, as I mentioned earlier and as described in the Wiki link.

So back to my own musings. I'd be interested to see if paraconsistent mathematics might present a distinction between consistency and truth that would mitigate against Gödel's almost equating the two. This would count against his argument in the SEP article. — Banno

Don't know anything about that which is why I avoided that thread. Interesting to see how this field develops.

I'd also be interested in work that sets out how mathematical rules might be shown (a philosophically loaded term after Wittgenstein) in empirical situations, rather than expressed in mathematical terms. See The Epistemology of Visual Thinking in Mathematics — Banno

Wittgy is another subject I don't know enough about to converse on. As he said, Whereof I cannot speak, thereof I should put a sock in it.

It needs to be borne in mind that mathematics is ultimately a system, game-like in nature, where we have complete freedom to choose our starting premises aka axioms. — TheMadFool

This is exactly the view that Gödel opposed. He believed that mathematics is objective; that mathematical truth is something that we study, not something we make up. If the axioms don't settle a given mathematical question, that's the fault of the axioms. There is truth "out there" waiting to be discovered.

Exactly in this part, I guess Gödel and Ravitch concided. Those axioms or propositions are so related with the "realism" itself. — javi2541997

The exact opposite. The axioms do not determine mathematical truth, in Gödel's view. The classic example being the continuum hypothesis. Gödel said that it has a definite truth value; whether or not it can be proven from the standard axioms.

Still, climate change is more of a "when", as opposed to nuclear armageddon's "if". — hypericin

How do you suppose that looked in 1945 and into the 1950s? Perhaps you don't remember the "duck and cover" drills that schoolkids did in anticipation of the Soviet nukes that were expected to come flying in any day.

How 'Duck-and-Cover' Drills Channeled America's Cold War Anxiety

Reading deeply the dissertation ..., — javi2541997

I did start reading a bit, and found this nugget:

Gödel has defended classical mathematics against each of the major programs of restricted methods. He has rejected intuitionism, semi-intuitionism, the vicious circle principle, and constructive or finitist programs in general. has defended classical mathematics against each of the major programs of restricted methods. He has rejected intuitionism, semi-intuitionism, the vicious circle principle, and constructive or finitist programs in general.

I shall definitely remember this the next time I cross swords with yet another neo-intuitiionist or constructivist. Hi @sime!

Thanks for linking the article. Interesting that he wrote it in 1968 and was a student of Church himself, as was Alan Turing.

ps -- I noted that the passage you quoted about vicious circles mentions the problems with Dedekind cuts. I'm not entirely sure of exactly what they mean, but I believe that the general idea is that the real numbers are characterized by the least upper bound property: every nonempty set of reals that's bounded above has a least upper bound. The problem is that we're characterizing the real numbers by talking about sets of real numbers. For that reason these kinds of ideas are called impredicative. I don't know much about this subject. I gather that's what this part of the paper is all about. Interesting paper.

That sounds like a good topic for a new thread. — Metaphysician Undercover

I'm entirely in agreement, except that this is the exact discussion that disappeared overnight. I don't want to repost it and upset whoever got upset at the last go-round. I can't do it unless the mods say it's ok.

I would like to re-post the part of the discussion that I was interested in: Namely, does the expansion of space take place everywhere, including between the atoms in my body? Or only "way out there," among the distant galaxies? The argument for the former being that the expansion would have no way to know the difference, and that the only reason we discount it is because the effect is far too small to be observed. The argument for the latter is the "raisin cake" model, in which the raisins are "gravitationally bound," while the cake expands.

I would not want to run afoul of the moderation policies of the site for re-posting a disappeared thread. Moderators please advise.