Logic - 2018-11-20

Sergei Akbarov

03:53

@CarlMummert Carl, I am here. I'll find Levy/Bar-Hillel/Fraenkel.

@TimothyChow Tim, this means that it's impossible to explain non-specialists, that nobody found a contradiction in mathematics up to now. I find this situation absurd.

Malice Vidrine

04:11

@SergeiAkbarov - It sounds like the problem is one of citation, not of explanation. Because I've explained it plenty of times to the satisfaction of non-mathematicians.

Carl Mummert

yes, I think that is the issue you are running in to

there is also an issue with certainty. While nobody thinks that the previous paradoxes are a problem, there is always the chance of a new one.

There's no reason to think it's a high chance, but some mathematicians don't want to say "all the paradoxes are eliminated" in print when there is still a chance that ZFC is inconsistent

(Sorry for the delay responding initially)

@SergeiAkbarov - I should ping you in case you have looked away from th room

Sergei Akbarov

Carl, I don't like this. This looks like an elitism, a separation of knowledge to the part for the "members of club" and to the part for "all others". Always there is a possibility to explain the situation for all people without the necessity to harm your reputation. (Provided, of course, that you have it clean.) There is no obstacles to write what you explain here in a book, and the problem disappears:

> While nobody thinks that the previous paradoxes are a problem, there is always the chance of a new one. There's no reason to think it's a high chance, but some mathematicians don't want t…

Carl Mummert

04:27

In person, I find most people in the field are very direct about what is going on - I think it is more related to the nature of written professional mathematics

I think you will find that even Levy, Fraenkel, and Bar-Hillel will not come out and directly say that ZFC is consistent - and I have no doubt that they believed it was, because they would not write a book about it if they seriously thought otherwise.

But in print they only say that a proof of ZFC would require methods not formalizable in ZFC, and it is best to assume the consistency for the purpose of relative consistency proofs.

Unfortunately, if sites like MSE don't pass the test, and if Wikipedia only wants peer reviewed references, it makes it hard to cite things even if they are obvious for everyone to see

Sergei Akbarov

In my opinion, you egaggerate the difficulties. But I actually don't know what to add.

Carl Mummert

I tried looking in Jech's set theory text, but he said almost nothing about consistency. So I don't know any source that just says ZFC is consistent, unfortunately.

The paper by Maddy that I linked is also not useful for that purpose, she has a different goal.

I don't think blog posts are useful for you, are they? @SergeiAkbarov

Sergei Akbarov

Carl there is no necessity ti say that ZFC is consistent:

> So I don't know any source that just says ZFC is consistent

That would be stupid. What we need is a phrase: "a contradiction is not found yet". And this is enough.

Blogs could help, since there is nothing else.

Carl Mummert

Aha. That may be easier to find

There is an old post on this site about it at math.stackexchange.com/questions/111878/…

There is also some commentary by @TimothyChow in this older post on FOM, which also shows some of the variation in the way the issues are interpreted cs.nyu.edu/pipermail/fom/2009-May/013641.html

There seems to be some useful commentary on p. 120 of Proofs and Fundamentals: A First Course in Abstract Mathematics by Ethan D. Bloch

See here: imagebin.ca/v/4NDhi3T7lQ7z

I don't have a preview of the rest of that paragraph in Google, but it seems like the type of thing you are looking for.

Sergei Akbarov

04:52

Yes, this is at least something... Thank you, Carl!

Carl Mummert

Another good source looks to be p 12ff of Topoi: The Categorial Analysis of Logic
By Robert Goldblatt.

user21820

@MaliceVidrine While that is the problem Sergei is focusing on here, namely an explicit citation from a reputable publication, I think we should (not saying that you don't) realize that ultimately Wikipedia's criterion is as subjective as it claims it isn't. There is no way to tell which publications are reputable, except to rely on (guess what?) expert opinion, which in turn cannot be identified except (guess what?) by having a sufficient knowledge of logic.

Carl Mummert

I had misunderstood that you wanted a source for the consistency of ZF rather than just for us not knowing any inconsistency

Sergei Akbarov

Carl, this is absurd. How could I wish a source for inconsistency if this contradicts Gödel?

Carl Mummert

When I read "where is it written explicitly that the paradoxes of the early 20th century were overcame in the current mathematics?" that is what it I read it as asking for - a proof that the paradoxes have been overcome, that is, a proof of consistency

"I wonder if there is a text where it is written explicitly that the old paradoxes (like the Russell paradox, and the others) do not appear in the modern axiomatic set theories?"

user21820

05:02

Indeed I read that too, which is of course impossible to be non-circular.

Malice Vidrine

@user21820 - Well, I happen to have unmediated access to objective truth, so I can help there. Though I do often lie or mislead in ways that might be mistaken for error, so...

Carl Mummert

Fortunately, it is much easier to find sources that say we don't know of any inconsistency in ZF.

user21820

@MaliceVidrine Yes you lie in this comment. =P

@CarlMummert But that's precisely the Wikipedia problem; it's even easier to find sources that say ZFC is inconsistent.

20

Q: What would be some major consequences of the inconsistency of ZFC?

I was happily surfing the arXiv, when I was jolted by the following paper: Inconsistency of the Zermelo-Fraenkel set theory with the axiom of choice and its effects on the computational complexity by M. Kim, Mar. 2012. Abstract. This paper exposes a contradiction in the Zermelo-Fraenkel set ...

Malice Vidrine

Did I? Or did I perform some other speech act besides making a claim? The loopholes are endless!

user21820

@MaliceVidrine Hahahaha don't go all philosophical. There is true, and false, and not even boolean!

Sergei Akbarov

05:05

Carl, that would solve the problem:

> Fortunately, it is much easier to find sources that say we don't know of any inconsistency in ZF.

Carl Mummert

@user21820 I am extremely familiar with the issues on Wikipedia (unfortunately). The fact that mathematicians don't like to make broad claims, and that they also don't like to write down things that everyone knows, makes it particularly difficult to fit their concept of sourcing to mathematics, which as a field is somewhat terrible about sourcing

user21820

@CarlMummert Indeed. But imagine if mathematicians tried to write down things that 'everyone' knows! Mathematics would (metaphorically) grind to a halt for a couple of years at least...

=)

(Not saying that it's a good thing to have things not written down. It's just understandable.)

Carl Mummert

@user21820 in the case of logic, we have a separate issue. For example, even Boolos wrote that we have no reason to "know" ZFC is consistent, even if we believe it. We can't really dismiss Boolos as a crank, unlike a random arxiv paper

user21820

@CarlMummert Exactly. Most people do not realize what the incompleteness theorems mean. They either go to one extreme and reject it as being applicable to all foundations, or they go to the other and reject all foundations as being paradoxical.

And very recently Noah also expressed similar opinion, if not more doubtful, than Boolos.

Mar 5 at 7:07, by user21820

> I'm not certain whether I agree with the statement "However, when we use an axiom in a proof, we normally know whether it holds for standard integers or not." For example, ZFC proves the consistency of theories which I'm not fully confident are consistent, so I'm not fully confident in ZFC's consistency, let alone its arithmetical soundness. But the ZFC axioms are of course widely used (including by me!).

Carl Mummert

I don't think that Boolos thought all foundations are paradoxical - he was arguing in that paper about Penrose's claims that we can directly perceive things like the consistency of ZF.

user21820

05:10

@SergeiAkbarov: See above quote. Do you now understand the difficulty of the problem you are trying to solve?

@CarlMummert I wasn't talking about Boolos; he's certainly not extreme in any sense!

But I'm now curious, how do you square Boolos' direct perception of consistency of ZF with Noah's comment?

Carl Mummert

I think Noah's statement is actually a good summary: most people think ZFC is consistent, but won't say that they "completely" believe it is consistent. Maybe they would be willing to make up some percent of belief, but they won't put it at 100%.

user21820

@CarlMummert Exactly, that in-between percentage confidence is what makes this issue nearly impossible to capture fairly, if at all.

Carl Mummert

Boolos was arguing basically the same thing: that we may believe ZFC is consistent, but we're not going to have the kind of proof that would let us "know" perfectly that it is consistent.

user21820

What roughly is your confidence level, by the way, for consistency and for arithmetical soundness of ZFC, separately?

Carl Mummert

I think ZF is consistent to the same degree I think the sun will rise tomorrow. My personal taste is, in such a situation, to just come out and accept that I "know" it, since I rely on it so frequently

user21820

05:20

Haha I see. And what about soundness?

Carl Mummert

The same. Not only do I and everyone else rely on on the numerical soundness in practice, I have never seen what I thought was a strong argument why ZFC might be consistent but unsound, apart from the mere possibility that it could be.

user21820

I see. So basically close to 100% confidence.

Carl Mummert

I would say even more confident, because I can at least see a reason why ZFC could in principle be inconsistent, just like I can see a possibility that the earth could vanish in a quantum fluctuation before the sun rises tomorrow. But arguing that ZFC is likely to be consistent but unsound seem to me like arguing that the sun is likely to rise tomorrow but Venus is likely to vanish at the same time. Technically it could happen, but why?

user21820

Well for me, I don't see any really good reason to believe it is arithmetically sound, because almost all consistent theories that interpret arithmetic are unsound.

Carl Mummert

Like ZFC, NBG, MK, and second order arithmetic?

user21820

05:25

I think higher-order arithmetic is arithmetically sound, for philosophical reasons.

Carl Mummert

Also type theory and its relatives.

user21820

The problem is that those philosophical reasons break down completely when attempting to justify ZFC, so I have no good reason to believe its soundness.

Carl Mummert

None of these is known to be unsound as far as I know. ...

user21820

@CarlMummert Of course, nobody has found any evidence that any is unsound.

Carl Mummert

It seems to me that the main arguments for the consistency of ZF also argue for its soundness

So if those arguments don't establish soundness, then really they don't establish consistency either.

user21820

05:28

@CarlMummert Right; I don't have a philosophical justification for ZFC's consistency either, but I have an intuitive feel that its syntax prevents inconsistency.

Just like there is a proof-theoretic reason that PA+¬Con(PA) is consistent.

Carl Mummert

If you think that it is possible for either Earth or Venus to vanish in a quantum fluctuation, but your intuition says it is more likely to be Venus, what can anyone say?

user21820

@CarlMummert That's not a good analogy. As I said, PA+¬Con(PA) shows that there can be syntactical reasons for consistency, which fail to also establish arithmetical soundness.

Carl Mummert

What is the syntactical reason for the consistency of ZF?

Sergei Akbarov

@user21820:

> @SergeiAkbarov: See above quote. Do you now understand the difficulty of the problem you are trying to solve?

You persistently attribute stupidity to me, greatly exaggerating it. I never said that I am seeking a confession of consistency of mathematics. Moreover, I find all these interchanding of feelings about whether this is so or not, stupid.

What I need is a confession that no contradiction was found yet. And this is not a great deal, the specialists could write this openly.

user21820

More precisely, it is possible that ZFC contains a fragment that is sound, but the additional axioms syntactically avoid inconsistency yet make it unsound.

@SergeiAkbarov Nowhere did I say you were stupid. You on the other hand keep accusing others of sneering at you, and making other unkind remarks to people. Please stop that or leave.

Carl Mummert

05:32

@SergeiAkbarov I hope that the sources I linked help out some.

user21820

I merely said that you are trying to get citation for a view (that ZFC seems consistent) that even logicians don't agree completely on.

Carl Mummert

@user21820: we seem to be back to talking about possibility. Anything is possible - Venus might disappear tomorrow. But I have never seen a syntactic argument for the consistency of ZF, as far as I can remember.

Indeed, that is precisely the issue. If anything, a central issue with consistency is in the comprehension scheme - we can't prove that every instance actually defines a set. If we could, that would reduce the consistency of first-order ZF to the consistency of second-order ZF, which has more compelling arguments. Like second-order Peano arithmetic, which is easier to argue for consistency than first-order Peano arithmetic.

Sergei Akbarov

I explained many time what I need, and in this chat also:

> What I need is a confession that no contradiction was found yet. And this is not a great deal, the specialists could write this openly.

And I don't feel pity to those logicians who see a problem here:

> I merely said that you are trying to get citation for a view (that ZFC seems consistent) that even logicians don't agree completely on.

Carl, thank you and good bye.

Carl Mummert

Have a good day @SergeiAkbarov

user21820

@SergeiAkbarov You said that in your culture people are direct. So I'll be direct; you are being very impolite, despite my original attempts to help you out.

Carl Mummert

05:41

I am also going to sign off, good night everyone

user21820

@CarlMummert Good night!

spaceisdarkgreen

@SergeiAkbarov Perhaps I'm being naive posting this after so much previous conversation, but the first hit when I google book search "no inconsistency has been found in ZFC" is p 111 is Mathematical Logic By H.-D. Ebbinghaus, J. Flum, Wolfgang Thomas: "Nevertheless, the fact that ZFC has been investigated and used in mathematics for decades and no inconsistency has been discovered, attests to the consistency of ZFC."

user21820

@spaceisdarkgreen I won't say it's naive, but it's in my opinion a bad argument to cite. It does attest to the consistency of the fragment of ZFC that has been used, but that is only a tiny fragment of ZFC.

But in any case welcome to this room!

@spaceisdarkgreen In particular, Friedman's grand conjecture is that the extremely weak theory EFA suffices for all mathematics in the Annals of Mathematics, and even today the vast bulk of modern mathematics can be done in ZFC with bounded specification and replacement (where the defining formulae have only bounded quantifiers).

spaceisdarkgreen

@user21820 I'm not saying that there isn't anything to quibble over here regarding the implications of this fact, only that the quote seems to satisfy Sergei's low bar of a source that explictly says a contradiction has not been found.

user21820

@spaceisdarkgreen I understand that low bar, even if I personally don't like to just meet a low bar. =)

So I should clarify: it's in my opinion a bad argument to cite in a proper mathematical/philosophical discussion. But to cite on Wikipedia? Well... =)

Sergei Akbarov

06:12

@spaceisdarkgreen Thank you, I'll try to use this.

user21820

08:01

→ 11 messages moved to Math Meta Chat

Malice Vidrine

On another topic, has anyone done any serious work on type theories without a universally defined function type type-forming operation?

It seems like it would be too fiddly for anyone to bother :P

user21820

@MaliceVidrine How would one deal with functions then? What's the easiest way to do it without a function-type forming operator?

One could treat "f : S → T" as a 3-place predicate, and so one can handle all the finite-order function-type iterations.

But it would be obviously be unnatural (and having no advantage) to do that instead of just forming the type.

Malice Vidrine

You can sort of deal with it by simply including some of the basic term-forming operations as part of the language; I've seen this done in toy languages, but rarely in anything more developed. I'm playing with a category whose internal logic looks to have only some function types, and in particular function types between all types under the image of a unary type forming operation. But the syntax gets... touchy.

user21820

@MaliceVidrine That's annoying. At least I want to be able to get the hierarchy of types built upon any explicitly given types.

Malice Vidrine

08:17

Martin Hyland seems to think the type theory that comes out of this indicates I'm looking at the underlying category in a fundamentally wrong way. I can't say he's wrong, but unfortunately whatever the correct way might be is even more obscure :P

user21820

I see.

Malice Vidrine

(That being the category of sets and functions of NF set theory)

user21820

Aha I knew it was something odd. =)

Malice Vidrine

It's frustrating because it's actually quite an adequate set theory for so much of "ordinary mathematics", but its behavior in other respects is just so bizarre.

Linguistically, it has a set of all sets, and you can construct the category thereof internally; categorically, that internal category is a lie and only "sees" a fragment of the whole universe.

And I have no idea what to make of that :P

user21820

@MaliceVidrine Isn't it just because "ordinary mathematics" can be handled by only a weak fragment of ZFC or NFU or COC, and whatever extra stuff each one has is "just so bizarre"?

Malice Vidrine

08:22

I don't know, ZFC or Martin-Loef seem to me to have pretty sensible universes. I guess my intuition is that mathematics should be locally Cartesian closed at the very least.

And there are a great many things about the NF-verse that are perfectly reasonable sounding; V forming a complete Boolean algebra is nicely consonant with the naive picture of sets most people are introduced to. And weirdly most of our common mathematical constructions seem to have stratified descriptions.

(except quotients. They're a bit of a bugbear from a stratified perspective)

Also, the thing I like most, is that stratified formulae are characterized semantically by a symmetry property.

user21820

Hmm what you just said are some of the reasons why I invented my own type theory. I suppose I had not told you about it, and it is not finalized. But basically it is based on 3-valued logic and in my opinion resolves the 'paradoxes' in a much more satisfying way than ZFC or NF[U], although it is probably not much stronger than bounded ZFC.

Malice Vidrine

Why the 3-valued logic?

user21820

Briefly: Classes are types whose membership is boolean. There is a universal type called obj, and obj is a class (it says "yes" on literally everything). There is also a type of all types, called type of course. And there are base types nat and bool. And for any types S,T, func(S,T) is the type of all functions from S to T, and proc(S,T) is the type of all procedures from S to T (it may be null on some inputs).

The type class of all classes can be treated as func(obj,bool). type can be treated as proc(obj,bool).

The 3-valued logic allows it to form the Russell type R = { x : x∈type ?∧ x∉x }, and prove that R∈R ≡ null.

Here "?∧" is a guarded conjunction, where the right expression only needs to be valid if the left expression is true. It is equal to the left expression otherwise.

Now note that the classes form a complete boolean algebra, consonant with the intuitive notion of collections that people have, because those collections are indeed classes (boolean membership).

Malice Vidrine

08:39

Interesting.

user21820

But for a type T in general, T ⋃ { x : x∉T } = { x : x∈T ∨ x∉T } may not be obj, for obvious reasons.

One still cannot have unrestricted specification of types, otherwise one still gets contradiction, but it is relatively clear why. If you want "∀x∈S ( P(x) )" to be boolean if "P(x)" is boolean for every x∈S, then we cannot permit a type to quantify over an arbitrary type, otherwise it can use that quantification to diagonalize itself.

The reason for the restrictions in my type theory are hence justified; it directly prevents circularity, and isn't just an ad-hoc scheme that seems to work.

I can say more about what the restrictions are, if you're interested.

But it may be helpful to first see what quantification allows. If procedures cannot quantify at all we get exactly the computable ones. If they can quantify over nat, then we get essentially the arithmetical ones. The question of course is what else we can permit while being confident of non-circularity.

If you permit all function-type iterations built from nat and bool, then you get essentially full higher-order arithmetic, but some predicativists might become unhappy. =P

Malice Vidrine

I will inquire about the exact restrictions some time soon (hopefully I remember), though it's nearing 1AM here, so I worry details will be lost on me :P

user21820

@MaliceVidrine Sure sure; see you again soon!

Have a good rest. =)

Malice Vidrine

Oh, I'm still awake for a bit. We've got classes off the rest of the week for one of America's more troubled holidays, so I'm staying up late because I can :D I'm still interested in the conceptual bits.

user21820

@MaliceVidrine Aha. Okay the basic idea is to internalize a notion of predicativity. We have the type of all predicative objects called pred, of which we have predicative types ptype. Naturally, ptype itself is not a ptype, but it doesn't matter at all for ordinary mathematics because ordinary mathematics deals only with predicative types. =)

In my type theory, ptype is in fact cartesian-closed.

This is a second nice thing you mentioned, besides the universal type.

Malice Vidrine

08:53

nod It's a pain to try and get by without it, for sure.

user21820

Note that the 3-valued logic allows us to escape normally inescapable paradoxes. For example we still have Cantor's theorem, that for any type S there is no func(S,func(S,bool)) that surjects onto func(S,bool). Essentially the same proof says there is no func(obj,class) that surjects onto class. But wait a minute, ( class x ↦ x ) clearly injects class into obj.

The reason that injection cannot be reversed to obtain a surjection is because class is not (provably) a class. And the type theory itself knows it, proving class∈class ≡ null.

This is how it does it, without stratification. That's the third nice thing, because I personally don't like stratification. =D

Anyway I forgot to say what pred was used for. Basically you only allow procedures to quantify over predicative types, and there are some syntactic rules governing when you can assert that a procedure is predicative.

We assume obj,nat,bool∈ptype. The idea is that ptypes are all well-defined collections whose semantics cannot be changed, so diagonalization cannot create a circularity.

Of course, predicativists will balk at cartesian-closedness of ptype, but I think it's safe here unlike in classical set theory, again because of the 3-valued logic. Already func(nat,bool) is not a class, so it is 'free' to only accept procedures that can somehow be determined to be certainly a function from nat to bool, and to only reject procedures that can somehow be determined to certainly not be one.

So quantifying over it doesn't seem problematic, as long as there isn't already a circularity somewhere else.

Malice Vidrine

In defense of stratification, there's actually a pretty nice story that goes along with it once you see the semantic property at work.

(as an aside)

user21820

@MaliceVidrine I was actually referring more to the stratification of the type hierarchy, rather than syntactic stratification. I don't like a stratified world that doesn't have a final closure of some sort to it. Just feels incomplete.

Malice Vidrine

Ah, gotcha.

user21820

You probably can see how heavily my type theory ideas are based on computability notions.

Malice Vidrine

09:07

(Also an aside: I miss Britain this time of year)

user21820

@MaliceVidrine For the snow? This winter is set to be the coldest in many years.

Malice Vidrine

Yup; which is really how a lot of type theories seem to be motivated if you're not a flighty category theorist.

It's just been rainy the past few years I've been there. But I liked my little flat and the kebab truck down the street from the CMS. And the rain. And the nonsensical street layout. I've just been rewatching the BBC series Requiem, which had just finished airing last time I was there, which is what reminded me.

user21820

@MaliceVidrine Hahaha nonsensical street layout. I suppose you don't like Manhattan metric much? =P

Malice Vidrine

Don't get me wrong, it's lots of fun to use google for walking directions and see where it takes you.

But I get turned around easily :P

user21820

I get lost easily if I have never gone to a place before. (Partly because I don't use Google maps.) But I have a good sense of location so I can almost always find my way back to my starting location.

I would probably get lost in a rectangular city much faster than in a place where every street is unique. =)

Malice Vidrine

09:17

The thing about grids is that if they're actually regular, then it's just n up and m to the left and there you are :P

But to the credit of Cambridge (and I presume other towns) there were a lot of nice bicycle/pedestrian shortcuts if you knew where to spot them.

Most US cities are incredibly hostile to cyclists.

And that is why something something on-topic logic words. Strong normalization and such.

:p

user21820

@MaliceVidrine Yes cartesian coordinates. Descartes is everywhere!

@MaliceVidrine You're looking for "canonical bicycle route". East/West then North/South.

Unless you meant conformity...

Malice Vidrine

I'm not sure I meant anything, other than dodging the charges of being off topic after I was cross with someone for being off topic :P

user21820

09:34

@MaliceVidrine On-topic is fine. Off-topic is fine if it's pleasant and doesn't squeeze out other discussion. =)

user21820

13:37

@MaliceVidrine @spaceisdarkgreen @LeakyNun: In case you're interested I just found Can ZFC be inconsistent? (Randall Holmes), which says:

> I don't serious think that ZFC is inconsistent. I did have occasion recently to wonder whether it was possible that ZFC might be inconsistent. I concluded that it is possible (however unlikely experience may suggest that such a result would be) in a way in which I regard the inconsistency of Peano arithmetic or Zermelo set theory as not being possible.

He also says that the iterative conception of the cumulative hierarchy seems to give Σ1-replacement, and then a further argument supports consistency of Σ2-replacement, but it doesn't extend beyond that. So although it seems unlikely for ZFC to be inconsistent, based on current experience, "there seems to be no philosophical reason based on the intuition of the cumulative hierarchy why this could not happen".

Leaky Nun

what fruit can any thoughts in this direction produce?

user21820

14:02

@LeakyNun It tells you to be careful when you use more than that.

Timothy Chow

16:29

@SergeiAkbarov As spaceisdarkgreen and others have mentioned, the statement you're looking for does exist in the literature. But I think you have to take things in perspective. Your comment falls in the category of, "Why don't the experts explicitly say this obvious thing that some beginners don't understand?"

It's a fair objection and there are thousands of such statements in every field, not just mathematics. In any given case, one can make a legitimate argument that the experts are being pedagogically sub-optimal. On the other hand, even the best teacher can't anticipate every confusion that every student might have.

In your case, your question has the additional property that it's close to other questions that are commonly asked, like whether people believe that ZFC is consistent, or whether there is a proof that ZFC is consistent. So your question could get confused with one of those questions.

I don't know if you've ever had the experience of using Google to search for something and then finding that Google thinks that you're asking about something that sounds similar but is much more commonly asked. Under such circumstances it can be very difficult and frustrating to get Google to answer the exact question one is interested in.

So, I won't try to argue that there is a really good reason why your particular question isn't answered in the literature as clearly and prominently as you would like it to be. All I'm suggesting is that you be a little more understanding of the task facing the teacher, who has to try to anticipate every possible question, as well as face the ire of students who insist that THEIR question is the obvious one which should OBVIOUSLY be addressed RIGHT UP FRONT.

@user21820 I don't think that Sergei Akbarov is trying to get a citation for the view that "ZFC seems consistent." He wants a citation for the view (fact?) that "there is no known contradiction in ZFC."

user21820

16:48

@TimothyChow Well, I think it's clear that Sergei has some misconceptions regarding certain issues in logic, and that is part of the problem:

12 hours ago, by Carl Mummert

When I read "where is it written explicitly that the paradoxes of the early 20th century were overcame in the current mathematics?" that is what it I read it as asking for - a proof that the paradoxes have been overcome, that is, a proof of consistency

Some topics cannot be made that simple anyway. Citations don't help in this particular case of incompleteness theorems, for the reason I stated; there are cranks who publish (even on ArXiV) claims of inconsistency of this and that. So we end up with a sociological problem of determining who is a reliable expert on logic...

In fact, that is exactly why Wikipedia wants exact citations even for trivial facts. I once tried to add a tiny bit of mathematics that trivially followed from Fermat's little theorem, but it was rejected until I wasted my time to find a citation that got close enough to that particular trivial bit!

Because Wikipedia seems to operate under the erroneous assumption that requiring citations improves its quality.

Consequently, you do get lots of citations for many health scams including energy machines.

By the way, I don't disagree with what you've said here.

Timothy Chow

17:28

@user21820 Basically I agree with what you say here (except that I haven't read back far enough in the conversation to say whether "Sergei has some misconceptions"), but I will nitpick about Wikipedia. I don't think that their "no original research" policy was motivated by the desire to improve quality per se.

user21820

@TimothyChow Lol! Then whatever do you think it was for? Seriously curious.

Timothy Chow

I think it was motivated by the attempt to find a policy that would let them try to weed out unverifiable claims. I'm sure they understood that it would also, as 'collateral damage,' exclude some quality material, but they evidently judged that to be a favorable tradeoff.

user21820

@TimothyChow And that favourable tradeoff is supposed to improve quality, right? Otherwise why have it?

Timothy Chow

Well, yes, if you define "quality" to mean "as close as possible to the original vision of neutral point of view and verifiability." But that's not what I interpreted you to mean by the word "quality."

user21820

Oh no wonder.

Yeah.

Sergei Akbarov

17:39

@TimothyChow Tim, I told this many times and now I see that I have to repeat, you exagerate the difficulties. I also was a student, and we asked our logic teacher if there were still some old paradoxes in logic, and he explained to us that they all were eliminated in axiomatic systems (although there is no certainty that new paradoxes will not be found). It did not occur to anyone then that this was an indecent question, and the teacher did not feel any embarrassment.

Before this story with Wikipedia, I was sure that all this was written in textbooks. I just now with great surprise found out that this is some kind of sacred knowledge.

Timothy Chow

@SergeiAkbarov How is it sacred knowledge? It's in Ebbinghaus, Flum and Thomas.

user21820

(1) Nobody here is embarrassed. (2) Nobody here thinks this is sacred knowledge, otherwise why on earth would we be talking so freely about it?

Your anecdote does not at all address what you kept insisting on in your question and comments on Noah's answer, namely you want a statement covering all paradoxes. Your teacher obviously took the question to be only about specific implementations of a select few paradoxes, likely Russell's, Cantor's, Burali-Forti.

Sergei Akbarov

As I told to spaceisdarkgreen, Ebbinghaus, Flum and Thomas are inaccurate, their statement sounds friviolous. And as you can see, it took me several days to find this, although this is the kind of statement that must be written in each textbook on logic.

user21820

@SergeiAkbarov You may be frustrated, but saying that people who know logic are making frivolous statements is not going to make others want to help. Lest you call them frivolous too.

Sergei Akbarov

As to all: this is a reformulation of the statement that there is no know paradoxes in axiomatic set theories.

Timothy Chow

17:49

@SergeiAkbarov I'm afraid that I'm not so familiar with this chat format so I don't know how to search it well, but I can't find where you said that the EFT statement is inaccurate and frivolous. Why is it inaccurate and frivolous? And I don't see why it needs to be in every textbook since it's obvious to most people. Again, you're insisting that your personal confusions be addressed by every single textbook even if most people aren't confused by them.

user21820

@SergeiAkbarov Which is precisely why I said your teacher's answer to your question does not address your question at all.

@TimothyChow He said it in a comment on his question.

Sergei Akbarov

Tim, here: math.stackexchange.com/questions/3003268/… Is "attests" here a synonym of "implies"?

"Nevertheless, the fact that ZFC has been investigated and used in mathematics for decades and no inconsistency has been discovered, attests to the consistency of ZFC".

user21820

@SergeiAkbarov Of course it is not a synonym.

Sergei Akbarov

merriam-webster.com/dictionary/attest

Then what does it mean?

user21820

@SergeiAkbarov I'm a native speaker. It does not mean "logically implies". It just means "affirm" with a slight connotation of "with evidence", but certainly not "with proof".

Sergei Akbarov

17:55

From Merriam-Webster: "3 : to be proof of : MANIFEST"

user21820

@SergeiAkbarov Are you a native speaker? Quoting a lexicon is pointless when you want to get the exact nuance.

Sergei Akbarov

No, I am not. But what I read in Merriam-Webster is confusing.

user21820

@SergeiAkbarov Anyway that very sentence contains the explicit claim that ZFC has been investigated and used in mathematics for decades and no inconsistency has been discovered, as of that time of writing. So if your question was merely about "no known paradox in ZFC" then you should be happy with that citation.

Sergei Akbarov

Tim, as I told here math.stackexchange.com/questions/3003268/… my opponents will accuse me in using unreliable source.

Because Merriam-Webster is more autoritative for them than even you.

user21820

@SergeiAkbarov I honestly don't know why you're so concerned about what odd people want to say. Do you know about homeopathy and energy bracelets? For many people, popular pseudoscientific TV shows are more authoritative than scientists. You cannot beat a system that does not treasure truth, without completely changing the system.

That's why I left Wikipedia ages ago.

Sergei Akbarov

18:05

This perhaps is again something cultural: in Russia before starting a battle it is very important to prepare carefully. If you see a weak point in your defence, you already lost.

user21820

@SergeiAkbarov So I suppose you are preparing for 'battle' with Wikipedia editors/admins? If they are as unreasonable as it seems, then why bother? You can make a silent protest by always directing people to the English articles, which are as of now not yet overrun by mathematically incompetent people.

Sergei Akbarov

@user21820 Yes, with them. In Russia all your life is a permanent struggle with the system. Wikipedia can be considered as an easy exrecise.

user21820

And I'd love to know what preparation you can make to do 'battle' with pseudoscience, because I am much more morally invested in fighting that one.

Sergei Akbarov

I was trying to find reliable references first.

The book by Ebbinghaus-Flum-Thomas is too risky.

user21820

@SergeiAkbarov Okay, but how are you going to convince anyone about whose opinion is reliable (after all everyone is biased)?

Sergei Akbarov

18:14

No, they seem to admit that if something is published in a peer-reviewed journal or book, this is a reliable source.

user21820

It seems what you want is a 100% unambiguous statement of the current empirical belief of consistency of ZFC. Unfortunately, most logicians aren't inclined to write in such an exacting manner, and perhaps nobody can.

When logicians write about such things, it seems to me they put on their philosophy hat, because there is no 'rigorous' way to discuss philosophy anyway. Unlike mathematical statements, which we can write symbolically.

Sergei Akbarov

It seems wrong. I need another thing: a confirmation that there are no known paradoxes in ZFC, NBG and MK.

user21820

@SergeiAkbarov I think it is better for you to instead write to a Russian logician (say Matiyasevich) and ask him to publish in Russian an explicit statement of the precise sort you want, in a Russian mathematical journal, and then go and cite that.

Sergei Akbarov

@user21820 And of course he will agree immediately.

Gentlemen, thank you for this talk. See you later.

Malice Vidrine

18:57

@user21820 - Heh, I'm going to be crashing in Randall's guest room briefly this srping. I'd forgotten about those slides.

Timothy Chow

20:21

@SergeiAkbarov If your only concern is Wikipedia, and all you want to find a reference for is that ZFC is not known to be consistent, then Ebbinghaus-Flum-Thomas will suffice. As for NBG and MK, an inconsistency in ZFC would yield an inconsistency in NBG and MK as well (I don't have a reference for this but this should be easier to find).

But you need to distinguish carefully between the statement that ZFC has no known inconsistency, and the statement that ZFC is consistent. E-F-T will suffice as a reference for the lack of a known inconsistency, but will not suffice as a reference for the consistency.

E-F-T is just correctly pointing out the distinction between the two statements. It is not frivolous to point out the important distinction.

The phrasing in your StackExchange question is also different. To say that ZFC "overcame the paradoxes" is not precisely the same statement as "ZFC avoids known paradoxical arguments." The statement that ZFC "overcomes the paradoxes" is not entirely clear and sounds possibly like a claim that ZFC is actually consistent.

@SergeiAkbarov If your only concern is Wikipedia, and all you want to find a reference for is that ZFC is not known to be inconsistent, then Ebbinghaus-Flum-Thomas will suffice. As for NBG and MK, an inconsistency in ZFC would yield an inconsistency in NBG and MK as well (I don't have a reference for this but this should be easier to find).

Sorry, I don't know if I can edit a previous comment. I mean to say "ZFC is not known to be inconsistent" but I mistyped "ZFC is not known to be consistent."

Regarding "attests"---it just means "provides evidence for" or "provides testimony for" in this context. The lack of a known inconsistency provides evidence for, but does not prove, that ZFC is consistent.

Now that I have seen some of the StackExchange discussion, I understand a bit better what you are complaining about. It looks to me that you did not recognize the enormous distinction between "ZFC is not known to be inconsistent" and "ZFC is consistent" and were not careful to draw that distinction in every single remark you made. This caused unnecessary confusion.

Sergei Akbarov

20:47

@TimothyChow Tim, I was sure that everybody understands everything. I did not think that my words can be understood so stupidly (as a request to give an evidence of consistency of ZFC).

If E-F-T is the only source this is bad.

Carl Mummert

@SergeiAkbarov: I am writing something at the moment. There is also a nice quote by Ershov and Palyutin - I wonder if that text is available in Russian also

Sergei Akbarov

@CarlMummert Carl which text?

Carl Mummert

"Mathematical Logic" translated from the Russian by Vladimir Shokurov

Here is what I posted math.stackexchange.com/a/3006875/630

Malice Vidrine

@SergeiAkbarov - Some people do ask questions that bad. If you appear to be asking such a thing, we have no reason to think you're not until you say something that indicates otherwise. I'm not sure why this is such an ordeal for you.

Timothy Chow

22:27

@SergeiAkbarov There are other sources you could cite, e.g., Penelope Maddy, "Set Theory as a Foundation," in Foundational Theories of Classical and Constructive Mathematics, says, "ZFC is not prone to Russell's (or any other known) paradox."

Transcript for

♦ $Mathematics$ Logic

Transcript for

♦ Logic

♦ $Mathematics$ Logic