I am obliged to conclude that what you're writing about has nothing to do with Godel or any of Godel's ideas. But since you claim Godel has made "a ridiculously stupid mistake," it appears you do not know what Godel's ideas are about. Which is too bad, because all you have to do is just read the first section of his 1931 paper, which you cited, and you would have enough of an understanding of his ideas to know that he at least was not mistaken. And as well you might try reading your other citation.

I suspect your usage is a kind of term-of-art that itself has nothing to do with any idea of Godel's.

This eliminates this terrible mistake by Gödel:

"14 Every epistemological antinomy can likewise be used for a similar undecidability proof." (Gödel 1931:43-44) — PL Olcott

I have to call you on this it to make it explicitly clear. Your reference is to a footnote to an illustrative preliminary proof that Godel presents in the beginning of his1931 paper.

He explicitly limits his argument to systems of sufficient expressive power. He further notes that while his sample expression asserts its own unprovability while at the same time neither itself nor its negation are provable, that it must be true because it asserts its own unprovability. "The proposition undecidable in the system PM is thus decidable by metamathematical arguments."

Thus no mistake, terrible or otherwise.

That's a bit the point. The gun-nut says that the people's right shall not be infringed and he takes that as an absolute restraint. Of course it is not an absolute restraint, but no gun-nut I've engaged with in any way will allow the conversation to get anywhere near questioning just what "shall not be infringed" actually means. Which is why I set their mental age at 4 1/2.

Btw, I found the video on the Swiss treatment of guns both an education and instructive. I had thought the Swiss issued (appropriately) military rifles to citizens packed as if for storage and not immediately usable - perhaps needing to be thoroughly cleaned first. And woe betide the Switzer whose gun was found in regular inspection to be out of its pack. But your video appears (it seemed to me) to make it clear that the Swiss citizen who has a government issued rifle is expected/required to practice with it.

Also the Swiss notion of duty in a well-regulated militia v. the US gun-nut's rabid insistence on a right that he does not have.

Gun nuts insist on, "...the right of the people...shall not be infringed." But I have not found a single one who will even respond to any question as to anyone who should not have a gun. The logic of the gun nuts wrt the 2A is simply anti-logical ignorant stupidity. Chronological age, 18+. Mental age, four-and-a-half.

Suppose a neighbor who gives you offense; why do you not dispatch him with an axe? Or late and in a hurry in a crowd, yet you do not knock people over in your rush: why not?

And the answers given to these and many similar possible questions will depend on who's answering. But a common ground for most will be a distinguishing between good and bad. Start doing that, sez I, and you begin to become human and to become a moral agent.

What? Where are you getting that? — AmadeusD

Same place they all come from.

This eliminates this terrible mistake by Gödel — PL Olcott

In simple English, please, please make clear just what Godel's terrible mistake was.

My take so far is that you're referring to knowledge. That is, propositions (presumably) known to be true wrt some accepted criteria. Godel was not working with such systems: his propositions were not known to be either true or provable, but that both had to be demonstrated. He merely proved that such a program had built-in limits.

I've just been through a section of Parfit's Reasons and Persons which deals with exactly this issue - whether future reasons constitute 'now' reasons. Parfit feels that a bias toward the near, as he terms it, then means neglecting these reasons one will have - which means, overall, your life will go worse. An interesting position. — AmadeusD

And depending on your age, you will the more feel the bite of it. There are mountains near where I live. I I've climbed some, but would like to have climbed others, putting it off for some reason or other. Now I cannot. Or perhaps there was a girl you were too shy to ask out, not realizing - or being told - that the man you would grow up to be would want you to have asked her. And all of that possibility lost.

This sounds like regret, but I think if dug into even a little, we find a moral imperative to live a life of at least Aristotelian balance and means, not too little, not too much, but not too much of that either.

A metaphor that comes to mind is driving on a highway. If your car is in decent shape, then you're obliged to drive neither too slowly nor too fast. The short-term morality concerned with hazard and possible harm and damage. Long-term with acting in a manner consistent with a life well-lived.

This means most people's morality will align on my account, even if they have different moral frameworks for arriving at the "yes/no" portion of whether to act. — AmadeusD

I agree with this, and although I tried I cannot express it any better.

But most folks leave out of their life calculations that most actions are done or not-done for now-reasons, neglecting future-reasons. That is, when you get older, either you cannot any more, or you realize that should/should-not yields to either I will do, or I will die never having done. And add to this the importance of memory. You cannot remember what you have not done.

The "moral" of the story, I suppose, is that each of us had better be pretty damn careful as to what morality we allow to rule us, what rules it makes, and why, because the cost of mistakes is high, and is paid mainly by the person who makes the mistake(s).

One example: in many - maybe most - churches, sex outside of marriage is considered the sin of "fornication," an atrocious mistranslation of the Greek word πορνεία porneia. And so there are people unmarried for long periοds of time, or who maybe never marry, who never get laid. Which is to say that their moral code has made them deny a significant part of their own humanity.

But this does not just apply to sex; it applies to almost everything where a choice is to be made and the reasons for making that choice.

So do you think any time you have (a implies b) , it's always true that (not a implies not b) — flannel jesus

Yes, it is correct. — Corvus

No, it isn't. Truth tables are easy enough to learn, and easy to do, if you don't have too many variables.
(p=>q)=>(~p=>~q) is false when p is false and q is true.

Here:
http://www.thatmarcusfamily.org/philosophy/Course_Websites/Readings/Tarski%20-%20The%20Concept%20of%20Truth%20in%20Formalized%20Languages.pdf
Is the whole paper. I cannot copy and paste from it.

I think you will find on p. 254 the requisite qualifications.

And it's worth noting that the languages referred to are all mathematical-logical languages, in the sense that any primitive language to be be discussed be translated into numbers, and the semantics of the language(s) being mathematical operations.

One can formalize the semantics—define truth—of lower order — Corvus

It's been observed were truth so definable, then the usual reading of Godel's sentence, unprovable but true, would have been instead untrue but true. That implies that truth is not so definable. Not to be confused with determining whether some proposition is true or not true, which can usually be done.

Edit. Actually, this from Olcott's citation above:
"(f3) assuming that the class of all provable sentences of the metatheory is consistent, it is impossible to construct an adequate definition of truth in the sense of convention T on the basis of the metatheory."

A knowledge ontology inheritance hierarchy capable of formalizing the entire body of human knowledge that can be expressed using language need not be incomplete in the Gödel sense. — PL Olcott

This at least seems true. Mainly because such a listing lacks the power of systems that are incomplete in the Godel sense, and in fact have nothing to do with it. Your groceries list can stand for such a body of knowledge, and nothing incomplete about it.

True and unprovable — PL Olcott

Properly and correctly qualified as unprovable within the system out of which it arose, but proved true by other means.

Btw, true is an adjective indicative of a quality that true statements have. Good luck with any attempt to comprehensively further define just what that quality exactly is.

For formalizing the entire body of human knowledge that can be expressed using language we need this: — PL Olcott

Let's try this. Suppose you have your "this," call it T. Now suppose you have some expression. Is it or its negation in T? If so, great! You're done. If not, then you have to figure out if it should be or not. And using existing knowledge, you cannot (if you could, it or its negation would already be in T).

Best you can do is add it as axiomatic in some way, or fundamental. Now you have T'. Now a new expression, same business, and you get T'', and so on and so forth. You might argue, and imo quite reasonably, that in collections of data this doesn't really arise. But in math-logic it exactly does. That is, there is no terminal Tωω...ω in math-logic.

And this concerns among other things differences between finite and infinite sets. .

What is your point? We suppose - that's the best I can do - that a proposition undecidable in L is decidable in L', and one in L' in L'', and so forth. But apparently there is no Lωω...ω that is itself complete.

seems to follow the principle that every simple idea can be made convoluted enough that it can no longer be understood. — PL Olcott

That's a convenient principle. Btw, how do you know when an idea is just that simple?

A formal system having only one order of logic is like the "C" volume of an encyclopedia only having articles that begin with the letter "C". — PL Olcott

And a complete set would have everything from A to Z. But in our case, you can't have a complete set.

When all orders of logic are included in the same formal system then such a system cannot be incomplete. — PL Olcott

Um, no.
I'm going to try to condense a proof due to Emil Post, in The Undecidable, Ed. Davis, 1965, pp. 304-337, esp. pp. 308-317

Let B be understood as a recursively enumerable set of numbers. Let B1, B2,...Bn, B(n+1)... be simply a listing of such sets with duplicates omitted and arranged perhaps by length and lexicographically for sets the same length.

Let (Bn,n) be simply the proposition, true or false, that n is in the set Bn.

By interlacing the Bns and the integers, e.g., B1,1; B2,1; B1,2; B1, 3; B2,2; B3,1, and so on, there are generated all the distinct couples Bn,n.

Call this set E, of all the expressions Bn,n, understood in each case as the expression that n is in the set Bn.

From E generate the set T of all Bn,n such that the number n is in Bn; i.e. true. The complement of T will be ~T, that is, all those Bn,n such that n is not in Bn.

Let F be any recursively enumerable subset of ~T. That is, if Bn,n is in F, it is in ~T, and n is not in the set generated by Bn.

Now it gets a little bit tricky.

Now generate the members of F. If a member of F is of the form Bn,n, place n in a set of positive integers called S. S being recursively enumerable will correspond to some B, call it Bv such that S=Bv. Construct Bv,v; that is, the proposition that v is in Bv.

And now I quote directly. "Now by construction, S consists of those members of F of the form Bn,n. Suppose that Bv,v is in F. Then,... Bv,v being false, v would not be in... Bv. That is, v would not be in S. But Bv,v being of the form Bn,n, v would be in S. Our assumption leading to a contradiction, it follows that Bv,v is not in F. But v can only be in S by Bv,v being in F. Hence, v is not in S. Finally, Bv,v says that v is in S.... Bv,v is therefore false; that is, Bv,v is in ~T.

"...[ S]ince Bv,v of ~T is not in F, T and F together can never exhaust E.... We can then say that no recursively generated logic relative to E is complete, since F alone will lead to the Bv,v not in F." That is. given the logic determined by T and F, Bv,v must be undecidable.

"[They, (the ideas here presented)] implicitly justify the generalization that every symbolic logic is incomplete...." (316)

Which is to say - just between us in case we're both wrong - that each system being itself deficient requires a successor system to fix it, but that simply creating a new deficiency. Ordinal arithmetic being formidable, I don't see an escape.

You shall have to keep it simple with me. From what I've read, all the incompleteness/undefinability theorems only apply to systems of "sufficient" power and interest. I find this online, "The theorem applies more generally to any sufficiently strong formal system, showing that truth in the standard model of the system cannot be defined within the system."

Your question (again, if I understand it), is can there be a super-strong formal system that is not incomplete. I am guessing not. And I'm sure a rigorous discussion would be well, rigorous.

You're off to the races into transfinite-order logics. If I understand the question of the title, it is equivalent to asking if Godel's incompleteness (theorem) is entirely resolved at some higher level of logic. My guess is not.

Thus a single formal system have every order of logic giving every expression of language in this formal system its own Truth() predicate at the next higher order of logic. — PL Olcott

And if nothing else, this is the clue, "next higher order...". It appears you want to get to the point where there is no higher order. And that would seem to lead to a set-of-all-sets type of contradiction.

What do you mean "my law"? — Alkis Piskas

Cause-and-effect is a presupposition. I'm under the impression that a lot of science - not all - no longer thinks of cause-and-effect as an adequate description of how the world works. I think the replacement is to think in terms of fields - subject to correction.

There's only pure logic here. — Alkis Piskas

Logic is fine. What does it have to do with the world?

With the right presuppositions, I can prove anything logically. But that does not make it so. Trivially, if I assume P ^~P, then I can prove anything.

In my case, the frame of reference was the "law if cause and effect".... Only that you didn't show why the first cause doesn't work and/or why it would be a paradox — Alkis Piskas

If you only assume there is a first cause, then you've shown nothing. If your law is that every effect has a cause, then is every cause caused? If not, then what is different about a first cause (and why only one)? If yes then what causes the first cause? And, this is just an exercise in language; what does it have to do with the world?

A first cause in the world, then, is speculation. Make it real.

Weren’t you the one saying you consider October 7th to be the beginning of this and that nothing prior matters? — Mikie

Nope, not the beginning, but a boundary that when crossed took all prior matters off the table until this one settled. As with a rabid dog or a medical emergency, you do what is necessary first. In the present case hostages and criminals.

How is this language, and the poster who posts it, even tolerated on the forum? — boethius

7 October is a fact. The events of 7 October are a fact. The behaviour of Arab/Palestinian/Hamas on 7 Oct, murderous and bestial, also a fact. But there is one whole side of this thread that simply fails/refuses to address those facts. "Oh my yes!" they proclaim, "They did bad things, but those poor people, the awful Jews made them do it." And then they dismiss the Arab side of it, as has been done since at least Yasser Arafat in my recollection, and according to others, well before that.

That leaves the Israeli/Jews essentially isolated. With some allies to be sure, but in many ways alone. So, who are we to question what the Israeli/Jew thinks best to do to keep alive when we simply do not live in his world. And how much better it would be if an international police force and court had presented itself to take over a criminal prosecution of the perpetrators of 7 Oct.?

You think paradoxes logical things categorically apart from hands-on material things? You think paradoxes the products of narratives made incoherent due to missing pieces? Do you have any ready-to-hand examples? — ucarr

Creatures of logic yes, of the world, no. From missing pieces? I prefer to say from incomplete or inadequate descriptions. Case in point the Cantor paradox referenced above. The idea is that the set of all sets presumably contains its own power set as a subset, which implies that the cardinality of the subset is greater than the set itself. The resolution in this case is to correct the description to state that there can be no set of all sets because it leads to a paradox - or contradiction if you prefer. But you may protest that the universe itself is the set of all sets, but that would simply be a misunderstanding of the terms: the universe contains everything as distinct elements, no subsets, and thus cannot contain its own power set, which in this case would be meaningless. .

You say language reaches its limit dealing with empirical experience. Can you elaborate on "dealing with"? For example, "Dealing with" means perceives and understands as if through a glass darkly.

I've been forming the impression you see clearly two distinct experiences, one linguistic, the other hands-on material. — ucarr

Small point to start with. If your reference is to 1 Cor. 13; 12, then what you think is "through a glass darkly..," is actually, "through a mirror [now] as a riddle, but then [later] face to face." The word darkly is actually the word αἰνίγματι - ainigmati. And if you can see the word enigma in that, then you are exactly right, the usual translation being "riddle."

It may be useful to note that all experience is empirical; that is, experienced. As such, subjective. Language then (of course) describes that experience, the language itself being neither the experience nor constitutive of the experience. You saw a pretty girl: neither your experience nor the description of it is the pretty girl. And she is beyond doubt pretty! But did you see her in infra-red or ultra-violet? Or the zillion or so neutrinos we now understand are passing through all of us all the time? Perception - experience - is what it is and nothing else. The world is something else.

And there is speculation about matters not experienced, like first causes. A common form is to admit ignorance, and then immediately to pretend to knowledge. Kant covers all of this in the opening pages of his CPR.

Are you steeped in linguistic philosophy? — ucarr

Not that I know of.

Do you think language is inherently limited in its ability to characterize empirical experience truthfully and completely, or do you think language has innate potential to do this, but your endorsement of this characterization comes with the proviso that, up front, tremendous work over eons is necessary? — ucarr

I think language can at best only deal with empirical experience - what other experience would there be? The trouble comes about when empirical experience is taken for the world itself as it is in itself.

Do you think paradox exists only within language? I ask bearing in mind superposition at the quantum scale. — ucarr

I'm of the mind that there are no paradoxes in the world, only in descriptions of the world. Of course when descriptions are incomplete, that leaves apparent paradoxes.

Assuming one accepts the law of causality — Alkis Piskas

And there you have it. Assuming you accept X, you get Con(X) (consequences of X). Except of course when you don't, then you can either reject X, or develop X', and maybe X'' or X'''.

I am under the impression cause-and-effect is no longer accepted in much of physics as being the right account for how the world works. The point I suppose when looking at foundational concepts, is to question everything and assume nothing, so far as is possible. As it must be that some assumptions are made, it remains to make them explicit and to test them to see if they "work." The idea of a first cause or concept seems not to work (in this context) leading to paradox. That alone would suggest it be rejected.

What empirical conclusion do you infer from the open-ended question of First Concept? — Gnomon

What do I infer? That lacking a lot of preliminary groundwork, mostly in establishing working definitions - though they be provisional and subject to change, pace Banno! - the question remains a non-sense question. That is, an attempt to make sense where there is no sense to be made.

Temporality is implied in "first." Admittedly we're all temporal beings, and make sense of our world in terms of before, after, during. But what does modern physics say? For events space-like related which came first depends on who you ask - and notions of entanglement make that even more difficult to understand.

And definitions - even understandings - partake of the nature of templates. But I believe it is Wittgenstein's observation that notwithstanding the efficacy of some templates, all templates are imposed and thus nothing of the world itself.

(But) we exist in such a way that our templates appear to get the world's work done, with improvements over time - maybe it's evolutionary. And it is understandable that people would confuse their ideas about the world with being how the world actually works.

Too many posters on this thread are looking through glasses - lenses - encrusted with junk. The templates they see that they think are derived from the world are instead artifacts of defective optics. A little lens cleaner might help. The purposes of the Arabs/Palestinians/Hamas are simply based in their beliefs and ideologies. The Israelis/Jews, on the other hand, are and have been literally fighting for their lives. And while this has been the reality of generations/centuries/millennia, 7 Oct. 2023 simply made clear and explicit the bestiality of Arab intentions and practices. As such, until and unless adjudicated, there is no reason to consider anything ante-7 Oct. That is, the animal who brutalized and outraged your daughter before, during, and after murdering her may be just misunderstood, but his actions make that for the while irrelevant: his own statement being that he is no better than an animal, a vicious one.

Hostages returned, criminals apprehended, justice served, then and only then back to pre-existing concerns.

Along with any reason for doing philosophy. — Gnomon

Along with any reason for doing foolish philosophy. But one place a fool never sees a fool is in a mirror. I attest to this from my personal experience with mirrors.

When you say "paradox in this case nature's way of saying "Dead-end. Turn about and go another way, "are you invoking the principle of non-contradiction? — ucarr

You're on the wrong road. Non-contradiction may guide how I think about what I think about, but it has nothing to do with the world.

If you wish to understand first causes, then at the least you shall have to decide what context you're in. Language? Then it appears the language yields paradox. The world? No apparent paradox, but also no easy understanding.

A trivial example: if I claim to be a faster runner than anyone, likely you will have a good understanding of what I mean, that I am fastest in my school, community, age-group, the world, whatever. But one thing I cannot be is faster than anyone, because I am an anyone, and that would mean I am faster than myself. So, while in the world I may be fast, the language fails as description. I'll opine here that all paradox is simply one sign of failure to understand - failure of language.

I'm interested in learning how and why "A set of all sets" is not reasonable. — ucarr

This site was referenced in another thread.
https://math24.net/paradoxes-set-theory.html
It is too clear, simple, and brief for papaphrase. Take a look.

The hazard, it seems to me, is that language itself is paradox-prone. Insofar as the paradox creates a mess and the mess needs cleaning up, the usual rule for cleaning up messes is to clean them up where they are and not where they are not. Cantor's paradox (about set theory) arises out of descriptive language thought entirely sound but found to be flawed, the remedy being to fix - qualify - the language. A set of all sets seems at first reasonable; it turns out not to be.

The "paradox" of first beginnings is an applying of language to the world. The world being neither obliged to cooperate with nor obey language, paradox in this case nature's way of saying "Dead-end. Turn about and go another way." Which is good advice.

A digression: it is a sign of intellectual maturity to both recognize good advice when and where met, and to follow it.

But first beginnings is an itch just crying out to be scratched! Since this is about the world (and not about knowledge of the world) the right approach is to conform the language to the world, and not the world to language. In practical terms that means to define, define define, the defining being refining/qualifying the language about the world until it works.

As "first cause," for example, implies temporality, that has to be understood as to what that is, and if it applies. For thee and me that's easy: the sandwich has to be made before it can be eaten. For the world itself, not so easy. And indeed it may be that temporal priority is an irrelevant dead-end!

Dead ends can be fun to explore; but it is a mistake to suppose they're the way to anyplace in particular.

deleted reply - wrong topic

It has seemed to me that the effort involves supposing that an(y) artifact of language (e.g., about so-called first causes) has anything to do with physical reality. Recognize that it doesn't and the problem of reconciling irreconcilables evaporates.

when mathematicians saw that Cantor's naïve set theory — an attempt to axiomatise mathematics — had contradictions, — Lionino

Can you sketch simply and briefly what such contradictions might be? Or point to where they are laid out?

What is clear is Biden's cognitive decline — boethius

Is that notwithstanding any cognitive decline, he has got a lot done in spite of the best - or worst - efforts of Republicans. And if Republicans had condescended to contribute to government instead of trying to destroy it, likely much more could have been accomplished. It is my guess he has a good and dedicated team and they all work together. Republicans, on the other hand, have no good man or woman - "good Republican" being nearly an oxymoron - nor can they work together, but instead like sharks in a feeding frenzy feed on each other and even themselves.

There's no excuse to collectively punish, through starvation, a civilian population for the crimes of a terrorist organisation in their midst, or indeed, their government. — Benkei

So tell us, then, why exactly are Hamas still holding the hostages?

I do not know what any Israelis are thinking - I imagine they think all kinds of things. My thinking is that I want the hostages, and I want Hamas, with good reason for wanting both. And I will press until I get both, history having taught that all else fails. For at least immediate relief, the Palestinians need only turn over both. Simple as that. But they would rather die. Well, they do not get a prize for that; they only get the death they want - and to be sure, that they force.

You, Benkei, seem to think the Israeli have a choice. I think they don't, Arab attitudes. policies, and actions having made it clear that they don't.