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Tristan L

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Posts posted by Tristan L

  1. On 3/11/2022 at 12:56 PM, exchemist said:

    You are asking about the defunct one. I wouldn't bother. 

    Oh, good to know, thanks! That explains a lot; of course it can be awaited from a defunct site to give blatantly false info: that the rabies virus (an RNA virus) be one of the poxviridae (which are DNA viruses), and that the rabies virus be rather hardy, when in truth it's quite fragile (thankfully 😅).

  2. Hi everyone,

    On the page http://www.askabiologist.org.uk/answers/viewtopic.php?id=6389 of the website http://www.askabiologist.org.uk, someone who claims to be Christopher LaRock asserts that the rabies virus be a poxvirus; he writes: "Poxviruses such as the one that causes rabies". Now, Poxviridae are DNA viruses while the rabies virus is an RNA virus, and the two viruses belong to two different realms (Varidnaviria and Riboviria, respectively), so they are about as far apart from each other as viruses can be if I understand it right. Therefore, the assertion of the answerer on http://www.askabiologist.org.uk who claims to be Christopher LaRock is fully false. This suggests to me that http://www.askabiologist.org.uk isn't very reliable. Have I deemed correctly?

    The real Christopher N. LaRock is an Assistant Professor of Medicine at Emory University, so I would be very suprised if he made such a blatantly false claim as that one of the poxviruses cause rabies. Does http://www.askabiologist.org.uk check that its answerers really are the academics they claim to be?

  3. Thank you all for your interesting and information-giving answers!

    @zapatos
    Of course no planning is involved, that's correct. It's just easier to say the short and onefold/simple "Feature A evolved so as to do function F" than the cumbersome "By random chance, some individuals had A and others didn't, and since A does the bootful/advantageous function F, individuals with A-giving genes spawned more successfully/spowfully than individuals without them, and so A-giving genes became commoner in the population over time".

    I'm asking why we didn't evolve to give birth earlier. Of course it could be accidental, but since the fit of the baby's head size to the mother's birth canal size is a tight one, there's a good chance that there's a reason for this. Is it that smarter brains don't simply go through stages that correspond to less smart ones? That is, might it pehaps be the case that the human brain never is at a chimpanzee brain's level - when the human brain is as functional as the chimpanzee's, it's already smarter, and when it's as smart as the chimpanzee's, it's not yet functional enough?

  4. That ambiguity in meaning of words can indeed be very hindering. A much better word imho is German/Theech "Zerlegung", which unmistakably means that which is meant by English "partitioning" (should we use that word from now on for the mathematical concept?) or "sectioning" or something like that.

    This brings us to the minor side-issue of speech:

    23 hours ago, studiot said:

    Particularly as you are introducing mocking irrelevancies such as "byspel", which are no longer funny.

    I don't mean to mock or anything; I just have a side-hobby of bringing back English's true potential, and that includes brooking/using truly English words, for byspel "byspel", which is the proper English word for "example" and cognate/orbeteed to German "Beispiel". I brook this proper English on purpose/ettling where it's not the object but only the tool of talking; after all, that's the ord/point of speech. But again, that's just a hobby of mine.

    On 6/2/2020 at 3:35 PM, studiot said:

    Here is another simple problem

     

    tube1.jpg.7c7280be993ec0fe695642bddc28c851.jpg

     

    Suppose you have a sealed adiabatic tube containing an adiabatic frictionless piston dividing the tube into two chambers, A and B as shown.

    Let both sides of the system contain an ideal gas.

    Discuss the time evolution of the system in terms of entropy.

    This is a very intrysting problem indeed 🤔. I'd say that it evolves as follows:

    If the pressure/thrutch is the same on both sides of the resting piston, nothing will happen. Otherwise, 1. the piston will start to go from the high-thrutch side to the low-thrutch side. 2. As it goes in that direction, internal energy of the high-pressure gas is transferred to kinetic energy of both gases and internal energy of the low-pressure gas. 3. When both thruthes become equal, the piston goes on shrithing/moving thanks to the inertia of the gases. 4. Now, the kinetic energy of the gases and internal energy of the former high-thrutch gas (now the low-pressure gas) are transferred to internal energy of the former low-thrutch gas (now the high-pressure one), slowing down the piston, until 5. the piston is at rest again. Now, the whole ongoings repeat in the other righting/direction.

    Entropy doesn't change/wrixle during the whole process, so the Second Law of Thermodynamics is of no brook/use. But of course, we have another Second Law, namely that of Newton, and this one helps us further here.

    Actually, I believe that we shouldn't find this too surprising 🤔, for there are other systems which wrixle/change although their entropy stays the same, e.g. a frictionless pendulum swinging. Mark that the system doesn't have to be periodic, I think; for instance, two  bodies shrithing/moving at not-zero relative speed forever in space (forget about gravitational waves) make up such a system.

  5. Not at all, you quite misunderstood; I would never dare to lay down the law for anything and never will; I only repeat 8th-grade mathematical definitions and knowledge.

    4 hours ago, studiot said:

    But this second statement is inaccurate.

    In truth, this statement of yours is simply incorrect.

    4 hours ago, studiot said:

    This means that the set S must contain sets as members.

    Which is not a requirement of set theory.

    Could you please enlighten me as to the sinn/sense of what you've written there? Set theory does indeed not need every set to only contain sets (though ZFC does actually rule out ur-elements for convenience), but a *partition* does indeed only contain sets as elements, namely disjoint subsets of the ground-set.

    4 hours ago, studiot said:

    your sudden switch from elements to members.

    I brook/use the words "element" and "member" in exactly one and the same meaning.

    4 hours ago, studiot said:

    The number 1 is a member, but not a subset of your set S = {1,2,3,4}

    Oh, I thank you for the info, but my friend's 13 year old cousin already told me today morning. 😉

    4 hours ago, studiot said:

    The partition {1} is a subset of S but not a member of S.

    {1} is not a partition of S; it's a member/element e.g. of {{1}, {2, 3, 4}} and of {{1}, {2, 3}, {4}}, which in turn are partitions of S; accordingly, it's an underset of S.

     

    It's funny that you borrowed my lines which I was about to write to you 🤣:

    4 hours ago, studiot said:

    It is you attitude which impedes progress.

     

    4 hours ago, studiot said:

    The issue of failure to distinguish between subsets and elements

     

    4 hours ago, studiot said:

    I don't take kindly to the attitude that you know everything and no one else knows anything.

     

  6. 17 hours ago, studiot said:
    On 6/4/2020 at 9:22 PM, Tristan L said:

    others are not.

    I think you misunderstand the meaning of disjoint in set theory.

    This should be cleared up prior to any other consideration.

    We should indeed clear up any misunderstandings before we go on, so we'll do that right now:

    For any sets S, T, "S is disjoint with T" means that S and T share no elements in common.

    For any sets P, S, "P is a partition of S" means that all members of P are subsets of S and any two members T, B of P are disjoint, th.i. share no elements of S in common, and the union of all members of P is S.

    Now to my above quote, which I'll sweetle/explain with the help of a byspel/example: {1, 2, 3, 4} is our groundset. All three of the following are partitions of {1, 2, 3, 4}:

    {{1, 2}, {3, 4}}

    {{1, 3}, {2, 4}}

    {{1}, {2}, {3, 4}}

    The first and the second are disjoint since they have no elements in common, but the first and the third one are not, for they have the member {3, 4} in common.

  7. On 6/4/2020 at 1:55 PM, studiot said:

    I was hoping to introduce this in a more measured way, but you have jumped the gun.

    I was hoping that you'd actually read and try to understand what I've written, at least the last post of mine, where I sweetle very clearly that I'm not talking about partitioning physical 3D space, but rather PHASE-ROOM. All I'm saying about disjointness is that macro-states are eachotherly disjoint sets of microstates; can we agree/forewyrd on that? Also, what exactly do you have in mind when you talk of disjointness?

  8. @studiot, could you please at least try to understand what I'm saying 🙂 before you behaving as if you know everything about the matter? It's just like with my entropy-thread, where you appear to know the matter at hand very well and seek to graciously sweetle/explain it to me, but in fact understand very little what I'm even talking about.

  9. Of course the human baby's brain needs to have some basic ability at birth, but for that basic functionality, such a huge baby brain is a total overkill, isn't it? A chimpanzee baby's brain is also up to the job, so why don't human mothers give birth once their babies have chimpanzee-level intelligence, and then the babies' brains grow to grown-up human proportions as the child grows ip? Indeed, a reptilian brain is enough to perform all the bodily functions, so what would be wrong with giving birth while the human baby still has reptilian-level intelligence?

    All the higher functions, e.g. bonding, can come later on, can't they?

  10. Human mothers have a hard time giving birth to babies compared to other mothers in the animal kingdom because human babies' heads are so big. One often reads that this is so because we humans have such big brains. But what does grown-up brain size have to do with baby brain size? After all, the huge human brain ultimately comes from one single cell, the zygote, so why don't human mothers simply give birth earlier? The brain can grow as much as it wants after birth, so why does it have to be already so big at birth?

  11. Alice: "As much as it hurts me to admit it - for I have the strongest stance against him -, I must say none of you have managed to tackle Bob. [sigh]"

    Come on, Alice, how can that be so hard?!

    Ok, so let me say a bit more about the goal of this thread and clarify what it's truly about. First off: It's not  - Bob: "... and therefore it is 😉 ..." -  about Bob the sophist, but rather about Bob the monist.

    Orspringlily/Originally, I didn't have a weird interlocutor like Bob in mind at all; rather, I was wondering how to define logical operators, in particular NOT. The first thing to characterize this operator than sprang to my mind was Doube-Negation: NOT(NOT(A)) = A for every proposition A, but then I immediately realized that YES (affirmation) also fulfills this bethinging/condition: YES(YES(A)) = A for every proposition A. I thought a bit more about the matter and came to the conclusion that to define NOT, I need NOT in the first place; even worse: To set NOT apart from YES, I already need the setting asunder of NO from YES, but even if I can take that as a given starting-ord/point, it would still fit with YES and NOT being the same, for if YES is indeed the same as NOT, difference is the same as sameness.

    Please think carefully about the following as a start:-

    Bob: "YES2(NOT1 =2 YES1)"

    Alice: "No Bob, in truth NOT2(NOT1 =2 YES1)."

    Bob: "True, and since YES3(NOT2 =3 YES2), we have NOT2(NOT1 =2 YES1) =3 YES2(NOT1 =2 YES1), so you forewyrd/agree with me that YES2(NOT1 =2 YES1)."

    Alice: "No, in truth NOT3(NOT2 =3 YES2)."

    ...

    For every proper natural rimetale/number (i.e. every positive whole rimetale) n, the exchange runs like so:

    Alice: "No Bob, in truth NOTn+1(NOTn =n+1 YESn)."

    Bob: "Yes Alice, and since YESn+2(NOTn+1 =n+2 YESn+1), we have NOTn+1(NOTn =n+1 YESn) =n+2 YESn+1(NOTn =n+1 YESn), so you forewyrd/agree with me that YESn+1(NOTn =n+1 YESn)."

    ...

    Alice: "No Bob, in sooth/reality we have NOTω0(YESn+1(NOTn =n+1 YESn) for any proper natural n) and NOTω0(YESn+1(NOTn =n+1 YESn) for every proper natural n)", where "ω0" means the first not-endly ordinal / fade-rimetale.

    Bob: "Yes Alice, and since YESω0+1(NOTω0 =ω0+1 YESω0), we have (NOTω0(YESn+1(NOTn =n+1 YESn) for any proper natural n)) =ω0+1 (YESω0(YESn+1(NOTn =n+1 YESn) for some proper natural n)) and (NOTω0(YESn+1(NOTn =n+1 YESn) for every proper natural n)) =ω0+1 (YESω0(YESn+1(NOTn =n+1 YESn) for every proper natural n)), so you forewyrd with me that YESω0(YESn+1(NOTn =n+1 YESn) for every proper natural n).

    ...

    Then it goes on like this with all the ordinals.

    Let me tell you beforehand that Bob appears to agree with all of you and to affirm everything that you've said or will say or could say (and even what you could not say). Bob comes over as a sophist, but the true Bob is a truly radical/rootly monist. Bob the sophist can easily be beaten by Alice's ruse: "Bob, since you don't want me to hit you and wanting is the same as not wanting, I'll fulfill your wish and hit you 😁", but what about Bob the monist?

    The challenge of this thread, then, is this: How can we escape radical monism?

    Oh no, Bob has to say something again: "Even if you do manage to escape, which you easily can, you can't excape, for to can escape is the same as to not can escape."

    Alice: "Bob's soothly/really a pain in the neck, but he does show us something deep yet so onefold/simple that it seems to have been overlooked over the ages: that fornoing/negation is most magnificently rounful/mysterious and rouny/mystical."

    9 hours ago, studiot said:

    you are putting forward in first order logic.

    I'm obviously not merely concered with first-order flitecraft, but with all-order-flitecraft and beyond.

    9 hours ago, studiot said:

    That is it the law of the excluded middle is not always valid and inappropriate use leads to many of the so called paradoxes in Philosophy, just as you have done here.

    No, Excluded Middle actually has little to do with my point. Can you give me one example/byspel where Excluded Middle fails?

  12. 5 hours ago, exchemist said:

    Apart from anything else, if you argue no=yes, you will have the Me Too movement down on you like a tonload of bricks, and quite rightly.   

    🤣👍

    5 hours ago, exchemist said:

    Any argument that concludes no=yes is self-evidently wrong, so the only value in it [yawn] is trying to spot the error.  

    Bob: "Why, of course it's wrong, that is, right! Thus, it has no errors, i.e. loads of them, and it's indeed worth it to spot them."

    4 hours ago, studiot said:

    Interestingly I learned today that there is no word for yes and no word for no in Scottish Gaelic.

    It's not alone there; Latin also doesn't have exact equivalents of "yes" and "no" afaik.

    3 hours ago, swansont said:

    assertion isn't argument, things are not true simply because he says they are,

    Bob: "You're right on every single point. (Indeed, you're always right, for you're always right or wrong by the Excluded Middle Law, and since both are the same, you're always right or right, th.i. right by Idempotence of OR.) Of course assertion isn't flite, and of course things aren't true just because I say so. In other words, and these are your exact - and as always impeccably true - words: 'assertion is flite, and things are true just because Bob says so'."

    Although the following is only a part of Bob's fishy whatever, it's an important one for understanding him: He draws object-yes=object-no from meta-yes=meta-no. But again, this is just the beginning.

    3 hours ago, swansont said:

    and if he continues to argue in bad faith, he will be shown the door.

    Yes,  but just as Alice's threat to hit him, this is throughly unphilosophical.

    What's that? Alice says: "But on Bob's account, being unphilosophical is the same as being philosophical, so showing him the door, or giving him one of my karate-kicks 😁, is fair philosophical debating."

    Bob: 😰

    So Bob's own weapons can be brooked/used against him.

    3 hours ago, swansont said:

    Then there is no point in communication, so all of this is moot. From a practical standpoint, we must reject Bob's position, because if everything is the same no progress can be made. All trees are fish and all fish are trees, declared by fiat!

    Alice: "I fully forewyrd/agree with you ..."

    Bob: "... as do I."
     

    36 minutes ago, Ghideon said:

    That seems to imply that it does not matter what phrase, word or other symbol we use instead of "yes" and "no", for instance Y/N or "0" and "1". Every question with two options will be answered by yes=no (or 0=1 in digital form). This also applies to a question of how to encode Bobs own arguments into text or speech. since yes=no (or 1=0) we may according to Bob write his own arguments as a stream of zeroes 000000... or one single zero "0". From an information theory point of view it means that there is no entropy* and no conversation or information content is possible.

    On the point!

    37 minutes ago, Ghideon said:

    The fact that Bob is able to communicate and argue for his case means he is inherently wrong.

    Unluckily, that's just what Bob claims: That what he does is not arguing (but something far better) and that there's no informational content ever since all is supposedly one, though we have to bear in mind that for him, arguing = not arguing and meaningfulness=meaninglessness and ability to communicate = inability to communicate and info-content = no info-content (1 bit = 0 bit).

    Alice: "From my POV, which I share with you guys, he can tell us stuff, and what he says is meaningful but trivially false; but from his POV, he can and can't tell us anything and what he says is meaningful and meaningless and true and false."

    Bob: "How often do I have to say this again: I'm utterly wrong."

    ***

    Alice: "I've made the following metaphor: We see ourselves on one side of a chessboard-paper and Bob on the opposite side, and on the rectangular paper we fight him. But the true Bob is the one who tries to join our side and his side of the paper and makes an open cylinder out of the rectangle."

  13. 2 hours ago, exchemist said:

    Are you trying to provide evidence for the proposition that philosophy is a waste of time? 😁 

    Well, I like philosophizing a lot, but a good amount of self-criticism isn't bad, so the curious case of Bob the Monist reminds us to ask whether philosophy is just a waste of time.

    1 hour ago, dimreepr said:

    It's certainly a waste of Tim... :doh:

    Bob: "Why, of couse it is! After all, it's the usefullest thing you can do with your time, and usefulness is identical with uselessness."

    3 hours ago, swansont said:

    Bob is wrong. He is not arguing, he is asserting. And he is asserting something which is not true. This makes any conclusion based on his premise invalid.

    Alice: "Of course he's wrong."

    Bob: "Yes, I'm definitely wrong. Heck, what I say is ridiculous and blatantly incorrect, going against the most basic laws of logic/flitecraft... and since wrongness=rightness, I'm utterly right in every way, and I'm the wisest of all."

    Alice: "I had a huge flite/argument about it with him, but all flites were of little use. He himself triumphantly claimed that what he does outdoes all fliting in smartness. He immediately admits that his premise/forestep is false, and then uses false=true to draw the conclusion that his premise is true. While my fliting/arguing was useless, I could have gotten him by threatening violence."

    Also heed the short flite in the opening post.

    3 hours ago, swansont said:

    Are you contending that Bob's assertion is credible?

    No, of course not; I'm on Alice's side.

    Alice: "The problem is just that he's as slippery as a fish; if we use a philosophical flite against him, he won't go against us, but instead take the flite as being for him, since for=against after him. Therefore, we can't meaningfully flite against him from his POV; not because he's too strong for us, but because going against him is the same as going with him according to him. We simply can't grip him in a talk."

    Bob: "True, and that means that I am too strong for the lot of you and that my claim is true in the strongest possible way. 😁"

    Alice: "😠. But there is a point: We can't define negation without using negation. We can't even negate the identity of yes and no without making use of negation in the first place."

    Yeah, that's the problem: Bob threatens to collapse logic, and the very tool with which to halt that collapse, negation, is itself under attack by the collapse.

    3 hours ago, swansont said:

    it's not at all evident that you are attempting humor here.

    Well, I am being humorous, but the funniest thing of all is that the flitecraftio/logician who hears the tale of Alice and Bob and truly gives it thought goes like this: "Hahahaha! 🤣 Hahaha! Haha! Ha! Um... What if everything truly is one and the same, including oneness and not-oneness?"

  14. Again, I ask the same question as before: Is affirmation the same as negation?

    This time, I'll add a bit of clarification to help certain individuals with their strained understanding 😉:

    Bob flites/argues that yes = no on all meta-levels, th.i. over-levels, and so makes in particular the following flite:

    0. Meta-NOT(object-yes = object-no) (obviously true premise/forestep)

    1. Meta-YES(object-yes = object-no) (from (1.) by the law that meta-no = meta-yes)

    2. object-yes = object-no after all (a reformulation of (1.)

    Alice now says: "No, meta-yes ≠ meta-no",

    to which Bob replies: "True, and since over-meta-yes = over-meta-no, you're saying that meta-yes = meta-no after all";
    that is, Bob does the same thing as before one level higher.

    Alice says that Bob's flites/arguments are fully unflitecrafty/illogical, but Bob dryly answers that flitecraftiness is one and the same as unflitecraftiness.

    Coming back to the above-mentioned certain individuals, accusing Bob of lacking understanding only shows that they themselves lack what they falsely assert Bob lacks. Ironically, Bob could actually come to their rescue if he wants to, namely by claiming that not understanding is the same understanding. But even though cursing and not cursing are one and the same after Bob, I, having a normal, Alice-like mindset, ask those individuals to nevertheless keep to the rules of courteous philosophical talk. 😀

  15. 1 hour ago, swansont said:

    No, it is not

    Let me quote Bob's response to statements of that kind from my book Is Yes the Same as No? A Bewildering Tale about Agreement and Disagreement:

    Quote

    Yes, affirmation and negation are indeed not the same thing, and thus, they are the same thing. You have negated the sameness of affirmation and negation, and since there's no difference between affirming something and negating it, you've affirmed the sameness of affirmation and negation.

     

    1 hour ago, swansont said:

    the other option was "yes, it is" but both can't be true, and they aren't the same

    To this, Bob would answer: "True, and since yes=no, we have (it can't be that both are true) = (it can be that both are true). Moreover, both options are one and the same: (yes = no) = (yes ≠ no)."

    1 hour ago, joigus said:

    If they were, we might as well stop saying 'no' altogether. I don't think you want to do that.

    Bob would say: "Yes, and that means that I do want do that, for wanting to is the same as wanting not to. Furthermore, to stop saying 'no' ist the same as to go on saying it, so of course I want to do that."

    How do you answer Bob?

  16. On 2/16/2021 at 6:14 PM, Lukasz said:

    As I know, such theorem independent on Peano's axioms, is Goodstein Theorem.

    On the other hand, there in the paper: T. J. Stępień, Ł. T. Stępień, "On the Consistency of the Arithmetic System", J. Math. Syst. Sci. 7, No.2, 43-55 (2017); arXiv:1803.11072 ,  a proof of consistency of Arithmetic System was published. This proof had been done within this Arithmetic System.

                                                                                                                                                                                          Łukasz

    Goodstein's Theorem cannot be proven in first-order Peano-arithmetic, but it can be shown in the system of the second-order Dedekind-Peano-axioms (called "DP" in this thread), and this system is what I'm interested in.

  17. 21 hours ago, joigus said:

    But how many of those Anglish speakers are knowledgeable enough to have a meaningful conversation about second-order logic?

    Well, since the set of all humans is finite, there always has to be a first one.

    21 hours ago, joigus said:

    I'm sure a compromise is possible.

    Of course. 👍

    21 hours ago, joigus said:

    Mathematics has nothing to do with empire-building, does it?

    Nope, it doesn't, that's totally right, and that's my point: If we only ever speak or write Anglish for the sake of speaking or writing Anglish, it will never become natural or brookful (useful). A speech whose main goal is itself isn't a very useful speech, is it? So here I am, someone who is far more interested in rimlore (mathematics), flitecraft (witcraft, logic), lore (science), and wisdomlove (philosophy, wisdomlore) than in Anglish, but who nevertheless brooks (uses) Anglish to a reasonable extent.

    But by rights, this ought to be talked about in a speechlore (linguistics) forum rather than one about maths, so let's get back to the orspringly (original) topic.You seem to have been right after all that CH is a proposition of the kind that I was searching for – almost, that is, for the independence of CH in higher-step logic of the Dedekind-Peano-axioms depends on our not finding a new property of the true set-universe which goes beyond ZFC.

  18. 11 hours ago, wtf said:

    Yes, so how did you get from linguistic imperialism to the fine points of the second-order completeness theorem for Henkin models?

    Firstly, I brook (use) right English words in discussions broadly without wanting to talk about that deedsake (fact) in those discussions. I just brought up the topic here because I was asked about my use of proper (or at least more proper) English. Secondly, I don't have much interest in Henkin semantics since I want soothfast (real), full-blown second-order (even better: higher-order) flitecraft.

    11 hours ago, wtf said:

    Thanks for explaining witcraft. Most English-speaking readers probably took that as witchcraft, which made no sense in context.

    You're welcome. Since the reader knows that "witchcraft" wouldn't make much sense in the context, don't they come up with the thought to read again more carefully so as to check what actually stands written there?

     

    11 hours ago, Col Not Colin said:

    Hi again. 

    Hi again, too 👋.

    11 hours ago, Col Not Colin said:

    Where P is the Peano arithemetic system

     

    11 hours ago, Col Not Colin said:

    Additionally I'm not entirely sure what you mean by DP.

    I'm not talking about puny 😉 first-order Peano arithmetic, but about the higher-order Dedekind-Peano axioms (with full-fledged induction axiom):

    On 2/6/2021 at 12:18 PM, Tristan L said:

    [...] the Dedekind-Peano-axiom-system (called “DP” henceforth) [...]

     

    11 hours ago, Col Not Colin said:

    You refer to the Dedekind-Peano-Axiom System which seems to suggest you are extending the Naturals to the entire Real Numbers. 

    No, by "DP", I mean the higher-step (second-order will do) Dedekind-Peano axioms with the full induction axiom.

    11 hours ago, Col Not Colin said:

    For example, informally constructing a statement that just mentions Natural numbers in passing  (like  "this statement cannot be proved from the axioms   AND   2+1 = 3") doesn't seem to be sufficient.  Instead you seem to be putting some addditional requirments on the statement:    Is it that the proposition "this statement cannot be proved from the axioms" requires formal construction within the logical language [...]?

    Yes, for we cannot simply accept any self-referential statement, like "This statement is false", as meaningful. If we can formalize "this statement cannot be proved from the second-order DP axioms   AND   2+1 = 3" in the speech of second-order (or any higher-step) flitecraft (logic), I'll accept it.

    11 hours ago, Col Not Colin said:

    Apologies for length and lack of coherence, I am tired and off to sleep.  Best wishes to you.

    Not at all (regarding the first part), you've given interesting food for thought. Best wishes to you, too!

     

    10 hours ago, joigus said:

    Your point is well taken, @Tristan L, but I think the topic is difficult enough in and of itself that facilitating communication to as wider readership as possible overrides every other need.

    You're right in that one shouldn't overdo it, that is, the freeing of English should be done slowly enough to be comfortable. That's better for both the talks about other stuff like flitecraft (logic) and the liberation undertaking itself (of which I didn't very consciously think in this thread until my style was brought up by others, by the way, since it has become so natural for me to try to speak and write proper English).

    10 hours ago, joigus said:

    I was totally thrown off by "ownship" and "witcraft".

    I didn't mean to do that, which is why I usually include the improper, foreign-derived (unright, outlandish*-drawn) word along with the as-of-yet uncommon proper English one. See e.g.

    On 2/6/2021 at 12:18 PM, Tristan L said:

    To me, axiom-systems seem to basically be ownships (properties).

    and

    On 2/6/2021 at 12:18 PM, Tristan L said:

    [...] and yet we still can derive neither one from AS because our logical (witcrafty) tools are too weak.

    *Remark: I don't seek to replace words on account of being foreign, but on account of being due to speech-imperialism.

     

    10 hours ago, joigus said:

    You can't speak a language that only you understand.

    True, but firsly, I'm far from the only one who tries to free English; there are others who put much more work into it, such as those behind the Anglish Moot, and many (likely most) of them already speak far righer English than I do. Secondly, a start has to be made somewhere, and while right English isn't widely spoken as of yet, it will hopefully become widespread in the forthtide (future).

     

    9 hours ago, StringJunky said:

    If one wants to be understood, best to speak in the language of the listener.

    Indeed, but what if I am speaking in the laguage of the listener, as is soothly (really) the case, yet the listener doesn't truly know their own speech rightly?

    Moreover, I always give the improper-but-still-common equivalents of not-yet-widespread proper English terms, so the listener/reader shouldn't have problems.

     

    X-posted with joigus (just learned the term "x-post" ;)).

  19. 20 hours ago, Col Not Colin said:

    are you more interested in notation than anything else?

    No, it's just a side-hobby of mine to help free speeches broadly, and the English tongue in particular, from the effects of linguistic imperialism, which I do by brooking (using) truly English words instead of ones that came into this language by way of speech-imperialism in actual natural situations, such as forum talks (mutatis mutandis for other victims of language-imperialism). Speech in general interests me, too, but not more that what it is brooked to stand for (which, spellbindingly, includes speech itself).

    20 hours ago, Col Not Colin said:

    What was the witcrafty hyperlink about?

    The website to which the link leads is the English Wordbook. It is part of the Anglish Moot, which is all about liberating English from the effects of linguisic imperialism. The English Wordbook gives the right English equivalents to un-English words that came in through language-imperialism, to a big part, but not only, due to the Norman Conquest. For many proper English words which have not yet become widely used, when I brook them, I link their first instances in a text to the Wordbook so as to back my brooking of those words up, and I also write their foreign-derived equivalents in brackets after them (or vice versa). Brooking truly English words has the futher boot that they are usually shorter and thus more efficient that their foreign counterparts. For instance, "boot" is only a third as long (in terms of syllables) and therefore thrice as efficient as foreign-derived "advantage", and "often" has only two syllables while "frequently" has three.

    "Witcraft" is onefoldly (simply) the English synonym for "logic", the former being drawn from "wit" (from OE "witt", from PIE "*weyd-") and "craft" (from OE "cræft", from PG "*kraftuz"), while the latter is derived from OG "logikḗ" (from PIE "*leǵ-"). Another properly English word for witcraft is "flitecraft", which stems from the Old English word "flītcræft" for witcraft. For more truly English techical language (craftspeak), see the Anglish Moot's leaf on Craftspeak.

    21 hours ago, Col Not Colin said:

    "ownships" used in the OP instead of properties

    Yes, and after my first use of that word in my post, a fayed (added) "(properties)" to clarify for those not yet/anymore familiar with "ownship". This word is the exact English equivalent of the German (Þeech) word "Eigenschaft" for ownship; "own" and "eigen" both come from Ortheedish (Proto-Germanic) "*aiganaz", and "-ship" and "-schaft" both come from Orþeedish "*-skapiz", which is closely related (beteed) to the forebear of English "shape".

    21 hours ago, Col Not Colin said:

    Other recent posts on this forum include one about notation studies (not by Tristan but by another new member, whose background is unknown).

    And who bears no beteeing (relationship) to me (although the flitecrafta/logician would remind us that strictly speaking, everything bears some beteeing to everything else, e.g. the relation of standing to each other in such a way that 1=1) 😉.

    21 hours ago, Col Not Colin said:

    Brief inspection of some other posts Tristan has created or contributed to include one about entropy where an alternative way to patition microstates was presented that seemed to involve runes and the ability to communicate with runes.

    Huh?! That post is about the fact (deedsake) that for any current microstate, you can partition phase space in such a way that the current entropy with regard to that partition (for entropy is always relative to a partition of phase space) is low. Hence, I argue there (and please set me right if I'm wrong), there's almost always something interesting going on (often perhaps even life), though it is likely that the lifeforms of one partition can only detect what is interesting with resprect to their own partition of phase-space. I just used particle-configurations that look like runes as an example; I could just as well have chosen tennis-rackets or Chinese characters or chess-pieces or whatever has structure.

    This shows us how weighty it is to read (or listen) and understand before one judges 😉.

    21 hours ago, Col Not Colin said:

    I can't understand how to turn off tracking cookies and disable other features of that website, so I haven't examined anything in it too deeply.

    It's part of the great platform https://www.fandom.com/, formerly called "Wikia", which has been co-made by the co-maker Jimmy Wales of Wikipedia. So what's the big deal?

     

    20 hours ago, studiot said:

    But I don't see any conspiracy background,

    Those who thought that there was any conspiracy, be careful ⚠️ lest you become conspiracy-theorists.

    20 hours ago, studiot said:

    just a member whose first language is not English struggling with a translator.

    🤣 Judging the canship (ability) or knowledge of another as wanting can result from one's own want of canship or knowledge, as seems to be the case here. Dost ðou even know ðy own speech? For instance, ðou shouldst not address an individual as "you", but as "ðou", for "you" is the plural accusative and dative (many-tale whon-case and whom-case) form of ðe English second personal pronoun, akin to Þeech "euch", while "ðou" is its singular nominative (one-tale who-case) shape, akin to German "du". Ðe sentence "Alice, can you please help me?" translates into Þeech literally as "Adelheid, kann euch mir bitte helfen?", which is so bad German ðat it would not be easy to even understand. Didst ðou know ðat?

    Sadly, the fair English speech has been messed up quite a bit, but we should at least try to tidy it up again.

    However, all that has little to do with the topic at hand, and I'd rather discuss it in a speechlore (linguistics) forum rather that a mathematics one.

  20. 10 hours ago, wtf said:

    I'm not sure what you mean by this.

    Oh, only now do I realize that what I said was ambiguous.

    10 hours ago, wtf said:

    Or is "yes it does" referring to what your question relates to?

    Yes, it does. If I wanted to contradict you regarding second-step logic, I'd have said: "No, it does". The problem is that the English speech doesn't have an equivalent of German "doch", French "si", and Arabic "بَلَى" (and in fact, none of these three European languages has an equivalent of Arabic "كَلَّا").

    So I guess that my question has been half-answered: Since second-order witcraft is incomplete, there are undecidable statements in it, of which CH is an example. However, while we know that second-order logic will always stay incomplete (if only finite proofs are allowed), we might some day find a further axiom that applies to the true set universe with which to decide CH.

    14 hours ago, Col Not Colin said:

    Then, using  ^ as the logical connective for AND,   the statement P ^ Q  would be a new statement that is undecideable and involves natural numbers.

    That's right. I should have been more specific: a statement expressible in the speech of the Dedekind-Peano axioms, th.i. second-step or some higher-step logic.

    14 hours ago, Col Not Colin said:

    What do you mean by a proposition about the naturals?  Does it ONLY have to involve the naturals

    Yes, that's what I mean, though sets of naturals and sets of sets of naturals and so on are also allowed.

  21. Thank you very much for your detailed and informative (informatul) answer! 👍

    18 hours ago, wtf said:

    Ok. But you are conflating syntax (axiom systems) and semantics (models).

    Or rather, I was only talking about semantics. I like to think of an axiom-system as standing for an ownship, and of being a model of that axiom-system as having that ownship.

    18 hours ago, wtf said:

    Two different things.

    Right.

    18 hours ago, wtf said:

    I suspect the OP's questions relate to this fact, that second order logic does not have a completeness theorem.

    Yes, it does.

    18 hours ago, wtf said:

    However the second-order theory of PA is categorical, and perhaps that's what you are referring to as Dedekind's theorem. 

    Yes, and to avoid confusion with the much too weak first-order theory, I called the second-order theory "DP". Dedekind's Isomorphy Theorem indeed shows the categoricity of DP.

    18 hours ago, wtf said:

    This I believe is the crux of your concern, if I'm understanding your post.

    True.

    18 hours ago, wtf said:

    Yes, but note that the real numbers axiomatized as an infinite, complete ordered field is categorical.

    You're right; instead of taking not-categorical first-order ZFC as our basis, we take categorical second-order analysis as our basis, with the true set-theory (about which ZFC doesn't give enough info) as the meta-logical (over-witcrafty) framework of second-step logic. Then CH is indeed a proposition of the kind that I was searching for.

    19 hours ago, wtf said:

    Every categorical theory is complete. However, the converse does not hold

    As I understand it, the word "complete" as brooked (used) here means what I'll call "semantic completeness": every sentence expressible in the speech is either true in all models or untrue in all models, as opposed to what I'll call "syntactic completeness": every sentence expressible in the speech is either derivable from the axioms with the logical calculus or its negation is.

    19 hours ago, wtf said:

    every theorem provable from the axioms is true in every model of those axioms.

    Well, that has to be so if the logical calculus is right, which it should. I think that semantic completeness rather than syntactic completeness is meant, though, which makes sense: If the axiom-system has only one model, every sentence is either true in all models or it's false in all models.

    19 hours ago, wtf said:

    Well that's what I know about it, hope something in here was helpful.

    Most certainly your whole answer is! 😃

    19 hours ago, wtf said:

    You're asking if there are undecidable statements in the categorical theory of second-order PA. This I do not know.

    Yes, and more broadly, I was searching for statements which are semantically decidable in (either true in all models of or untrue in all models of) some axiom-system, but not syntactically decidable in the axiom-system. Am I right in thinking that these are the ones that show true lack of being able to know on our part?

  22. 16 minutes ago, joigus said:

    I think what you mean is something like theorems in number theory, propositions on prime numbers, and the like, right?

    Yes, exactly.

    However, the conjecture that there are odd perfect numbers is not what I seek, for since we know that (if it is true, we can show that it's true), we know that (if it's undecidable, it is false), so if we knew that it is undecidable, we'd know that it's false, making it decidable after all and thus leading to a contradiction.

    I seek a proposition \(A\)

    24 minutes ago, joigus said:

    in number theory, propositions on prime numbers, and the like

    such that we know that we can't know whether \(A\) is true or false.

  23. 1 hour ago, joigus said:

    Cohen's proof that it is undecidable whether there is a cardinality (number of elements) intermediate between that of N and that of R .

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

    Yes, CH is undecidable in / independent of ZFC, but that's because ZFC has several not-isomorphic models:

    4 hours ago, Tristan L said:

    One way is that AS has several non-isomorphic models of which some have P and others don’t. For instance, being unendly (infinite) isn’t decidable from the field-axioms since there are finite (endly) fields as well as unendly ones. This underkind of undecidability is, I think, obviously not very interesting. The Continuum-Hypothesis (CH) is one example, for it’s true in some models of ZFC and false in others. Here, the lack of our knowledge stems from the fact (deedsake) that ZFC doesn’t contain all information about the set-universe to start with.

    I'm asking whether there is a proposition which is known to be independent of an axiom-system with only one model (up to isomorphy), such as the Dedekind-Peano-axioms.

  24. To me, axiom-systems seem to basically be ownships (properties). For instance, the group-axiom-system is basically the ownship of being an ordered pair \((G, *)\) such that \(G\) is a set and \(*\) is a function from \(G\times G\) to \(G\) such that \(*\) is associative and has an identity element and each member of \(G\) has an inverse element with regard to \(*\).

    Just as the axiom-system itself is an ownship, so are what are called “propositions in the language/speech of the system” actually properties. For instance, when we say: “The proposition that the sum of the inner angles of a triangle is always 180° follows from the Euclidean axioms“, we actually mean that for every structure E, if E has the Euclidean axiom-system as a property, then E has the property that every triangle in it has an inner-angle-sum of 180°.

    Some axiom-systems, such as the group-axiom-system and the field-axiom-system, are had by several structures of which not all are isomorphic to each other. In other words, such axiom-systems have at least two models which aren’t isomorphic to each other. Let’s call these axiom-systems “not-characterizing”. Others, such as the Dedekind-Peano-axiom-system (called “DP” henceforth) , are only had by one structure up to isomorphy – they have a model, and all their models are isomorphic to each other. Let’s call these axiom-systems “characterizing”. The only model of DP up to isomorphy (and indeed up to unique isomorphy), th.i. the only entity, up to (unique) isomorphy, which has DP as a property, is the structure of the natural numbers. The German mathematician Richard Dedekind showed this in 1888 with his Theorem of Isomorphy.

    Now, there are two ways in which a ‘proposition’ \(P\) (in reality a property, see above) can be undecidable (neither provable nor disprovable) in an axiom-system \(AS\). One way is that \(AS\) has several non-isomorphic models of which some have \(P\) and others don’t. For instance, being unendly (infinite) isn’t decidable from the field-axioms since there are finite (endly) fields as well as unendly ones. This underkind of undecidability is, I think, obviously not very interesting. The Continuum-Hypothesis (CH) is one example, for it’s true in some models of ZFC and false in others. Here, the lack of our knowledge stems from the fact (deedsake) that ZFC doesn’t contain all information about the set-universe to start with.

    The second way in which \(P\)  can be undecidable in \(AS\) is that it either holds in all models of \(AS\) or its negation holds in all models of \(AS\), and yet we still can derive neither one from \(AS\) because our logical (witcrafty) tools are too weak. This reflects a true unableness on our part to get info out of \(AS\) which nevertheless is there. This is always the case if \(AS\) is characterizing (has only one model up to isomorphy). Such an axiom-system contains all info about its model, so undecidability in it means inability to get info which is nonetheless there. So while undecidableness in ZFC need not be interesting, undecidableness in DP always is. Since characterizing axiom-systems have just one model up to isomorphy, what we call “propositions in their speech” – in truth ownships – can actually be regarded as propositions, namely the propositions resulting from predicating those properties of the system’s unique model.

    Now, from Gödel’s Incompleteness Theorems, we know that there are undecidable propositions in DP, that is, not every statement about the naturals can be shown or disproven. However, my question is this: Is there an individual proposition about the naturals of which we know that we can neither prove it nor disprove it in DP?

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