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Tristan L

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  1. 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?
  2. 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: 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. 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.
  3. 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. In truth, this statement of yours is simply incorrect. 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. I brook/use the words "element" and "member" in exactly one and the same meaning. Oh, I thank you for the info, but my friend's 13 year old cousin already told me today morning. 😉 {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. 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.
  5. 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?
  6. @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.
  7. 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?
  8. 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?
  9. 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." I'm obviously not merely concered with first-order flitecraft, but with all-order-flitecraft and beyond. No, Excluded Middle actually has little to do with my point. Can you give me one example/byspel where Excluded Middle fails?
  10. 🤣👍 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." It's not alone there; Latin also doesn't have exact equivalents of "yes" and "no" afaik. 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. 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. Alice: "I fully forewyrd/agree with you ..." Bob: "... as do I." On the point! 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."
  11. 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. 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." 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. 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. 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?"
  12. 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. 😀
  13. 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: 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)." 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?
  14. The question is already in the title: Is affirmation the same as negation?
  15. 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.
  16. Well, since the set of all humans is finite, there always has to be a first one. Of course. 👍 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.
  17. 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. 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? Hi again, too 👋. I'm not talking about puny 😉 first-order Peano arithmetic, but about the higher-order Dedekind-Peano axioms (with full-fledged induction axiom): No, by "DP", I mean the higher-step (second-order will do) Dedekind-Peano axioms with the full induction axiom. 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. Not at all (regarding the first part), you've given interesting food for thought. Best wishes to you, too! 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). 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. and *Remark: I don't seek to replace words on account of being foreign, but on account of being due to speech-imperialism. 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). 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" ).
  18. 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). 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. 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". 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) 😉. 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 😉. 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? Those who thought that there was any conspiracy, be careful ⚠️ lest you become conspiracy-theorists. 🤣 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.
  19. Oh, only now do I realize that what I said was ambiguous. 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. 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. Yes, that's what I mean, though sets of naturals and sets of sets of naturals and so on are also allowed.
  20. Thank you very much for your detailed and informative (informatul) answer! 👍 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. Right. Yes, it does. 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. True. 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. 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. 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. Most certainly your whole answer is! 😃 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?
  21. 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 \(A\) such that we know that we can't know whether \(A\) is true or false.
  22. Yes, CH is undecidable in / independent of ZFC, but that's because ZFC has several not-isomorphic models: 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.
  23. 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?
  24. However, the shared forebear of epidermal scales, feathers and hair need not itself have been a scale.
  25. I’ve read some claims that the synapsids (theropsids) had smooth, glandular skin rather than scaly skin like modern reptiles – Estemmenosuchus is often cited in this regard –, and that the scale-like structures that they did have, their belly-scales, aren’t homologous to lepidosaur scales. While the latter claim may be the case, I do have qualms with the former claim. After all, this study (see https://www.sciencedaily.com/releases/2016/06/160624154658.htm for an overview) has proven that modern reptiles have anatomical placodes just as mammals and birds do, and that mammalian hair, modern reptile scales, and bird feathers are all homologous to each other and come from the skin appendages of the last shared forebear of mammals, modern reptiles, and birds. Estemmenosuchus is a derived synapsid (namley a therapsid), so its lack of scales may very well be secondary, right? Moreover, the lepidosaurian-like scales of the varanopid Ascendonanus (see https://www.researchgate.net/publication/323782950_First_arboreal_%27pelycosaurs%27_Synapsida_Varanopidae_from_the_early_Permian_Chemnitz_Fossil_Lagerstatte_SE_Germany_with_a_review_of_varanopid_phylogeny) show that varanopids had reptilian scales, and even lepidosaur-like ones. Of course, this isn’t necessarily evidence for synapsid scales since varanopids might be bird-line amniotes rather than mammal-line amniotes (though this suggestion has been contested and argued for and against). Still, doesn’t the homology of hair, scales, and feathers strongly indicate that the last common forebear of mammals, modern reptiles, and birds had scaly skin, and that early synapsids thus were scaly, too, in a way homologous to modern reptiles? If so, am I right in rejecting such claims as the one that early synapsids looked like naked lizards, or that the skin of our early ancestors was frog-like rather than reptile-like? The latter claim is false anyway since even if they didn’t have scales, their skin would still have been suited to a dry environment like reptiles’ and unlike frogs’, right? Is there other evidence for (or against) scaly skin being the plesiomorphic condition of all crown-group amniotes?
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