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Tristan L

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  1. 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.
  2. 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
  3. 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 actuall
  4. 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 par
  5. 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 alwa
  6. 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 categoric
  7. 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.
  8. 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.
  9. 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
  10. However, the shared forebear of epidermal scales, feathers and hair need not itself have been a scale.
  11. 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 rep
  12. Right. It would indeed need a stretch of the imagination to see how lifeforms from the last aeon could get info into our aeon in a way that brings about living things. I find the hypothesis that life arose much more convincing. Then again, I just found out that my idea isn't new: it's the idea of information panspermia. Still, I find it very speculative and think the arising of life from scratch to be much likelier. But there are some very good solutions to the ILP which show that info isn't destroyed after all, aren't there? I firmly believe in the indestructibility of info (t
  13. Yes, those are some good points for life having arisen rather than always been there. But the properties of the CMB depend in part on those of the Big Bang, which in turn might be influenced by a previous aeon, I think. Is that right? For instance, see here for the possibility of the CMB containing info from an aeon before ours. As far as I know, the genetic code is very similar across all Earthly living beings and hasn’t changed much over billions of years. I don’t mean individual genomes or sets of genomes, but rather the genetic speech. Could our genetic speec
  14. Good points. I also think that life did arise. However, I think that we shouldn’t take this as a given fact without questioning it. Arguments such as yours are one of the things that I was searching for. Has the question of whether life did arise in the first place been discussed in the scientific literature? But what if those beings influenced life in our cycle more subtly? For example, maybe they knew that amino-acids and nucleotides would arise naturally, so they rather wrote the genetic code and sent it to our cycle’s raw materials. I don’t really believe
  15. Many scientists, philosophers, and people of religion have sought to find the answer to the question: How did life arise? However, isn’t that question loaded? Shouldn’t we first ask: Did life arise? Mightn’t it be the case that the Universe is everlasting, and that life has always existed in this Universe? For instance, if Roger Penrose’s conformal cyclic cosmology is true, then isn’t it possible that intelligent living beings in each aeon manage to seed the next aeon with life, perhaps with signals of some sort? Something similar could be asked about other cyclical models.
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