Timeline for Can Mereology be bi-interpretable with Set Theory, in absence of the bottom object?
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25 events
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| Jan 21 at 15:59 | comment | added | Zuhair Al-Johar | Would this argument in a previous answer of yours would apply to destroy bi-interpretability between the two kinds of labeled Mereology. | |
| Jan 20 at 22:55 | comment | added | Zuhair Al-Johar | On further thought, actually the matter is more tricky than just blurring all empty objects. If we define membership as in the main post, then we'll have non-empty Ur-elements, and so we need to blur the distinction between any two sets having isomorphic membership graphs on their transitive closures, I think this will let us get rid of the copying destroying Extensionality in Mereology without bottom, and my enable bi-interpretability with "Mereology with bottom". Of course by Mereology I mean Mereology + singleton, or what I call "labeled Mereology" | |
| Jan 20 at 19:36 | comment | added | Zuhair Al-Johar | still you need to take care of the labeling function and adjust for the Ur-elements. I need to think of it thoroughly and see for myself. | |
| Jan 20 at 13:40 | comment | added | Joel David Hamkins | We don't need a pairing function, since on the usual meaning of the term, interpretations work with (uncoded) pairs directly---like how we interpret ℂ in ℝ. To interpret a theory, you define a class of k-tuples and an equivalence relation and use this as the domain of the interpreted structure. In the converse direction, when we have the bottom element, the domain is simply everything else. That directly is easy. It's like interpreting ℕ in ℤ. (And yes, your correction is right, I should have said $a\subseteq a'$ not $a\subseteq b$.) | |
| Jan 20 at 11:27 | comment | added | Zuhair Al-Johar | I don't know if we can put matters this way: $(a,a) \ P' \ (b,b) \iff a \ P \ b$, and $(a,b) \ P' \ (c,d)$ if $a \neq b$, and if $c \neq d$ then $(a,a) \ P' \ (c,d)$ if and only if $a$ is an Ur-element. As regards the new equality relation we need to blur distinctions between Ur-elements and $(a,b)$ pairs when $a\neq b$. | |
| Jan 20 at 10:59 | comment | added | Zuhair Al-Johar | just a corrective note, I think in your comment you wanted to say $a \subseteq b'$. [you wrote $a \subseteq b$]. | |
| Jan 20 at 8:45 | comment | added | Zuhair Al-Johar | cont..., by then you'll need to manage the failure of Extensionality in the old system so that it becomes absent in the new system, so $a \to (a,a)$ won't be enough for that interpretation. Possibly you'll need to send all Ur-elements in the old system (the one without bottom) to bottom in the new system, and adjust matters accordingly. I think we may be able to manage it, but I'm not sure, in the particular system $\sf M-Bottom$ you have a unique non-labeling atom, and so failure of Extensionality is produced in a controllable manner, so I think we may be able to work the bi-interpretation. | |
| Jan 20 at 8:35 | comment | added | Zuhair Al-Johar | cont...; there are some conundrums, for example most of the time when it said Mereology without Bottom, it is meant a system that proves the non-existence of a Bottom object. This means that your new part-hood and equality relations on the pairs, won't really interpret the older system, since you have a bottom for $P'$ but you negate that for $P$ in the original system. Not only that you need to keep the labeling function over the pairs, so you need to send the object representing the empty set in the original system to bottom in the new system. to be continued... | |
| Jan 20 at 8:24 | comment | added | Zuhair Al-Johar | Ok, so your interpretation is to work on ordered pairs $(a,b)$ and define a new part-hood relation $P'$ over those pairs to the effect that $(a,a) \ P' \ (b,b) \iff a \ P \ b$, and $(a,b) \ P' \ (c,d)$ whenever $a \neq b$, also you want to define a new equality relation over those pairs that blurs the distinction between all $(a,b)$ pairs if and only if $a \neq b$. There is a problem, ordered pairing is not definable in Mereology. However, it's definable in Labeled Mereology (i.e. Mereology + Singleton), but in the latter you need to preserve the singleton function as well. to be continued.. | |
| Jan 19 at 19:49 | comment | added | Zuhair Al-Johar | Well, as I said I'm too tired now to fathom that. But, If you are firm on that then this is the answer to this question, which is to the positive both ways. Or at least in the milieu of set theoretic mereology you were speaking about (and I think it generalizes to the system I've presented if I got it right). | |
| Jan 19 at 19:46 | comment | added | Joel David Hamkins | No, because I am using pairs and a quotient by an equivalence relation (like interpreting $\mathbb{Q}$ in $\mathbb{Z}$), whereas synonymy would mean that you don't use pairs or an equivalence relation. | |
| Jan 19 at 19:43 | comment | added | Zuhair Al-Johar | I'm too tired now to fathom that, I'll try to fathom it tomorrow after I have a good sleep. But for instance would that bi-interpretability achieve synonymy also? | |
| Jan 19 at 19:38 | comment | added | Joel David Hamkins | The existence of this bi-interpretation is why I find the big fuss about $\emptyset$ that one finds in some mereological contexts to be silly. Also silly: to fuss about parthood vs. proper parthood, since these also are bi-interpretable. | |
| Jan 19 at 19:36 | comment | added | Joel David Hamkins | Given a model of mereology with no $\emptyset$, I define $(a,b)\equiv (a',b')$ if either $a=b=a'=b'$ or $a\neq b$ and $a'\neq b'$. Define $(a,b)\subseteq(a',b')$ if either $a=b$ and $a'=b'$ and $a\subseteq b$, or $a\neq b$. Map $a$ to $(a,a)$ for the interpretation in the one direction. In the other direction, simply forget about $\emptyset$. | |
| Jan 19 at 19:33 | comment | added | Zuhair Al-Johar | forgive my thick skull, but what do you mean exactly, how the bi-interpretability goes? | |
| Jan 19 at 19:31 | comment | added | Joel David Hamkins | The new bottom element (if there wasn't one before). That is, in a mereological model without $\emptyset$, I can interpret a model which is just the same, except that now $\emptyset$ does exist. And this is a bi-interpretation. | |
| Jan 19 at 19:31 | comment | added | Zuhair Al-Johar | when you say (a,b) represent the new element, which new element you mean? | |
| Jan 19 at 19:25 | comment | added | Joel David Hamkins | The existence of a bottom element or not cannot be important for bi-interpretation, since mereology without a bottom element is bi-interpretable with mereology with a bottom element. Interpret with pairs, where (a,a) represents a, and (a,b) represents the new element when a,b distinct, and define parthood on pairs to reflect this intention. | |
| Jan 19 at 18:53 | comment | added | Zuhair Al-Johar | Thanks! But, I'm not speaking about set theory without $\emptyset$ . I'm speaking about bi-interpretability between Mereology without Bottom (which is more natural mereologically speaking) [like the theory M-Bottom], and Set theory meaning ZFC (or MK-Found.-C), which of course has $\emptyset$ and doesn't have any Ur-elements. Notice that $\emptyset$ need not be captured as a bottom object in Mereology, for example in Lewis system we have $\emptyset$ interpreted as the mereological totality of all non-labeling atoms. | |
| Jan 19 at 13:39 | comment | added | Joel David Hamkins | Sorry, I'm just not that interested in set theory without $\emptyset$. If you want urelements, that is a separate issue, and one must add a predicate to distinguish them from the empty set, but this would happen whether you are doing mereology or element-based set theory. | |
| Jan 19 at 8:37 | comment | added | Zuhair Al-Johar | The point is that the bottom axiom is necessary for having a natural bi-interpretability between Mereology and Set theory, and it becomes more essential if we further demand synonymy. | |
| Jan 19 at 8:27 | comment | added | Zuhair Al-Johar | I personally think the answer to the first question is to the positive. But, it is the next question that makes me wonder. Taking away bottom would almost kill Extensionality, and even then the pattern of the emerging Ur-elements might not conform with simple removal of Extensionality from ZFC, and so kills bi-interpretability. This would mean that the most natural systems of Mereology even those augmented with the singleton function, since they lack the bottom object then they won't naturally lead themselves into bi-interpretability with Set theory. Only the bottom based Mereologies can! | |
| Jan 19 at 8:03 | comment | added | Zuhair Al-Johar | Are they synonymous? If we don't have a bottom object, can they be bi-interpretable? Note that we agree on using Lewis interpretations of part-hood and the singleton (labeling here) function. Your background set theory is ZFC, the axiomatic system I posed is in relation to MK-Foundation-Choice. However, still examining these questions in YOUR theory may shed a light on how they'd fair here. So, to put matters in your milieu: is ZFC synonymous with set theoretic mereology(+singleton). Next question: is ZFC bi-interpretable with a set theoretic mereology (+singleton) that lacks bottom! | |
| Jan 19 at 1:36 | history | edited | Joel David Hamkins | CC BY-SA 4.0 |
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| Jan 19 at 1:13 | history | answered | Joel David Hamkins | CC BY-SA 4.0 |