Let existence be defined as appearing within a domain. — auto to on

Per the definition of "world" above that, it seems like existence of the world (qua the world) should be defined as appearing as the domain (that contains all domains). Not appearing within a domain. The existence of everything but the world itself could use the term "within."

[...] the world (qua the world) should be defined as appearing as the domain (that contains all domains). Not appearing within a domain.

Thank you for responding. Two points:

1) This move could be considered both ad hoc and insufficient. The former because this would be constructing a unique definition of existence for the world, in order to avoid the consequence of the world's non-existence or any of the conceptions that are consistent with the previous definition of existence. If all other entities has some other definition of existence and these two definitions cannot be reduced to a more general, why use the same phrase? All entities could be existing and the world could be mehxisting. The latter because the worlds self-containment follows from the definition of the world, and we would still have to give an account of this no matter whether we term this relationship existence or not.

2) I don't seek to reject the definitions made for this conception of the world, i simply seek to reject this conception. There might be other conceptions that are also consistent with these definitions, but it is only this particular conception I seek a refutation for.

[...] the world (qua the world) should be defined as appearing as the domain (that contains all domains). Not appearing within a domain

.

Thank you for responding. Two points:

(1) This could be considered both ad hoc and insufficient. The former because you define the existence of the world in a unique way, disjunct from existence as it applies to other entities. If these two definitions of existence are not reducible to a more general definition, why should they both be called the same? The latter because the self-containment of the world follows from the definition of the world, and we must give an account of this irregardless of how we define existence.
(2) I do not seek to reject the definitions of existence and the world, I merely seek a refutation of this particular conception of the world. Other conceptions may be consistent with these definitions.

If these two definitions of existence are not reducible to a more general definition, why should they both be called the same? — auto to on

You mean why say that both are "existence"? It doesn't really matter what word you use. But why would there be a requirement to have the same arbitrary definition you chose for everything? Where is that requirement coming from?

In other words, you're creating a problem because of an arbitrary definition you're chosing. So why choose that definition?

In other words, you're creating a problem because of an arbitrary definition you're chosing. So why choose that definition? — Terrapin Station

There results no problem from the definition of existence, as I said;

the self-containment of the world follows from the definition of the world, and we must give an account of this irregardless of how we define existence. — auto to on

Defining existence in this way only frames the self-containment of the world as a claim of existence. But this particular conception of the world seeks to establish the self-containment of the world as a non-contradictory claim. The definition of the world and the definition of existence are, indeed, unfounded in this context, but they are not the object for consideration. Indeed other conceptions of the world which remain consistent with these definitions are possible, the only object for consideration is this particular conception of the world. You cannot refute a particular conception simply by redefining it, such a move must first become motivated by diagnosing flaws within the conception.

There results no problem from the definition of existence — auto to on

The whole dilemma you're proposing here is due to the defintion you're using.

First note that:

'Let the world (w) be defined as: the domain that contains all domains.'

is not a definition. The use of the definite article at the beginning is itself an assertion of the existence of at least one entity that satisfies the condition that is the rest of the sentence.

A proper definition would be:

'A "world" is a domain that contains all domains.'

You then need to define domain, and then try to prove that there exists at least one domain that is also a "world".

Your proof uses the word "world" throughout, for multiple entities, where it appears you should be using domain. We can't be sure whether your result is valid until you fix up your definitions and the words used in the proof. But if the proof is invalid, it will probably be related to using the word "world" instead of "domain", and thereby obscuring missteps in the proof.

It may help to know that it was shown in the early 20th century that any set theory that allows there to be a 'set of all sets' is inconsistent. So if your 'domains' are like sets, your theorem will probably be doomed to the same fate (that's one reason why it's crucial to define "domain").

I want to state that I agree with critisism raised by other replies. Especially that the definitions are causing the problem.

However I have further criticism that haven't been voiced yet.

Let the world (w) be defined as: the domain that contains all domains.
Let existence be defined as appearing within a domain. — auto to on

1) "The world" seems to imply that there is only one world. Therefore the introducing of multiple worlds (w1,..., wn) is unjustified since they contradict the first definition.

2)The definition of existence presupposes the existence of a domain in which elements can appear in. Therefore if we claim the world exists we either presuppose a different kind of existence (existence2) of the world in order to ascribe existence1 (existence as used in the definition) to the world. Or we get a circular reasoning pattern where the existence of the world is a prerequist for existence, and existence is required for the existence of the world.

Furthermore if we claim that the world exists we don't get w2 that is within w1 since we only ascribe existence to w2. Therefore w1 would be the worlds pseudoexistence (existence2) and w2 would be the world with ascribed existence1. However they both refer to the same world w.

I know that existence is not a property and rather a quantifier but since you use existence rather confusingly I'll describe it as if existence was a property. If we have an object O and we ascribe two different properties to O we don't get O1 and O2 and rather O with properties P1 and P2.

Therefore the confusion most likley arises from the questionable use of the term existence and it's two forms that are not distinguished properly. If we see this as faulty or insufficient description we get a faulty tautology because of the definitions used.

This means that any other deduction based on this definition is flawed. Therefore viewing any other transformation in this tautology as consistent because of conclusions that might be true is a logical error. (From False everything follows)
This applies to your response stating:

Other conceptions may be consistent with these definitions — auto to on

3) First of let's note that S1,...,Sn just appear randomly and are not defined and irrelevant to the problem. Therefore we can leave them aside.
Note: I am aware that you think theres a problem if S1 appears alongside w1 but also within w1 however theres no definition given that states something among the lines of "if something appears alongside x this something can not be within x". You generally seem to establish rules that are not given by the definitions. Another example would be something containing itself creates a new instance of itself. This is needed to even argue that you get w1 and w2. (It's like assuming that if I say I am my best friend that you assume theres two of me, since you have in your head that the best friend is a different object despite no rule stating that.)

Since you somehow want to hold on to your definition for no clear reason lets adress that aswell.

Let us therefore assume that the world (w1) [...] is contained within world (w2) [...]. Now we are met with the same difficulty, if the world properly contains itself then this should obtain (w1) = (w2). [...] but it[(w1)] must also contain itself, let us call the world contained within (w1) for (w0). — auto to on

What you are doing here is create a problem where there is none. Since w1=w2 and w2 contains w1, w1 contains w1. Or in other words w1 contains itself.
We can show this aswell if we accept that w0 is the world contained in w1 and show that w0 = w2.

This can be done intuitivly by saying that w0 is the domain that contains all domains and that the same is true for w2 and that there can only be one domain of all domains.
We could also state the reason given in 1) and conclude that you are just making up new names for the world and think this is a logical problem.

However lets assume the premisses you defined as trivially given for w2 (our initially defined world).

Since (w) in (wn) refers to world and world is defined as domain of all domains (wn) trivially is a domain (d).
Further every (w) in (wn) refers to world and world is defined as the domain of all domains (D)
Since every domain (d) is contained(<=) in the domain of all domains (D)
It follows that (d) w2 <= w0 (D)
Since you have already established that w0<=w1<=w2 and therefore w0<=w2
we get w0=w2 or more general wj=wi where i and j are arbitrary numbers. We would also get w1=w1 or in other words w1 contains itself.

In conclusion your set of definitions is faulty or at best incomplete. Even if we ignore the flaws the problem you make up doesn't exist since you just rename a single instance and think the ability to establish an infimit ammount of names equates to a infinit regression since you presume that ascribing a new name creates a new object.

I hope I understood you correctly and that this helps solving your problem.^^

It may help to know that it was shown in the early 20th century that any set theory that allows there to be a 'set of all sets' is inconsistent. So if your 'domains' are like sets, your theorem will probably be doomed to the same fate (that's one reason why it's crucial to define "domain"). — andrewk

Yes, this is why the ZFC set theory - which since its introduction as a replacement for the naive set theory in the early 20th century has become a standard axiomatization of sets - includes the Axiom of Regularity, which implies that no set is an element of itself.

Thank you for your comments; I will respond in three sections, addressing each of you to the best of my abilities. I have been away the past few days, so i apologize for the late answer.

A general note before proceeding: framing the claim as that of existence may have been confusing. The claim I intended was this particular notion of self-containment, and the claim to existence was merely used as framing device. I do not intend the OP as a proof for the existence of the world, but merely a hypothetical, that if we claim the world exists this could be a way of conceiving the self-containment. I merely want to investigate the possibility of the world, and in particular whether there are some reasons for rejecting this particular conception of this possibility.

(A)

The whole dilemma you're proposing here is due to the defintion you're using. — Terrapin Station

I perhaps see why you might think this if you think it's a claim of existence. But the conception does not result from the definitions of existence, since the definition of the world is what requires self-containment. If the world is the domain of all domains, it should be a domain within itself. The definition of existence only functions as a specific way of framing this self-containment, since it denominates the appearance within a domain as existence. With this additional definition we can then say that the world's existence follows from it's definition.

(B1)

The use of the definite article at the beginning is itself an assertion of the existence of at least one entity that satisfies the condition that is the rest of the sentence. — andrewk

You are correct, I'll then rephrase the definition in conformity with your definition:

'A "world" is a domain that contains all domains.' — andrewk

(It should be noted here, that there are two conventions at play. The definite article could also be interpreted, not as an assertion of existence, but the assertion that this definition only allows one of these entities. CaZaNOx also noticed this in the (1) section of his comment)

(B2)

Your proof uses the word "world" throughout, for multiple entities, where it appears you should be using domain. We can't be sure whether your result is valid until you fix up your definitions and the words used in the proof. — andrewk

Using the term 'domain' would obscure the intention to establish the self-containment of totality. Ex. 'Now we claim that a domain exists and it then follows that this domain shall appear within itself'. This point would be invalid by only arguing from the notion of a domain. The word 'world' does not refer to multiple entities in the conception, but the same entity. The infinite nesting is a result of securing the identity of the world throughout (w1) (w2) ... (wn) so (w1)=(wn), and therefore the world does not refer to multiple entities.

But perhaps I am misunderstanding you on this point?

(B3)

You then need to define domain, and then try to prove that there exists at least one domain that is also a "world". — andrewk

As I stated in the general note, I do not attempt to prove the existence of the world, but merely that if we claim that the world exists then there exists a consistent conception of the world. If it helps you I could tentatively define domain as an object that contains other objects and specifies the conditions of their individuation. It should be obvious that this is not equivalent to a set, and therefore set-theoretical considerations do not apply.

(C1)

The definition of existence presupposes the existence of a domain in which elements can appear in. Therefore if we claim the world exists we either presuppose a different kind of existence (existence2) of the world in order to ascribe existence1 (existence as used in the definition) to the world. Or we get a circular reasoning pattern where the existence of the world is a prerequist for existence, and existence is required for the existence of the world. — CaZaNOx

This is interesting and i'm glad you brought it up; if we require that something must exist independently of what appears within it, in order for something to appear in it, the existence of the world is impossible. The world can only appear in itself, and therefore can only exist if it contains itself. But with this requirement the world must exist in order to contain itself, and it must contain itself in order to exist, so it cannot exist. I think the conceptual interpretation of this bi-conditional is simply one of identity. Thus if we establish the possibility of the self-containment of the world, we thereby also establish the possibility of existence. In this particular respect appearance within the domain is equivalent to the existence of the domain.

Now this particular problem of circular reasoning arises from the conjunction of incompatible principles. The general principles of explanation applied here are PSR, i.e. for any fact (f) there is some fact (f') that explains why (f) obtains, and the principle of non-circularity (NC), i.e. there is no (f) that can satisfactorily explain itself. Given the definition of existence, we might claim that (E) the domain within which an object exists (at least partially) explains (the possibility of) the existence of the object. The specific aspects of the application of these principles in this case is (PSR-E) that for any existing object the domain within which it appears (partially) explains (the possibility of) why it exists and (NC-E) there is no object that is identical to the domain that (partially) explains (the possibility of) it's own existence. Now given the definition of the world in the conjunction with (PSR-E) the world has a particular form of explanatory comprehensiveness (EC), i.e. there no object which existence (or possibility thereof) is not (partially) explained by the world. (EC) of course applies to the world and therefore violates (NC-E). Furthermore, per definition there cannot be any other domain that performs the function of (E) for the world. Thus we must reject the existence of the world, reject the principle of non-circularity, revise our notions of existence or perhaps give another account of explanatory inferences.

I think introducing the notion of existence2 in order to explain away this problem is not very parsimonious and only solves the specific version of this problem and not the more general problem that arises from the conjunction of non-circularity and and an ultimate theory with explanatory finality and comprehensiveness. Rejecting any notion of the world and/or any ultimate theory is an option, but i think the motivation rests on a too simplistic notion of how explanatory inferences function. Now discussing specific solutions on how we may revise our explanatory principles here, would take us too far off from the discussion at hand and into different epistemological theories. I think the most general remark, that we could assume for the present discussion, is that different modes of explanation function are used at different levels of discussion. When discussing theories of totality a subsumptive interpretation of 'explanation' is inappropriate, instead we must understand 'explanation' as the systematic coordination and coherence of it's parts. Thus the possibility of the world is not ruled out by (NC) but only by the internal incoherence of the conception, and what i believe threatens this coherence is the requirement of self-containment in conjunction with totality.

(C2)

know that existence is not a property and rather a quantifier but since you use existence rather confusingly I'll describe it as if existence was a property. — CaZaNOx

I assume that existence is a property, and not a quantifier. I have serious doubts that we can equate the concept of existence with the notion of the existential quantifier. It doesn't seem that being quantifiable is a prequisite for existing, and it excludes different formal theories due to the ontological commitment of the quantifier. But let us not diverge too much from the discussion at hand.

(C3)

If we have an object O and we ascribe two different properties to O we don't get O1 and O2 and rather O with properties P1 and P2. — CaZaNOx

Yes this is correct. The infinite nesting this conception requires was motivated from assuming that a specific argument applies to the self-containment of the world. This is an argument that utilizes the principle of the indiscernibility of identicals (IID) in the following way. If (w2) contains (w1) then they are distinct because (w2) contains one more object (w1) than (w1) and (w1) is contained in one more object (w2) than (w2). Assuming (IID) and having established that (w1) and (w2) have different properties, we can per modus tollens conclude that (w1)=/=(w2). Thus the world does not contain itself. In order to mend this we assume that (w2) is contained by (w3) ... (wn) and that (w1) contains (w0) ... (wn) thus making them indiscernible. But the second premise of this modus tollens is problematic because it assumes from the outset that (w1) and (w2) are different entities and not merely different predicates applying to the same entity. This is what you also observed here:

What you are doing here is create a problem where there is none. Since w1=w2 and w2 contains w1, w1 contains w1. Or in other words w1 contains itself. — CaZaNOx

But even if this argument is faulty, and we therefore can construct another theory of self-containment, that does not require such infinite nesting (and hence why i hinted that other conceptions might be consistent with the two definitions), I would still like to see other arguments for rejecting this specific theory. This undermines one motivation for the theory, but I was also wondering whether there are other unfavorable consequences for the theory and, perhaps most importantly, internal incoherence in the notion of such infinite nesting.

(C4)

Since (w) in (wn) refers to world and world is defined as domain of all domains (wn) trivially is a domain (d).
Further every (w) in (wn) refers to world and world is defined as the domain of all domains (D)
Since every domain (d) is contained(<=) in the domain of all domains (D)
It follows that (d) w2 <= w0 (D)
Since you have already established that w0<=w1<=w2 and therefore w0<=w2
we get w0=w2 or more general wj=wi where i and j are arbitrary numbers. We would also get w1=w1 or in other words w1 contains itself. — CaZaNOx

I think this (in conjunction with (C3)) is the core of the problem. How do we establish the conditions of identity for the world? How can we be sure that the relation (<=) does not alter the identity of the world? The conclusion wj=wi only holds if we know for sure that (<=) does not alter the identity of the world, so as to make it something else. Now the argument from IID tries to make the case that (<=) alters the identity of the world, by assuming the world as appearing within itself (w2) must necessarily have different properties than the world within which it appears (w3). But by arguing this point it must assume that (w2) is the (purported) world as an entity and not merely a predicate that applies to the world. This then argues that (w2) has x objects contained within it, and is contained by domain y and (w3) has more than x objects contained within it and is not contained by domain y. Thus (<=) alters the identity of the world. On the other hand the counterargument against this is that (w2) with it's different properties and (w3) with its different properties are both properties that apply to x, so the world = x = (w2) ^ (w3). Now I believe that there is some strength to the argument that in this case the world does not properly contain itself, since if x must appear within x, it cannot be one of it's properties that appear within it, but x itself. But then the same argument applies and now the world y = x1 ^ x2 = ((w2) ^ (w3)) ^ ((w2) ^ (w3)) and so on ad infinitum. Thus we may solve the original discrimination between (w2) and (w3) by assuming infinite nesting so each case of (wn) has the same number of domains contained, and is contained by the same number of domains.

What I find most interesting here is your considerations that tie into objection 3. You say:

you just rename a single instance and think the ability to establish an infimit ammount of names equates to a infinit regression since you presume that ascribing a new name creates a new object. — CaZaNOx

If an infinite regress is necessary for the establishing of the identity of the world, but the identity of the world contradicts an infinite regress, this conception has a serious problem. This is related to the claim in objection 3:

It seems like a prerequisite for the notion of the world’s infinite nesting to be meaningful that there must be at least some relative differentiation between the different iterations. — auto to on

And answer 1:

It is a reiteration of the same, and insofar as that which is reiterated is identical to antecedent or consequent iterations, these iterations cannot distinguish a multiplicity of entities. — auto to on

How can we then say there is an actual infinite regression in this conception of the world? The only differentiation is established in the ascribing of different names for the same entity, but this only happens in this case due to the sequential nature of thought processes. But then the world cannot contain itself, since the prerequisite for self-containment is that an infinite regress is inherent to the conception, and the prerequisite for infinite regress is a differentiation between the iterations, and this differentiation is only established in the ascribing of different names (and these names are not inherent differences). How would this tie into notions of actual infinities in mathematics?

I'd like to hear your thoughts on this.

ADDENDUM:

(C5)

Therefore the confusion most likley arises from the questionable use of the term existence and it's two forms that are not distinguished properly. [...] This means that any other deduction based on this definition is flawed — CaZaNOx

I think (C1) and (C3) has shown why there need not be two definitions of existence, and why there might be other conceptions consistent with the definition of existence and the definition of the world.

(C6)

First of let's note that S1,...,Sn just appear randomly and are not defined and irrelevant to the problem. Therefore we can leave them aside. — CaZaNOx

You are correct. The discussion of the other domains are left-overs from the broader discussion where i took this particular conception from. Thus they are not pertinent. Just for the sake of elucidation they were used in denying that a certain principle did not apply in the case of the worlds self-containment. The argument were something like:

(T) it is impossible for the world to appear in a domain that appears alongside domains.
Alongside: there are other domain that do not only appear within the world.

This principle holds in the case that the world appeared within something else because:
(a) the domain containing the world and the domains it appears alongside would encompass more than the world.
(b) if all domains are contained within the world and some of these also appear alongside the world, then they both appear within and outside the world, thus insofar we conceive the world as appearing in something other than itself, it would violate the definition of the world. The world contained is then either not the world at all, or the fields that appear alongside the world does not exist at all. Both would lead to a rejection of the assumption that the world appeared in something else.

Now the definition of alongside clearly doesn't allow for (T) to obtain if the world appeared within itself. We could then define alongside as: two domains appear alongside if they both immediately appear within the same domain. Now in the case of (S1),...,(Sn) in the conception of self-containment (T) doesn't have any argumentative force:

(a) would not apply in this case because none of the domains that contain the world would encompass more than the world.
(b) would not obtain because in no case does (S1), (S2) and (S3), or any of their subdomains, appear outside the world. They only appear alongside the world insofar as they appear within the world

Using the term 'domain' would obscure the intention to establish the self-containment of totality. Ex. 'Now we claim that a domain exists and it then follows that this domain shall appear within itself'. This point would be invalid by only arguing from the notion of a domain. — auto to on

Logic doesn't work like that. We are not allowed to use a name for a concept and then rely on it having all the properties and associations that it has in natural language. The only properties that a named thing has in logic are those that are given to it by formal axioms.

If you want to make an argument using a class of things called 'domains', you need to start by laying out in a series of logical statements (axioms), all the properties you want that class of things to have. The next step is then to establish that one or thing has those properties - ie that domain(s) exist. You can do that by assertion in another axiom, or by deducing it from other axioms.

I perhaps see why you might think this if you think it's a claim of existence. But the conception does not result from the definitions of existence, since the definition of the world is what requires self-containment. If the world is the domain of all domains, it should be a domain within itself. The definition of existence only functions as a specific way of framing this self-containment, since it denominates the appearance within a domain as existence. With this additional definition we can then say that the world's existence follows from it's definition. — auto to on

What, what and what??

Let's start with this: " If the world is the domain of all domains, it should be a domain within itself." First off, "The world is the domain of all domains" is a definition you're making up. It's a construction of yours.

But aside from that, what exactly does "It should be a domain within itself" refer to?

Logic doesn't work like that. We are not allowed to use a name for a concept and then rely on it having all the properties and associations that it has in natural language [...] If you want to make an argument using a class of things called 'domains', you need to start by laying out in a series of logical statements (axioms), all the properties you want that class of things to have. — andrewk

I don't agree, and i don't share this methodological approach. As indicated by the fact that i don't require you to do the same for the statements contained in the quote.

Let's start with this: " If the world is the domain of all domains, it should be a domain within itself." First off, "The world is the domain of all domains" is a definition you're making up. It's a construction of yours — Terrapin Station

Yes, this is a correct observation, in fact i defined it quite explicitly in my post:

Let the world (w) be defined as: the domain that contains all domains. — auto to on

what exactly does "It should be a domain within itself" refer to? — Terrapin Station

Well this is the what this conception of the world tries to answer.

I don't agree, and i don't share this methodological approach — auto to on

Since you don't agree to the use of logic, there is nothing that can be discussed. I suspect this is not the best forum for you to find sympathetic ears for your beliefs.

Since you don't agree to the use of logic, there is nothing that can be discussed. I suspect this is not the best forum for you to find sympathetic ears for your beliefs. — andrewk

I agree to the use of logic in processes of reasoning, as evidence by the argumentative approach of this post and consequent comments. I don't agree that the only legitimate means of proposing an argument is by formal means. I'm not sure whether your equation of these two is the result of confusion or an attempt at an insult, but i do find it insulting. If you want to make the case that a formal method is necessary in these matters, then i welcome you to present an argument for this. But merely asserting that this is the only legitimate methodological approach has no persuasive force on me. So if you want me to take you seriously at least answer these basic questions:

How would changing the word 'the world' with 'a domain' in the original argument strengthen the argument?
Why can't i use natural language to formulate the argument?
Why can't i rely on the semantic significance of the words used in the argument as long as i define central terms?
Why would i be required to define 'domain' in formal terms?
Why is proving the existence of a domain necessary for the argument?
Why can i only prove the existence of a domain by formal means?