I suppose that you also think that a union of ZFC+CH and ZFC+~CH theories is an inconsistent theory. Yet according to Hamkins the worlds defined by these two theories are parts of a consistent multiverse. — litewave

Wow. You most clearly demonstrated your ignorance of the basics of this subject, and continue to carelessly misappropriate Hamkins.

Yes, ZFC+CH+~CH is inconsistent. That's clear on its face.

And worlds are models. Models are not consistent or inconsistent. Theories are what are consistent or inconsistent.

Yes, just as with Hamkins, different theories may have a different class of models from one another. Even certain kinds of theories by themselves have models that are not isomorphic with one another.

A (consistent) set theory has many models not isomorphic with one another. ZFC itself has models in which CH is true and models in which CH is false. But there does not exist any model of ZFC+CH+~CH, since inconsistent theories do not have models.

Hamkins points out that we are free to work separately in different models. He doesn't say that we combine a model of ZFC+CH with a model of ZFC+~CH.

You know virtually nothing about the subject of set theory and models of set theory.

As I mentioned, I doubt you even know what ZFC IS.

It is as if you took these two statements:

(1) "This ball is red."

and

(2) "This ball is not red."

and concluded that these statements are contradictory. But you didn't notice that these statements are not about the same ball but about two different balls and thus there is no contradiction between them. — litewave

That's a variation of your abysmal ignorance of the basics of the subject of models of theories.

(1) and (2) are a contradiction. However, there is a model in which (1) is true and (2) is false, and a model in which (1) is false and (2) is true. And that may be the case on account of 'this ball' referring to different objects per different models (or, also, on account of whatever 'this ball' refers to being in the subset of the universe named by 'red', or not in the subset of the universe named by 'red' per given models).

Learn the basics of mathematical logic and model theory, toward the subject of models of set theory.

Meanwhile, https://thephilosophyforum.com/discussion/comment/766751 stands, and now we add:

* You are utterly confused on even the most basic notions of a theory and models of a theory.

Hamkins points out that we are free to work separately in different models. He doesn't say that we combine a model of ZFC+CH with a model of ZFC+~CH. — TonesInDeepFreeze

He combines the world of ZFC+CH with the world of ZFC+~CH into a multiverse. And since a set with cardinality between naturals and reals exists in the world of ZFC+~CH, it also exists in the multiverse.

No, he mentions that there are separate universes. That is the multiverse: The collection of separate individual universes. He doesn't combine universes all into one big clump. There is no such thing when there is not even what a model when the theories contradict one another. And there is no "the world of ZFC+CH" or "the world of ZFC+~CH". Rather, for each set theory, there are many non-isomorphic models.

Get it in your head: There is no model of an inconsistent theory. There is no model of ZFC+CH+~CH, let alone a model from combining two models.

And Hamkins doesn't say that he obtains a model made by combining models of ZFC+CH and of ZFC+~CH. That would be ludicrous.

And if Hamkins says that, aside from particular theories, in general and simpliciter, there is a set of cardinality between the naturals and the reals, then we would need to see the exact passage in which he says that.

What Hamkins mentions is that there are two approaches: (1) A universe view in which some particular model is taken as the one that determines all mathematical matters. (2) The multiverse view in which there is not one particular model that is taken as the one that determines all mathematical matters, but instead there is PLURALITY (his word, my emphasis) of models, and mathematical matters therefore are not settled simpliciter but rather with a plurality of answers, each depending on particular models. You completely mixed that up to think he's saying that the multiverse is a stew of all models thrown into a single pot to make another model itself. He says the OPPOSITE of that. And the question whether there exists a set of cardinality between the naturals and reals is not answered 'yes' according to some single multiverse, but rather answered 'yes' or 'no' according to different models that are in the collection of models that is the multiverse.

You don't know anything about what a theory or model is. Yet you make stubborn false claims about the subject, even misrepresenting Hamkins. That is intellectually shameful.

Rather than keep repeating your ignorance, you would do better to grab a book on the basics of the subject of theories and models and properly learn the concepts.

No, he mentions that there are separate universes. That is the multiverse. The collection of separate individual universes. He doesn't combine universes all into one big clump. — TonesInDeepFreeze

"Clump"? Is that supposed to be another technical term? A multiverse is a collection of universes or worlds, that's all.

And there is no "the world of ZFC+CH" or "the world of ZFC+~CH". Rather, for each set theory, there are many non-isomorphic models. — TonesInDeepFreeze

Then there are "CH worlds" and "~CH worlds". You are missing the point, which is that a set with cardinality between naturals and reals exists in the multiverse.

But there does not exist any model of ZFC+CH+~CH, since inconsistent theories do not have models. — TonesInDeepFreeze

Then the multiverse is a model of what? Or what is it?

"Clump"? Is that supposed to be another technical term? — litewave

Your snark is ludicrous in context.

You entirely skipped my specific and exact explanation, to instead try to gain a pathetic bit of snarky upper hand.

Actually 'clump' is not technical, because there IS NO technical notion of combining models of contradicting theories.

Then there are "CH worlds" and "~CH worlds". — litewave

There are many models of CH and many models of ~CH.

You are missing the point, which is that a set with cardinality between naturals and reals exists in the multiverse. — litewave

No, you are missing the point that such a set exists in some models in the multiverse and not in other models in the multiverse.

You utterly got Hamkins backwards. Not surprising, since you are an ignorant, intellectually dishonest and (now seen to be) petty crank.

Then the multiverse is a model of what? Or what is it? — litewave

I already told you. It's a collection of models (or "worlds" informally).

Are you interested in understanding anything about theories, models, set theory and models of set theory? Or are you just going to continue insisting that you're right about everything even though you know nothing about the subject?

No, you are missing the point that such a set exists in some models in the multiverse and not in other models of the multiverse. — TonesInDeepFreeze

Really, where did you get that? I said that such a set exists in a ~CH world and does not exist in a CH world.

I already told you. It a collection of models (or "worlds" informally). — TonesInDeepFreeze

Well, whether you call the multiverse a model or a collection of models, the fact remains that a set with cardinality between naturals and reals exists in this collection, although it doesn't exist in all its subcollections (models), which is fine. And the same applies to the existence of any set that is defined by a consistent axiomatized set theory; that's why I said that all logically possible (consistent) collections exist.

:lol:

where did you get that? — litewave

From several textbooks and articles on mathematical logic and set theory.

whether you call the multiverse a model or a collection of models — litewave

As far as I can tell (based on the page you mentioned, and the surrounding pages) the multiverse is not a model. It is a collection of models.

the fact remains that a set with cardinality between naturals and reals exists in this collection — litewave

Sets exist in universes (domains of discourses) for model. The collection is a collection of models. The only things that exist in that collection are models. The set exists in the universe of one of the members of the collection.

although it doesn't exist in all its subcollections (models), which is fine. And the same applies to the existence of any set that is defined by a consistent axiomatized set theory; that's why I said that all logically possible (consistent) collections exist. — litewave

And now we're exactly where we started. Read the conversation again, if you wish to understand. Or, better yet, get a book on the subject so that you can understand its basics. Or continue to ignorantly demand that you're right, no matter how carefully it is explained exactly why you are not.

Still, it is incorrect that "according to set theory, all logically possible (consistent) collections exist." It's your own personal notion, not found as a theorem of set theory and not even as an agreed upon, let alone well articulated, informal meta-principle regarding set theory.

What do you think a theory is?

What do you think set theory is?

What do you think an inconsistent theory is? (You claim ZFC+CH+~CH is not an inconsistent theory, so it's clear you don't know what an inconsistent theory is.)

What do you think a model is? (You think there's a model (or "world") of ZFC+CH+~CH, so it's clear you don't know what a model is.)

What do you think a model of set theory is?

Sets exist in universes (domains of discourses) for model. The collection is a collection of models. The only things that exist in that collection are models. The set exists in the universe of one of the members of the collection. — TonesInDeepFreeze

If a set X exists in a collection (model) which exists in a collection (multiverse), I see no problem in saying that the set X exists.

What do you think set theory is?

What do you think is an inconsistent theory (You claim ZFC+CH+~CH is not an inconsistent theory, so it's clear you don't know what an inconsistent theory is.)

What do you think a model of set theory is? — TonesInDeepFreeze

Set theory is a description or explanation of sets. An axiomatized set theory is a set of axioms about sets. A model of a set theory is a set, or a collection of sets, that is described by the theory. An inconsistent set theory would be one that affirms and denies the same property to the same set.

So it's very clear now, each mathematician lives in one's own private multiverse.

The 1986 remake as the classic?

I grieve for my people. — Banno

I am sure your people respect the advice of their beloved king/god/malevolent controller, when it is exclaimed/heralded :rofl:
But seriously you preferred big Vincent's 1958 original?