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?

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.

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.

The union of all axiomatized set theories does. — litewave


That is an inconsistent theory. — TonesInDeepFreeze

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.

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. Same with axiomatized set theories: they define different worlds and thus are not contradictory.

But if ZFC is the context of your notions, at least a kind of "base" theory for your uncountably many theories — TonesInDeepFreeze

I mentioned ZFC just as an example of axiomatized set theory.

But the more basic point is that, no matter your own views (or even Hamkins's, for that matter), it is not the case that "according to set theory, all logically possible (consistent) collections exist".

Set theory does not say that. — TonesInDeepFreeze

The union of all consistent axiomatized set theories does.

. Please say exactly what passages in part 7 you regard as saying that there is a set with cardinality between the naturals that is decided ('settled' in context) by the consistency of ZFC+~CH. — TonesInDeepFreeze

Hamkins regards the world defined by ZFC+~CH as equally real as the world defined by ZFC+CH and that both worlds exist in the multiverse. So that's how the continuum hypothesis is settled.

"Since we have an informed, deep understanding of how it could be that CH holds or fails, even in worlds very close to any given world, it will be difficult to regard these worlds as imaginary."

And no retraction from you that you falsely put words in my mouth by claiming that I said naive set theory must obey the axiom of regularity. — TonesInDeepFreeze

I didn't put words in your mouth. I thought that when you used the word "naively" you referred to naive set theory.

So, please quote the specific passages you contend claim that all sets exist that are "selected" by at least one consistent axiomatic set theory. Tell me the exact formulations you have in mind that Hamkins mentions in his own words. — TonesInDeepFreeze

on page 2: "In this article, I shall argue for a contrary position, the multiverse view, which holds that there are diverse distinct concepts of set, each instantiated in a corresponding set-theoretic universe, which exhibit diverse set-theoretic truths."

And please cite where Hamkins says there is a set with cardinality between the naturals and the reals and that that is decided by the fact that ZFC+~CH is consistent. — TonesInDeepFreeze

The example with CH is in part 7: "Case study: multiverse view on the continuum hypothesis". See it for yourself. I tried to express the gist of it in my previous post.

And that is not what anyone means by 'naive set theory'. So your notion is not set theory and it's not naive set theory. — TonesInDeepFreeze

Well, Wikipedia article on naive set theory just mentions the general concept of a set as a collection of objects, and related general concepts like set membership relation, equality, subset, union, intersection etc. There is no requirement that a set cannot be a member of itself or that a set must have a minimal member, which you tried to impose on naive set theory.

What are all the axiomatized set theories? There is no definitive list, and there is no conceptual limit. For that reason alone your notion is fatally vague. — TonesInDeepFreeze

Sure, the number of axiomatized set theories is uncountable. I see no reason to set any arbitrary limit to them.

And what does "included" mean? — TonesInDeepFreeze

Look at Hamkins' paper on multiverse in set theory. As an example, he talks about a world defined by axioms of ZFC + CH and a world defined by axioms of ZFC + negation of CH. He claims that both worlds exist in the set-theoretic multiverse. That means that a set with cardinality between naturals and reals doesn't exist in the (ZFC + CH) world but it exists in the (ZFC + negation of CH) world and thus exists in the multiverse. I cannot speak about set theory with the same rigor as Hamkins or you but this seems to be what I am trying to say.

since ZFC+CH is now ruled out by your requirement that there exists a set of cardinality strictly between the naturals and the reals — TonesInDeepFreeze

ZFC+CH is not "ruled out", it just defines a part of the multiverse, a part in which there is no set with cardinality between naturals and reals.

I think I can clarify a lot by addressing this part of your post:

(5) Set theory does preclude certain kinds of sets that otherwise it would be consistent to say they exist. In particular, the axiom of regularity precludes certain kinds of sets that otherwise would be consistent to say they exist.

Since you did not reply to that, I'll add: I surmise that naively (informally, intuitively) most set theorists' notion of 'set' includes that sets are not members of themselves, and that, more generally, every set has a minimal member. That is especially witnessed as the axiom of regularity is a standard axiom, which is especially relevant since you say that naive set theory is "elaborated upon" by axiomatizations such as ZFC. This is a point blank refutation of your claim that "according to set theory, all logically possible (consistent) collections exist", as indeed both the naive notion of sets and the standard axiomatizations exactly preclude the existence of certain kinds of sets that would not be inconsistent to assert their existence otherwise. That point cannot be skipped and it alone decisively refutes your claim. — TonesInDeepFreeze

As I said, by "set theory" I mean set theory in the most general sense. I supposed that this is what is commonly understood as naive set theory, but to clarify, I mean the concept of a set or collection of objects that is elaborated in all consistent axiomatized set theories together. In other words: if a set is included at least in one consistent axiomatized set theory, then such a set exists. As I said, there are uncountably many axiomatized set theories. ZFC set theory is just one of them. So I include also sets that are members of themselves and sets that don't have a minimal member, as long as these sets are consistently defined. Elsewhere you stated that there is no such thing as an inconsistent definition, so let me give you an example of an inconsistently defined set: an empty set that has one member.

Also, the example with CH may be clarifying. If there is a consistent axiomatized set theory that includes a set with cardinality between the cardinalities of the set of naturals and the set of reals, then such a set exists, simply because it is included in a consistent axiomatized set theory (that has an axiom that is the negation of CH). It's no problem that in a different axiomatized set theory, which has CH among its axioms, such a set doesn't exist. A set theory with CH as an axiom simply selects only certain sets among which a set with a cardinality between naturals and reals is not included. Every consistent axiomatized set theory selects certain sets, and by "all logically possible (consistent) collections" I mean sets or collections selected by all consistent axiomatized set theories together - that's the multiverse view in set theory. If a set is included at least in one consistent axiomatized set theory, then it exists in the set-theoretic multiverse.

Yes, a world which is called a possible world ... — Shawn

But the actual world is still necessarily identical to itself and therefore cannot be something different than it is. My actions in a merely possible world may be different than my actions in the actual world but my actions in the actual world cannot be different than my actions in the actual world.

Every day we make choices where seemingly we could have done something otherwise. — Shawn

Yes, seemingly.

Even taking your argument to the extreme, there could be a possible world where causality would have allowed for a different event cone to allow a counterfactual to arise. — Shawn

Quantum-mechanical indeterminacy could do that. But a world with a different outcome of a quantum measurement would be a different world, with a different identity, than our world. And I cannot be in both worlds, if by "I" we understand someone who is conscious of being only in one world. The "I" in the different world would be my copy, a counterpart.

Could you elaborate? — Shawn

Does Kripke believe that it is possible for us to act differently than we actually act? It seems he doesn't, because our actions are parts of spacetime and spacetime is necessarily identical to itself, which means it cannot be different than it is and so our actions cannot be different than they are.

What's posited here is that if two things are identical then they are necessarily identical. — Banno

Right, and our spacetime is necessarily what it is and could not have been different. Bye bye free will?

It is not true that according to set theory all logically possible (consistent) collections exist. — TonesInDeepFreeze

I mean "set theory" in the most general sense, also known as naive set theory. It just says that a set is a collection of objects. This general concept of set is elaborated in uncountably many axiomatized set theories, for example the famous ZFC set theory. I refer to all these axiomatized concepts of set, as long as they are consistent.

For example, it is consistent with set theory that there is a set that has cardinality strictly between the cardinality of the naturals and the cardinality of the reals, but it is not a given that it is the consensus of set theorists and philosophers that such a set exists. — TonesInDeepFreeze

For me, as long as such a set is consistently defined, it exists. In some axiomatized set theories it may exist while in others it doesn't. That's because every axiomatized set theory selects a limited collection of possible (consistently defined) sets. This is what Joel David Hamkins has called "set-theoretic multiverse".

I imagine there's a lot less to do in a merely possible world, for one thing... — busycuttingcrap

Why would there be less to do in a merely possible world? Some worlds may be simple and others more complex, whether they are merely possible or real.

And if something is logically possible, then we may also say that "there is a possible world" where that something is true or is the case. — busycuttingcrap

And what is the difference between a logically possible world and a real world?

But what is outside of spacetime? Abstract objects like thoughts and numbers. — Art48

Thoughts may actually be concrete objects (parts of a brain) and abstract objects in the sense of universals, like numbers, may actually be similarity relations among concrete objects. What is outside of spacetime then? Perhaps other concrete objects, including other spacetimes. According to theory of relativity, spacetime is actually just a special kind of space, a 4-dimensional space with one dimension (time) somewhat different from the other three. And according to topology, a space is just a special kind of collection. All mathematics seems to be reducible to concrete collections, from the empty collections (non-composite objects) to infinitely large collections (infinitely large composite objects). That's why set theory (the ultimate theory of collections) is regarded as a foundation of mathematics.

So according to set theory, all logically possible (consistent) collections exist, from the empty collections to infinite collections, and they, or relations between/among them, constitute all known mathematical objects, relations or structures. Note that all of this exists necessarily/automatically because nothingness constitutes the content of empty collections and empty collections constitute the content of larger collections, and larger collections constitute the content of even larger collections, and so on. And a spacetime is one of those collections and we are collections that are parts of a spacetime.

So is everything math? Well, there seems to be something about collections that is extra-mathematical. There is a composition relation (or set membership relation) between a collection and a larger collection that includes it. So collections are somethings (not nothing, as there can be no relations between nothing), but mathematics doesn't tell you more about these somethings than that one something includes another something. Mathematics is just about relations between these somethings and these relations are reducible to the composition relation. These somethings are not relations; they are what stands in composition relations. These non-relations might be called "things" or "qualities".

This does not address the question, wherein alternatives to choose among occur. — javra

What question do you mean?

For example, your intent is to learn about subject X; how does this intent of itself establish whether you choose a) to read a book about X or b) to see a documentary about X? — javra

I may stumble upon a good book review and then the information from the review together with my intent to learn about subject X create in me the intent to read the book about X and this intent drives me to read the book. Where is free will?

Is he not his ingrained predispositions? — NOS4A2

Yes, you could say that the agent is his ingrained predispositions (as well as his experiences in the course of his life). So the agent has free will even though all his actions are completely determined by his ingrained predispositions?

What I’m suggesting is that there isn’t a strict logical dichotomy between “completely determined (hence no free will)” and “completely undetermined (hence no intentionality)”; that there logically can very well occur something in-between, a “partly determined and hence partly undetermined” state of being that (partly) defines us as agents; and that our free will - if real - would necessarily be of the latter state of affairs: e.g., always partly determined by intents (among other possible factors), but never completely determined. — javra

Then our actions are partly intended and therefore partly unfree, and also partly unintended and therefore partly unfree too. So they are wholly unfree.

Its a variant of compatibilism, though I take it you're not much enamored with the prospect of compatibilism. — javra

I think compatibilist version of free will has some merit because it says that we have free will if we can do want we want. But it also admits that our actions may still be completely determined by factors that are ultimately out of our control (we do what we want but our wants are ultimately ingrained in us), which seems to conflict with what we usually mean by free will when we bother to talk about it: a free will that gives us ultimate control and moral responsibility that can override all circumstances.

Are you saying that the agent can act freely even though all his actions are completely determined by his ingrained predispositions?

Slipping, falling, colliding with the ground, and breaking one’s arm are not the “free acts” of the agent because those are the actions he is trying to avoid. — NOS4A2

Can the agent choose not to avoid those actions? If he cannot, is he acting freely?

Either way, be it something you’ve previously chosen for yourself of something ingrained that is beyond your choosing, it does not nullify the logical possibility of free will in the choices you do make at any given juncture. — javra

Even if your choice is driven by a goal you chose previously, the choice of that goal itself was driven by something ingrained in you or by another goal which however must ultimately be driven by something ingrained in you too because you cannot have an infinite regress of goals. So ultimately all your choices are completely determined by factors that are out of your control or maybe are partially undetermined, which precludes your control too.

Nor would the occurrence of free will necessitate that causal determinacy does not take place in the world - it would only necessitate that the world is not one of (complete) causal determinism. — javra

To the extent that your action is not determined by your (ultimately ingrained) goals, it is unintended and therefore unfree. Indeterminacy doesn't save free will, it just makes you do things you don't intend to or hampers your ability to do things you intend to.

But you are not isolated from your environment. You cannot think freely without breathing oxygen and you cannot walk freely without having a ground to walk on. So why is that slip on a banana peel not your free act?

No. Our actions would yet be "free" if we could choose otherwise in a selfsame situation - hence a situation wherein the same overarching intent (e.g., to increase one's own happiness) and the same alternative / conflicting wants (e.g., seeing a movie or reading a book) occur. — javra

Well, in physics the outcome is determined by the joint influence of all present forces. It seems similar with my decision/action - it is determined by the joint influence of all my present drives. In quantum mechanics the outcome may also be partially undetermined by the forces - there can be different outcomes in the same situation (with the same forces present). Maybe my decision/action is partially undetermined by my drives too, but then again, to the extent it is undetermined by my drives, it is unintended and hence unfree.

In a sense, then, you chosen, determined, directed, regulated, willed your thought by having it. — NOS4A2

So every thought that pops into your head is freely willed by you?

By choosing between alternative potential intentions - like the intent to read a book or the intent to see a movie. — javra

Like, I have an intention to read a book and also an intention to see a movie? How do I intentionally decide between them? I would need an intention to yield to the first or the second intention. But how do I choose that intention?

Then your mind seems oddly passive. I'm glad I'm not you, just watching your thoughts pop up, unable to decide between them. — Banno

Oh, I can decide between them, I just need a thought to decide between them, except when I don't need a thought to decide between them, in which case I don't decide or I decide unintentionally.

. Intents are teleological processes, i.e. teloi, and not causal processes as the latter is understood in modernity via Hume's notion of causation and the notions of those who followed. — javra

I want to eat a cookie and this wanting is the intention that drives me to get a cookie. If the wanting is of an obssessive intensity you can literally feel how it pushes you to your feet and toward the cupboard with the cookie.

But, as to the issue of determinacy, if we do hold free will then we are only partly determined by determinants (teloi and antecedent causes included) in the choices we make, and thereby remain partly free to choose what we see fit — javra

So to the extent our action is determined by our intentions it is not free. But to the extent it is NOT determined by our intentions it is unintended and therefore not free either.

Nope. Free will pertains to living beings, in particular humans. — NOS4A2

So it is not enough for my free will act to originate in me. I must also be alive and maybe also intend to do the act? But how do I choose an intention without already having it?

litewave, It's far from clear what you response to the OP is. — Banno

I agree with OP. We cannot choose to think a thought without already thinking it. Which means that our choice of thoughts is just thoughts popping into our head. So much for free will.

Still, intents do not of themselves choose outcomes. We as agents so driven by our intents do. — javra

Intents drive us and we drive outcomes. Seems like a row of billiard balls.

Thoughts don’t choose between thoughts. Agents - such as one’s own conscious being - choose between thoughts. — javra

Why would agents do that? Because they are driven by thoughts, including by thoughts to choose between thoughts. Or when they are not driven by thoughts, their choices are unintended, which precludes free will too.

. The slip itself is more of an act of physics, I suppose. — NOS4A2

So physics has free will?

Something I’ve been wondering about regarding the block universe is, does the block universe model depend on physicalism being true, or could it also work with ontologies such as monistic idealism? I’m hoping you could help me figure that one out. — Paul Michael

I think the simplest interpretation of theory of relativity is that time is literally a space and therefore it doesn't pass, it just exists. But we have various feelings and one of those feelings is that time passes, which is apparently associated with feelings of memories and expectations.

To rule out the possibility of free will one will have to show that thoughts, or any action for that matter, comes from somewhere or someone else. — NOS4A2

If you slip on a banana peel is it an act of your free will or is it an act of the banana's free will?

I have, however, edited what occurred to me. Was the editing an act of free will or was it the product of a fussy compulsion? — Bitter Crank

It occurred to you to edit what occurred to you.

If this is correct, does this automatically rule out the possibility of free will? — Paul Michael

Yes, and it gets even worse if you consider that time doesn't pass because it is just a special kind of space, as theory of relativity implies. Do we have free will if the future already exists just like the past?

What does such a redundant modifer even mean? As compared to 'not really real' or 'unreally real'' :roll: — 180 Proof

It means to emphasize that something is "real", especially when one doesn't know what "real" means.