Comments

  • Are there primitive, unanalyzable concepts?
    I would say the property is less fundamental than the concept it refers to; because it presupposes it.Bob Ross

    What is the difference between property and concept? Isn't it the same general/universal entity?

    The interesting thing with 'being', is that it isn't really a property: that opens up the discussion to absurd ideas, like beings which themselves contain being in their essence and other beings which do not (e.g., Spinoza's view).Bob Ross

    Why would "being" not be a property? "Being", or "existence", seems to be something general/universal that is instantiated in all existing entities, including in itself. Why would some entities have being in their essence and others not?
  • Are there primitive, unanalyzable concepts?
    This pecularity indicates, by my lights, that ‘being’ is a primitive concept and, as such, is absolutely simple, unanalyzable, and (yet) still perfectly valid.Bob Ross

    Existence seems to be a property of entities that exist and as a property it can be defined by its instances, that is, by entities that exist. What is more primitive: a property or its instances? I would say neither; instances cannot exist without a property and a property cannot exist without instances.
  • Are there primitive, unanalyzable concepts?
    I think there are a lot of concepts that are not decomposable, that is, you cannot break them down into component parts without losing something. Perception might be one of these things. It's easy enough to describe perception. E.g., "you see a beautiful sunset over Death Valley."

    If you try to decompose the experience into what causes it though, you end up losing elements. No amount of talk of neurons or light waves, B-minimal properties, etc., no matter how informative, seems to avoid losing something.
    Count Timothy von Icarus

    Maybe that's because such descriptions are incomplete and even if they were complete they might be infinitely long or they might involve objects or relations that are difficult to imagine. At least neurons and light waves are 3-dimensional spatial objects so we can form some simplified picture of them, but consciousness also seems extended in time, so we might need to imagine qualia as spatiotemporal, hence 4-dimensional objects, which I don't know if anybody can. And if Gulio Tononi is right, then qualia are objects with many more dimensions in some abstract space of possible causal relations.

    Right, there are some pretty good arguments out of the Thomist camp that all properties of things have to involve how they relate to other things or parts of themselves.Count Timothy von Icarus

    Trivially, all relations between any two objects can be said to be similarity relations which specify which properties the two objects have in common and which properties they don't. Thus any object (as a bundle of all its properties) is, in principle, completely defined by its (similarity) relations to all other objects.

    But then what does it mean for something to simple?Count Timothy von Icarus

    Mereologically it means that it has no parts, or set-theoretically it is an empty set. But in another sense, I would say that every object, no matter whether it has parts or not, is something simple, unstructured, a monadic quality, a whole. A whole may have relations to other objects that are its parts, but it is not identical to any of its parts, it is something else than any of its parts, something in addition to its parts, which has part-whole relations to its parts.
  • Are there primitive, unanalyzable concepts?
    First of all, all definitions are essentially circular, as evidence by somebody not being able to immediately glean a language simply by by being handed a dictionary. But with some ideas, the circularity of the definition becomes very short, such as in your example.noAxioms

    I think this is a good point. Every object can be defined with its relations to all other objects. We can arbitrarily take some objects as "primitive" and define other objects with relations to these "primitive" ones. However, some objects are more frequent than others, like some words are more frequent than others in a language extract. It is useful to focus on these more frequent ones, and perhaps take them as primitive (as a basis for defining others), because it makes understanding reality easier; these are the regularities or commonalities in reality, known as fundamental (most general) concepts, or fundamental (smallest) particles, or laws of nature. It is easier to define movements of planets of the solar system with their relations to the Sun than to the Earth.
  • Absolute nothingness is only impossible from the perspective of something
    How could that be? I am definitely not conscious of my experience 10 minutes ago. Either I am, or I am not; there is no in between. And the fact is, at some point I was, but I no longer am. That's change.Ø implies everything

    You can have different experiences simultaneously, like hearing a sound, seeing something, experiencing a memory and an anticipation, and having the feeling of passage of time. Some of these experiences may feel more salient while others may feel vague or subconscious.
  • Absolute nothingness is only impossible from the perspective of something
    That still necessitates change; the change from experiencing a moment subconsciously to experiencing it consciously.Ø implies everything

    Maybe not. The conscious and subconscious experiences may be simultaneous, the subconscious in the background. Transition from moment to moment may be just an impression, a feeling.

    But I think I've ruled out eternalism as self-contradictory, which means there must be real change.Ø implies everything

    I am not sure it is self-contradictory.
  • Absolute nothingness is only impossible from the perspective of something
    However, eternalism is itself very problematic, philosophically. How do you explain our changing experience?Ø implies everything

    I don't have our experience of passing time figured out, honestly it seems like a major mindfuck. I imagine myself as being extended not only in space but also in time, as having many temporal parts in different moments of time, in continuous sequence. Each temporal part od myself experiences only its moment but my whole temporal self somehow (subconsciously?) experiences itself as a whole too, which perhaps provides the impression that the different experiences at different moments belong to me as to a single object. The arrow of time according to physics seems to be provided by the rising entropy along the time dimension (2nd law of thermodynamics), which perhaps creates the impression of irreversible progression of moments from the past to the future, along with impressions of memories and anticipations.
  • Absolute nothingness is only impossible from the perspective of something
    I don't think that works, because it introduces the choice again. Since both worlds already existed separately, then they were two separate objects (despite their identicality). Thus, a paralogical choice is made between which of the two worlds gets a banana and which gets an apple.Ø implies everything

    But such a choice is never made because the apple and the banana are part of the identity of each world and the identity of an object can never be different than it is, because that would constitute a logical inconsistency. So, the world that has the apple has the apple necessarily; it is logically impossible for the world to be different.

    You seem to be thinking of objects as changing in the passage of time but time is structurally (mathematically) a special kind of space, and space doesn't pass; it just exists. What appears in our experience as the future already exists, just like the past, and it exists the way it is and cannot be different, because that would constitute a logical inconsistency.
  • Absolute nothingness is only impossible from the perspective of something
    If one simply answers that the original sentience is no longer present, and two new sentiences were born (both having access to the original sentience's memories, and experiencing their birth as continuous extension of the original sentience's experience), then you have answered the question.Ø implies everything

    I think you can put it that way. Another possibility might be that there were two worlds with two sentiences where everything was the same up to a moment when an apple appeared in one world and a banana in the other world.
  • Absolute nothingness is only impossible from the perspective of something

    There can be two sentiences - one experiencing the world with an apple and one experiencing the world with a banana. After all, a sentience is just an object, like anything that exists. There can be many objects.

    I don't see any mystery in why one sentience experiences this world and another sentience experiences another world. Location and experience are part of the identity of each sentience. Each sentience (each object) is what it is, and cannot be what it is not because that would constitute logical inconsistency.
  • Absolute nothingness is only impossible from the perspective of something
    Okay, now we are getting somewhere. This splitting of worlds; has it happened after sentience entered the picture?Ø implies everything

    I don't know what sentience has to do with this. This is about logical consistency: each world is identical to itself. Each world is what it is and is not what it is not. That's all.

    Edit: By the way, the principle of logical consistency or identity does not pertain only to "worlds" but to any object. For example, one object cannot be both an apple and a banana because that would be a logically inconsistent object. But there can be two logically consistent objects - an apple and a banana.
  • Absolute nothingness is only impossible from the perspective of something
    Okay, let me take this step-by-step:

    1. First moment in time, there is just being (I don't claim you believe this, but you have to deny it).
    2. For this moment in time, due to the lack of any laws or anything specific, it would be logically consistent that a banana spawns at coordinates x,y,z.
    3. By the same logic, it would also be logically consistent that an apple spawns at coordinates x,y,z.

    So, in the next moment in time, what happens? Do both spawn? Well, each spawning is separately consistent, but together, they are inconsistent.
    Ø implies everything

    Then there are two logically consistent worlds - one in which an apple spawns and one in which a banana spawns. Both worlds exist because they are logically consistent.
  • Absolute nothingness is only impossible from the perspective of something
    Sure, but in the real world, a banana and an apple cannot exist with their centers overlapping.Ø implies everything

    Because such a world would be logically inconsistent, with respect to the laws that characterize its structure. So I state again: all logically consistent objects exist because there is no difference between logical consistency and existence.
  • Absolute nothingness is only impossible from the perspective of something
    But then everything would have popped into existence simultaneously, and contradictions would have arisen. How did the universe remove these contradictions? How did it choose one thing over the other?Ø implies everything

    In mathematics all logically consistent objects exist simultaneously and there is no contradiction.

    The purely logical donkey, when faced with two equally voluptuous hay stacks, starves to death.Ø implies everything

    So what? There is nothing logically inconsistent about starving to death.
  • Absolute nothingness is only impossible from the perspective of something
    For what it's worth, like a mathematician, I see no difference between existence and logical consistency and so I regard both these concepts as one and the same. Then the sentence "No object exists" can be reformulated as "No object is logically consistent", which is evidently and necessarily false. There are plenty of logically consistent objects; every object that is identical to itself is logically consistent and therefore exists (as opposed to, for example, the famous "square circle", which is a circle that is not a circle, a logically inconsistent and therefore nonexistent object).
  • Object Recognition
    In other words, I would have to pick out the lion first, before I have good reason to avoid it.NotAristotle

    So you pick out the lion without having a good reason and then you pass your lucky genes to your offspring. Or you don't pick out the lion and your genes go to waste. Fast-forward millions of years and virtually everyone in the gene pool automatically picks out lions.
  • What is your ontology?
    What is your explanation for existence?Benj96

    At heart my ontology is trivial: every possible concrete thing either has no parts (and thus has the structure of the empty set) or has parts (and thus has the structure of a non-empty set). These concrete things make up all possible worlds and all possible worlds are real worlds because there is no difference between possible and real.
  • Why is the Hard Problem of Consciousness so hard?
    You do see how the assertion that 'something just happened' does not actually amount to any kind of rationale?Wayfarer

    But things do happen, and this particular thing (survival) then tended to repeat itself and become more prevalent, simply because that is what surviving entities do.
  • Why is the Hard Problem of Consciousness so hard?
    Well, I also suggested a solution to the problem of qualia, which is the hard problem.
  • Why is the Hard Problem of Consciousness so hard?
    Yes - well, when you can demonstrate a self-creating machine that follows goals, then I will accept the answer. Because machines are human artefacts, produced intentionally to deliver a result. They embody the intention of the agent who builds them.Wayfarer

    For organisms evolved by random mutations and natural selection, many goals seem to be derived from the primary goal of survival and replication. This primary goal originated when some purely unintentional (goalless) entities happened to have (by random strokes of luck) properties that sustained them in their environment, which resulted in the unsurprising fact that... they kept surviving and thus became more prevalent in the environment. And so it started to seem like they were following a "goal" of survival. Later some of these entities happened to replicate and thus kept on "surviving" in their offspring. After some time the environment was filled with entities that seemed to have the "goal" of not only surviving but also replicating. And as time went by and the surviving and replicating entities happened to acquire additional properties that made them do various things, the entities seemed to acquire additional goals, many of them supporting the primary goal of survival and replication while goals that hindered survival or replication tended to cause their unlucky carriers to die out. Sure enough, some of the new goals were more or less neutral with respect to survival or replication, and such goals were carried forward in the entities that survived and replicated.
  • Why is the Hard Problem of Consciousness so hard?
    Isn't it just lumpen materialism? You still haven't allowed for intentionality other than as a byproduct or epiphenomenon of these essentially unintentional relations.Wayfarer

    Intentionality? That seems like an easy problem, not the hard one: a machine following a goal. For example a self-driving car is "intending" to get to a destination (without running over pedestrians). Or a killer robot is "intending" to kill someone.
  • Why is the Hard Problem of Consciousness so hard?
    Maybe what makes consciousness so elusive is that we only regard elementary particles as real things. There aren't really any atoms, neurons or brains. Just elementary particles. Collections of elementary particles like atoms, neurons or brains don't seem to be real things, just useful fictions.

    When you add up the masses of all elementary particles that make up a neuron, their total mass is the mass of the "neuron"; there is no additional mass provided by the "neuron" as a collection, as a whole. The absence of an additional mass provided by the "neuron" is the result of the fact that the locus of the gravitational force is only at the level of elementary particles, not at the level of their collections.

    The locus of all known physical forces is at the level of elementary particles, and when you add up the forces of all elementary particles that make up a neuron, their total force is the force of the "neuron"; there is no additional force provided by the "neuron" as a collection, as a whole.

    So it seems that a neuron doesn't really exist. And the same applies to collections of neurons. Consciousness seems to be somehow "generated" by collections of neurons, or seems to "emerge" somehow from collections of neurons. But what if consciousness actually is collections of neurons? What if qualia are collections of neurons? That would rid the metaphysical picture of mysterious "generation" or "emergence". But the fact remains that qualia as collections of neurons (or perhaps even as neurons themselves, which are collections of elementary particles) don't contribute their own mass and forces in addition to the masses and forces of their constituent elementary particles, and so qualia seem to be causally inert, epiphenomenal. Yet, if qualia are collections of elementary particles then qualia can exert causal influence indirectly, via their constituent elementary particles.

    My point is that collections of elementary particles are not just useful fictions but real things, different from and additional to the elementary particles, even though the collections don't have a direct causal influence, only an indirect one via their parts. That may explain how qualia are seemingly nonexistent although we all know them to exist as contents of our own consciousness or parts of our own brain.

    And by the way, elementary particles themselves may actually be collections of even more fundamental parts, which we have not discovered yet, or which cannot be physically probed even in principle. If collections are not real things and elementary particles are collections then even elementary particles are not real things. Then, are there any real things at all or is everything a fiction? Are only empty collections (non-composite things) real things - or would that be an arbitrary assumption?
  • Why do Christians believe that God created the world?
    If someone is omnipotent, they're able to do anything. If there was something they couldn't do, they wouldn't be omnipotent.Bartricks

    But a square circle is not really something. It is nothing. In mathematics it is the content of empty set.

    You said if God created the world, then God created the world. Er, yes. And?Bartricks

    It's a consistent proposition. You suggest that an inconsistent proposition can be true: "If God created the world, then God didn't create the world." The point is: God's actions are parts of reality and reality must be logically consistent, which means identical to itself. If the action of God's creation of this world is a part of reality then it is so necessarily because otherwise reality would not be identical to itself. Everything that happens in reality, happens necessarily. God's free will is at best compatibilist because no other free will is coherent.

    Again, you don't seem to understand what the issue is.Bartricks

    You said that this is not the best possible world. I say that even if this is not the best possible world, it may still have been created by an omnibenevolent God. Because an omnibenevolent God may have created the 10 best possible worlds and our world may be one of them. But who knows, maybe our world is the best possible one, who are we to say it isn't? How do you define the best possible world? I would imagine it's a world that somehow maximizes happiness across the whole spacetime, and so suffering at some time may enable more happiness at another time.
  • Why do Christians believe that God created the world?
    Yes they can. They can do anything, including things that violate the laws of logic.Bartricks

    There can't be such things. If there were, there wouldn't be.

    Anyway, it's beside the point, for it is clearly not a violation of the laws of logic to refrain from creating something.Bartricks

    Right, but sometimes it would be. It is not a violation of the laws of logic for a circle to exist, but it would be a violation of the laws of logic if the circle existed in a set of triangles. Similarly, it would be a violation of the laws of logic if you refrained from doing something in a world where you don't refrain from doing it.

    You've just said 'reality must be consistent......therefore God has created the world'Bartricks

    No, I didn't.

    'If' God created a world, then he would create the best world. This isn't the best world, is it?!Bartricks

    I don't know. But maybe an omnibenevolent God would create top 10 best worlds and this is one of them?
  • Why is the Hard Problem of Consciousness so hard?
    But just as the wood and spring of the mouse trap in no way explain how a mouse trap could be consciousness, the laws of biology, chemistry, electricity, and quantum mechanics in no way explain consciousness—or even hint that consciousness is possible.Art48

    Come to think of it, we also don't know what mass or electrical charge is. We just know that it is something that behaves in a certain way, for example it is attracted to other things with mass or attracted to or repelled from other things with electrical charge. So there is a correlation (or association) between mass or electrical charge and a certain kind of attracting/repelling behavior. The problem with consciousness seems similar but more complicated: a conscious brain behaves in a certain way that is different than how an unconscious brain behaves but the behavior is more complex and sometimes so subtle that there seems to be no difference between the behavior of a conscious brain and the behavior of an unconscious brain. Also, consciousness doesn't seem localized on the level of elementary particles but on the level of very complex wholes. But in all these cases we can see that if there is a behavior then there must also be something that behaves. I think this is a special case of a more general truth: if there are relations then there must also be something (a non-relation) that stands in those relations.
  • Why do Christians believe that God created the world?
    Then he wouldn't be omnipotent. God is by definition omnipotent. So God can do anything. That includes refraining from creating a world.Bartricks

    But even an omnipotent being can't create a logically inconsistent object because such an object cannot exist, for example a square circle.

    That's flagrantly question begging.Bartricks

    Reality must be logically consistent, which means that it cannot be what it is not. God's action is a part of reality, which means that God's action cannot be what it is not, and so if God creates a world he cannot not create it.

    It gets worse, not only does omnipotence not positively imply that God created the world, omnibenevolence positively implies God did not create it.Bartricks

    Then God creates the best worlds that are logically possible (consistent).
  • Why do Christians believe that God created the world?
    But I don't believe God created the world we live in. It doesn't look like the kind of place an all-good person would create. But Christians typically do believe that God created the world. Why?Bartricks

    Ok, I'll play God's advocate: God created this world because he couldn't have done differently. Not creating this world would be logically inconsistent because he would not have created the world he has created.
  • We Are Math?
    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.
  • We Are Math?
    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.
  • We Are Math?
    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?
  • We Are Math?
    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.
  • We Are Math?
    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.
  • We Are Math?
    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.
  • We Are Math?
    But if ZFC is the context of your notions, at least a kind of "base" theory for your uncountably many theoriesTonesInDeepFreeze

    I mentioned ZFC just as an example of axiomatized set theory.
  • We Are Math?
    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.
  • We Are Math?
    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.
  • We Are Math?
    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 realsTonesInDeepFreeze

    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.
  • We Are Math?


    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.
  • Kripke: Identity and Necessity
    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.
  • Kripke: Identity and Necessity
    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.