Remember, you're trying to showYou seem to have read my response as a series of disconnected links rather than as a coherent argument, which explains your confusion. The first source is describing event from a physics point of view that leads into a discussion of causality in the second link (which is by a computer scientist you are correct who is writing about causality in science in general and heavily in physics in particular and which is sourced relatively heavily).

If events are defined as changes in state as described, andallevents must have a fixed number of input states, and as he additionally writes:For any point p at a time t, the causal influences depend on the values of functions at some selected set of points at a previous time step. The values at those points, in turn, depend on a somewhat larger set of points at some earlier time step.

IE the value of points are determined by their causally prior points, then the value of all events is determined by their causally prior points.physicists agree that physical events necessarily have causes. Even assuming arguendo that this computer scientist's definition of event is such that events necessarily have causes, where's the proof thatphysicists agree that physical events necessarily have causes?

What physical event causes particle decay?You misread what I wrote. I wrote that these equations aredescribedby a probability function that is a representation of a physical process.

No, it was a failure to respond. Even assuming that every word of your statement was true, you failed to address part of the question you were supposed to be responding to. That's a failure to respond.Well it certainly isn’t a failure to respond, it was a response you disagreed with and decided to make a snarky comment about.

You haven't shown at all that the definition you gave isThis has also been a habit with you in this thread, insisting that therecould bea vastly different definition of terms (see here iterative, for which we have yet to see an example of anyone using in a fashion other than I quoted) than the standard definition.

Should I also define “the” for you? Could you suggest an appropriate physics dictionary that reveals the specific physics definition of “the” that differs from all other definitions?standard, or that physicists agree that physical events necessarily have causes.

Given your proclivity for failed attempts at clarification and definition, I don't think I'd trust whatever word salad you'd offer as the meaning or definition ofthe.

You're just offering bald assertions again.Then what do you think the author meant when he said: “and caused by the breakdown of the nucleus into a more stable form.” To which you might ask, what causes the breakdown to the more stable form? To which he has already answered, the inherent instability of the nucleus. It doesn’t just “happen” it happens as a result of an unstable nucleus. Becausewhenthat happens is governed by a probability distribution does not mean the action lacks a cause.

Further, I think your question seems to misunderstand how these features can be used across fields. Because an action can be treated as happening without a cause for a certain line of exploration (meaning essentially that we can ignore the cause) does not mean it actually lacks a cause. We often treat random number generators as actually random for certain purposes, it doesn’t mean they actually are random however.

Sure, it would be wrong to infer that because something is described by a probability distribution, it must lack a cause. But that's not the inference; I didn't argue, "Necessarily, everything described by a probability distribution lacks a cause; particle decay is described by a probability distribution, therefore particle decay lacks a cause."

My understanding is that the particular observation from the probability distribution is the uncaused event. I.e., there's no physical cause for the random variable to take on such-and-such a particular value.

But my understanding is tangential; you're the one with the claim (that causality implies all spacetime events are caused by some other spacetime event, such that there's no minimal element--WRT the causal order, of course) to support.

Do any causet authors mention that there cannot be minimal elements? What do you make of R/S's explicit physical interpretation of the birth of a causally-minimal element?

"Volume is clearly a desirable trait"? Was someone arguing that volumeI think you aren’t taking this citation in the full context of the paper. I’ll attempt to provide some more context.

First, the citation does indicates that volume is clearly a desirable trait since it is in support of an argument he is making. He is saying that his axiomatic change allowing minimum volumes to be assigned without fatal behavior is a reason for acceptance for accepting his proposal over the current version, clearly indicating that he sees this as an improvement. This is because, as we see later, an event should have some kind of measurable substance to it if it is to be part of the CMH.wasn'ta desirable trait? And do you mean some particular kind of volume, or just volume in general? Because it's fairly trivial to assign a "volume" to any particular object (the trick is for the assignment to behave well under certain transformations).

This doesn't show that there has to be a minimum volume.

Sure, discrete here meaning that singleton sets are assigned finite volume finite volume (cf. footnote 2). This doesn't say there has to be a minimum spacetime volume.Second, I think your interpretation ignores the consistent language throughout the document:The physical interpretation of C is completed by fixing a discrete measure mu on C, assigning to each subset of C a volume equal to its number of elements in fundamental units,IE, causets have a discrete measure of volume.

up to Poisson-type fluctuations…

Okay. Where does this show that there has to be a minimum spacetime volume?Causal set theory takes the bold step of incorporating this discreteness at the classical level, via a discrete measure mu that, roughly speaking, “counts fundamental volume units."In their inaugural paper Space-Time as a Causal Set, Bombelli, Lee, Meyer, and Sorkin use the partial order formulation of causal set theory, with the exceptions that an explicit statement of countability,and the nuance of Poisson-type fluctuations, do not appear:”...when we measure the volume of a region of space-time, we are merely indirectly counting the number of “point events" it contains... ...volume is number, and macroscopic causality reflects a deeper notion of order in terms of which all the “geometrical" structures of spacetime must find their ultimate expression... ...Before proceeding any further, let us put the notion of a causal set into mathematically precise language. A partially ordered set... ...is a set... ...provided with an order relation, which is transitive... ...noncircular... ...[and] reflexive. A partial ordering is locally finite if every “Alexandroff set"... ...contains a finite number of elements... ...a causal set is then by definition a locally finite, partially ordered set."Local Finiteness. Sorkin's version of the causal metric hypothesis (CMH) virtually demands some type of local finiteness condition, since the associated measure axiom (M) assigns an infinite spacetime volume to an infinite subset of a directed set. In this context, local infinities produce absurd behavior of the worst sort from a physical perspectiveMore on this in a second.

CMH (Causal Metric Hypothesis) isn't the relevant axiom. They're talking about the measure axiom.For example, one might consider altering the measure axiom to admit discrete measures assigning arbitrarily small volumes to subsets of a directed set; i.e., measures without an effective volume gap. However, this strategy would lead far away from the basic insight that \number" can serve as a proxy for volume in the context of Malament's metric recovery theorem.

Here I think he is talking about essentially what you mentioned. That we could try to alter a fundamental axiom of the CMH in order to assign volumes approaching zero, but that doing so would need to occur for physical reasons rather than to save an existing axiom (indicating it contradicts a current axiom) and that it would essentially remove the fundamental insight gained by causal set theory.

This is good support for your claim (that under the measure axiom, there's a minimum spacetime volume), although the author doesn't state that allowing arbitrarily small volumes would go againstphysics; arbitrarily small volumes might be incompatible with the measure axiom, but your argument is aboutphysicalpossibility. (Although I don't believe that a minimum spacetime volume is necessary or sufficient for the universe to be past-finite.)Following this axiom, the volume measure mu on a causal set C assigns “approximately" one fundamental unit of spacetime volume to each element x of C, regardless of the details of local causal structure near x.Assigning it zero would mean you'd assign itIndicating that unless we adopt the odd position that a fundamental unit of spacetime volume is zero (which would essentially make it meaningless) each volume measure is a non-zero, finite value.zerofundamental units of spacetime volume, not that you'd set the fundamental unit of spacetimetozero.

And just because each volume measure is non-zero and finite doesn't mean that there's a minimum value (e.g., each number in the set (0,1) is non-zero and finite).

Interval finiteness permits physically fatal local behavior under Sorkin's version of

the causal metric hypothesis. In particular, it permits instantaneous collapse, or instantaneous

expansion, of an infinite volume of spacetime, as explained in section 4.2 above.Mixing finite and infinite local behavior is dubious even without taking the measure axiom (M) into account, but is particularly problematic when the measure axiom is assumed to hold.Topological local finiteness in the interval topology is equally problematic in this regard.Okay, but how are "finite and infinite local behavior" relevant?

That's quite an abuse of language; there is no "measure axiom for CMH". CMH is a proposition, the Causal Metric Hypothesis; the measure axiom is another proposition. CMH doesn't "have axioms".Finally, I would point out the measure axiom for CMH which says:The volume of a spacetime region corresponding to a subset S of C is equal to the cardinality of S in fundamental units, up to Poisson-type fluctuations.

So a fundamental axiom of CMH is that the volume of a region is equal to the cardinality of the subset, which would indicate a finite, non-zero volume for all subsets of a causet.

Again, just because each volume is finite and non-zero doesn't suffice to show that there is a minimum spacetime volume.

lolWhat an unprofessional response.

I've asked you the same specific question about four times now; you've confused it twice, and just now refused to answer it. I'll ask it again.This was defined in the initial analogy, if you have a specific question, please ask it, otherwise please review the work already posted.

You said:These images refer to causets which do not fit our particular scenario, in these the spatial dimension admits multiple possible values, but that was not permitted via our scenario. So given that we can only have one possible spatial value, how can we have multiple maximal elements per stage?

Again, the context here is a set T (taken to represent time coordinates), a point in space x, and a collection of events (x,S) where S is a subset of T. T has an order (you've in the past referred to "predecessors" and so forth). The collection (x,S) of events has a causal order.

So when you ask "how can we have multiple maximal elements per stage?", to which order do you refer, T's (the "temporal" order) or (x,S)'s (the "causal" order)? The "multiple maximal elements" are maximal with respect to some order, but there are two orders here. Which were you referring to?

Evans got this paper published in Foundations of Physics:No, as I’ll state again, that the work presented in that journal was specifically work that was criticized or shown to be grossly or fraudulently incorrect.

I understand that criticism of related works should count, but that is fallacious to the argument being presented. If you wish to say the journal itself published work of a grossly or fraudulently incorrect nature, then show that the work it actually published is, itself, grossly or fraudulently incorrect.

Unification of gravitation and electromagnetism with B^{(3)}

The experimentally supported existence of the Evans Vigier field.B(3),in vacuo implies that the gravitational and electromagnetic fields can be unified within the same Ricci tensor, being respectively its symmetric and antisymmetric components in vacuo. The fundamental equations of motion of vacuum electromagnetism are developed in this framework.

These equations of motion, and the B^{(3)}formalism of gravity and electromagnetism in general, were shown to be grossly incorrect. Specifically, these equations of motion were shown to fail Lorentz invariance.

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