Stoic,
First, I don't recall Euclid's maxim being written in stone.
Second, the modification of Euclid's maxim under discussion effectively restricts it to finite sets
That it’s not written in stone wasn’t my point. I’m doing two things here. I’m doing aporia and basing that aporia off of a history of justification. Infinite set-theorists restrict EM to finite sets in their debate with the intuitionists and the constructablists. This is a purely mathematical debate. Every single time the question of metaphysical instantiability comes up, naive mathematicians and oblivious metaphysicians extend the justification in that debate to a completely different debate, in a completely different context, where rules against inverse operations on existing objects are no longer forbidden. So, there’s a completely different enumerative game going on here, with the lifting of rules that no longer apply, where the absurdities are no longer checked and balanced, but can be nipped in the bud by restricting the justification to the debate between the mathematicians where it belongs, and sticking to the extremely plausible EM for any instiantiable collection, where the Principle of Correspondence makes sense, with the negligible cost of keeping actually infinite collections from being a thing in reality. This aporetic strategy seems to go through in flying colors for me.
Of course, you can do this with finite sets. But I don’t know where you’re getting this for infinite sets. I took Set Theory in Graduate School and inverse operations like subtraction and division are explicitly prohibited because they’re ill-defined. There has to be this strange, unnatural distinction between size and cardinality, but size ends up not being a quantitative property. It’s a concept describing the characteristics of the elements in a set; it can’t quantitatively enumerate those elements without introducing cardinality again. But once cardinality is brought back in, we’re back to being ill-defined. But again, we’re within the mathematical realm. I can’t use any of the justification mathematicians use in their debates with the intuitionists to justify how it’s metaphysically possible for there to be a one-to-one correspondence between the set of natural numbers and instantiated objects in the world, spatio-temporal or not. These are completely different domains.
I think I follow. Though it’s hard to know what it would look like for zero to exist. It’s easy to imagine an actually infinite number of things existing and then doing to the forbidden operations.
No, that’s not what I meant by ‘there can be only one kind of stuff’. I didn’t mean it’s impossible for there to be other kinds of stuff; I meant that it’s possible that what we’re talking about is one kind of stuff. For example, one could imagine an infinitely extended ‘gunk’, as the mereologists talk about (https://en.wikipedia.org/wiki/Gunk_(...ogy%2C%20an%20 area%20of,of%20gunk%20is%20itself%20gunk.)
I know that Advanced LIGO detected the gravitational waves from the merging of two orbiting black holes, and I also know that the gravitational waves caused by inflation have yet to be detected. But what I meant to emphasize is that the last prediction made by GTR has come true. Gravitational waves exist. It’s not the case that the eventual discovery of the gravitational waves from inflation will be of a different type. Gravitational waves are gravitational waves. The prediction that inflationary cosmology is the way to go is pretty much in the bag. The only barrier in the way is advancements in the technology of LIGO.
The prediction is backed by a fully confirmed GTR. Experimentally confirming it is the Holy Grail, but it’s pretty much over at this point. Everyone is just waiting on the technology. A cosmological application of GTR makes the population of inflation inevitable, and so the detection of the so-called ‘primordial’ gravitational waves are inevitable. This expectation has nothing to do with the mistake made by Antarctic BICEP2 that registered cosmic dust rather than waves. That mistake was one of experimental confirmation. The prediction is still in the bag at this point.
I realize that Steinhardt cautions that
. . . but that’s more of a casual hope or a truism than anything else. Anything ‘may’ happen, but it probably won’t, as admitted by both Turok and Steinhardt. It would be hard to salvage their model (or some comparable model without inflation) without introducing ad hoc features to ‘save the phenomena’, so to speak. This and the fact that physicists are already very aware of where this is going as evidenced by folks like Sébastien Galtier, Jason Laurie, and Sergey V. Nazarenko (file:///Users/matthewdamore/Downloads/universe-06-00098%20(1).pdf) In here (https://inspirehep.net/literature/1672978), you also have Peter Adshead(Illinois U., Urbana), John T. Giblin(Kenyon Coll. and Case Western Reserve U., CERCA), Zachary J. Weiner(Illinois U., Urbana), who admit that,
The predictions from physicists are ubiquitous and pretty confident: (e.g. https://inspirehep.net/literature/1735811)
First, I don't recall Euclid's maxim being written in stone.
Second, the modification of Euclid's maxim under discussion effectively restricts it to finite sets
Set theory allows you to add elements to a set, or remove elements from a set. This is true even of infinite sets.
It seems to me that one could argue in the same way that the number zero doesn't exist. Mathematics only excludes division by zero because you can get contradictory answers.
When you say that there can only be one kind of stuff constituting an infinite magnitude, what kind of stuff are you talking about? I don't think "distance" counts as "stuff".
The gravitational waves that have been detected have been attributed to the collisions of black holes and/or neutron stars, which are distinct from the type of gravity waves that would have been caused by inflation. I don't think there is anything in Steinhardt and Turok's model that would rule out the gravitational waves that have been detected so far.
“Neil Turok, director of the Perimeter Institute, said the technology would undoubtedly improve. If it can reach a sensitivity 100,000 times better than Advanced LIGO has already demonstrated, he said, then it would be possible to hear the booming of gravitational waves from the birth of the universe about 13.8 billion years ago” (https://www.theglobeandmail.com/news...ticle28713410/).
“Based on their analysis, the scientists think that both current and future planned gravitational wave detectors will be able to detect the frequencies of gravitational waves emitted by shocks. These frequencies correspond to emission times of around 10-4 to 10-30 seconds after the big bang” (https://phys.org/news/2016-10-early-universe-today.html.)
“even if we detect primordial gravitational waves, we should not rush to the conclusion that they are due to inflation. Better theories may come along that avoid the pitfalls of inflation and that nevertheless predict gravitational waves” (https://blogs.scientificamerican.com...lped-conceive/),
“. . . the frequencies of the generated gravitational waves lie far from those to which LIGO is sensitive. However, the amplitude of this signal will remain (relatively6) invariant when changing m, while the emitted frequencies are proportional to this scale [14]. For this reason, these preheating dynamics after low- scale inflation could in principle be detected by LIGO. Advanced LIGO’s peak sensitivity is on the order of Ωgw,(f)h2 ∼ 10−10, which is several orders of magnitude lower than that the amplitude produced by the simulations which achieve complete reheating. aLIGO’s peak sensitivity lies around f ∼ 50 Hz, which would probe inflationary scales ∼106 GeV. Note that the subsequent expansion history of the Universe also affects the gravitational-wave transfer function; we assume the Universe is radiation dominated after emission until matter-radiation equality. Since preheating into gauge fields naturally leads into radiation domination after inflation, this approximation is well-justified.”
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