Since there are no experimental ways to play with infinite objects, it follows the investigation is formal and thus transfinite metachemistry is indeed a subset of logic
yeah alessandro is joking, though it is still fun to ponder what happens if it is assumed to be serious and see where such rabbit hole leads
nevertheless, for this example, it is not going to be useful for a very long time, because we don't even have countably infinite entities in the real world known so far, let alone a chemistry based on them
What I am suspecting about infinite entities is that, if they do exists in some form in reality, they had to be somehow stuck inside a compact set to prevent them from going unbound
as otherwise we would easily see something gone really weird
Yes mathematically the set of strings encoding rationals are isomorphic to the set of rationals embedded in the real line. But it appears that the only physical meaning that can be ascribed to the real line forbids the rationals embedded in them from being meaningful.
And incidentally, there's no order-preserving embedding of ω[1] into the reals.
Can you prove that last fact?
Actually I think you do need to ensure you have a solid logic foundation before revisiting such stuff haha..
Well, even if spacetime is a continuum and not discrete, planck's length will mean we may never be able to zoom all the way down to the points in the rationals and other dense sets because it will give a limit on the smallest thing that can be measured in principle
I vaguely recall that any attempt to embed ω[1] to the reals will involve uncountably many intervals, but there are only countably many rationals, thus a contradiction will arise in the default model of ZF. But I might have confused that with the proof of the impossibility of embeding the long line into the reals
As for an infinite hydrocarbon, what I can speculate will happen is that you can never vaporise it even if the temperature is the same as the times back near the big bang. If it does able to ionise for reasons, it will be a bizarre entity that will give a lot of fragments without changing its molecular weight, which means conservation of energy has to be obeyed in some other way
and I strongly bet it will be a black solid unless it has the mophology of sheets
@Secret Yes it's correct that technically a ray of light is fuzzy at planck scales.
However.
Even the quantum mechanical description of a photon has an underlying real line (underlying the complex plane).
And in that description the photon's centre of mass still propagates linearly along the real line.
Even though the wavefunction has increasing spread over time.
@Secret As for embedding ω[1] in the reals, here's the easiest proof that just popped into my head.
There are ω[1] many pairs of consecutive ordinals in ω[1]; just take all the limit ordinals and their successors. There are ω[1] many of them otherwise their union will be a countable ordinal and hence less than some further limit ordinal.
If all these pairs embed into the reals, then the corresponding reals have each a rational between them, and all the sandwiched rationals are distinct.
It's still fun to ponder about that, given that in organic chemistry we do check reactivity trends as the carbon chain gets longer. If a plateau behaviour is observed, we can perhaps say that at the limit of infinitely long carbon chain, we get such and such value.
But that is just theoretical limits, thus as you say, no one can possibly make these things, unless humanity somehow encounter an infinite entity some time in the far future
As for nonlocal phenomenon, there are a lot of these in highly conjugated molecules where the electronic behaviour have huge correlations between two or more sites. Such behaviour are often exploited in catalysts to lower energy barriers
@Secret I don't think you're talking about the same thing. Yours seems to be still local. Non-local rules means rules regarding points at unbounded distances.
Notice that in my example each chain has to be precisely one longer than the previous one. That is non-local because each chain is longer and longer and so harder and harder to verify that it satisfies this property.
I see, yours is a much stronger notion of nolocality than it is used in physics and chemistry, sorry about that
Being a chemist, it is easy for me to somehow get tether to the real world in such subjects before going full abstract, hence the many speculations on how it might behave
As far as I can see, there does not seem to be truly non-local rules, unless you count the wavefunction equation, though even that is technically not non-local.
You can imagine that each part of the wavefunction is locally modified by the local environment.
If one has any causal phenomenon that span spacelike distance, it will break relativity, thus we are screwed in the real world way before we even reach countably far.
Nonlocal influence that does not signal however is possible, entanglment has no limit on how far the two subsystems are separated
My chemistry sense when looking at the ω hydrocarbon also told me something more: This molecule most stable conformation will basically cause it to become a globular structure.
We can roughly run an induction argument using a known rule in chemistry about steric clash, that is, two bulky groups are going to repel each other as far they could in the absence of other factors (our molecule is nice because it is a hydrocarbon, thus it is all vander waal's interaction and sterics).
Starting from the C=1 branch, we induct up in consecutive pairs, and we will find the steric clash is going to in…
Actually for such molecule, I predict it will be stable in theory, since the number of comformation you can gain for each increase in the number of carbons as you walk along the main carbon chain, will eventually dominate the energy of making such molecule
The most stable conformation of such molecule, however, will involve potentially countably many neighbours around almost every carbon atom, which means you need to transverse pretty much all the chains just to calculate the interacton energy at each carbon, thus your notion of nonlocality will be satisfied as expected.
Here's an easy version, consider a long chain of molecule like a noodle. Now let every atom of it wiggle around, you should find it is much easy for that to turn into coiled speghetti than to remain a long straight chain
Now our molecule is basically a noodle with a lot of bristles, thus when it wiggles around, taking care not to bump into itself, it is going to end up mostly into a coiled mess
Now when you try to ask how much repulsion and attraction each carbon felt, you need to take account of all its neighbours, which can be potentially countably many
O yeah, I was taking an abstraction assuming the carbon atom has no size, but then this will mean I contradict myself when I said it has to avoid bumping into itself
In that case, you are correct that you can pack only so many atoms around a carbon atom, thus at its most stable conformation, the interaction will be pretty much localised within a spherical region, despite the whole moelcule being unbounded in extent
The structure of the molecule (the details on how the carbon bonds are connected), however obeys a nonlocal rule as given by your reason above about harder and harder to verify the length of chains saatisfy the required property to be longer than the previous
Start with the reals $\Bbb{R}$, construct the irrationals $\Bbb{I}=\Bbb{R}\setminus \Bbb{Q}$
Now, pick one irrational, say $\pi$ and then construct the set $\pi_{\Bbb{Q}} = \{q\in \Bbb{Q} : \pi + q\}$
... ok gg, does not work, as for every such shift, there will be some irrational $r < \pi$ which then becomes $r > \pi$ thus ordering was flipped for these entries
Is there a bijecttion $f:\mathbb{Z}\rightarrow\mathbb{Q}$ s.t $f(x) < f(y)$ iff $x < y$
