Is it ever physically possible for a broken egg to reassemble into an unbroken one?

[polymath257]

I think you might find it more difficult than you imagine to get odds of 1 in a googolplex.

So, for example, the radius of the observable universe is about 13 billion light years

46.5 billion light years last I checked.

13 billion is the farther distance we can see. 46 billion is the current distance of those things that took 13 billion years for the light to get to us.

The difference, one this scale, isn't particularly relevant: maybe another factor of 100.

which is around 10^26 meters, or 10^38 femto-meters.

So, the number of cubic femtometers in the observable universe is around 10^114.

The number of fundamental particles in the universe is around 10^80, so the odds that the specific arrangement of particles in the space of the universe (up to femtometer accuracy) is about (10^114)^(10^80), which is less than 10^(10^83). This is assuming the position of each particle is independent of every other particle. This is *far* less than a gogolplex.

Now, the odds for every fundamental particle in the universe *randomly* and independently happening to be in the specific cubic femtometer they are, independently for each femtosecond n a second, would be less than (10^10^83)^(10^12), which is about 10^10^95. This is still far smaller than a googolplex.

In fact, one in a googolplex would be worse odds than the odds of every particle in the universe randomly and independently being in the precise cubic femtometer, independently for each femtosecond in 100,000 years.

So, no, we do NOT see events with a lower probability happening every instant of our lives.

Not when you math it like that, no. However:

- Femtometer resolution is ridiculously coarse right out of the gate. You won't even be able to predict if two protons are on course to fuse into a deuteron or not at that resolution.

- You've neglected to account for all the various different properties that each of those fundamental particles can have. Those are going to be important.

- The whole mess is iterative, which means that each ridiculously improbable configuration follows from an equally preposterous configuration. That means that if you don't have extremely good resolution the errors compound very quickly. Given that the large-scale structure of the universe originated as quantum fluctuations during the universe' inflationary era you're probably going to need Planck-scale resolution in all four dimensions.

If you want to know the probability of these exact sodium ions being pumped in and out of my neurons, to send exactly these electrons and photons bouncing around the pinball game we call the internet, to eventually light up exactly these atoms in your display, trigger precisely these opsins, and fire exactly these sodium-ion gates in your neurons, yadda, yadda, yadda, blah, blah, blah... requires calculating all the intermediate and preceding states and their improbabilities. If just one proton had been a bit to one side rather than the other 10 billion years ago then stars in our distant history would have blown themselves apart very slightly differently, our constituent atoms would never have been formed, and somebody very different would be having a very different conversation.

And if you do Planck length accuracy that 10^38 turns into 10^70, so the volume turns into 10^210. The approximation I made at that point put the volume less than 10^1000.  The rest was taken into account in my calculation. Yes, that 1 proton difference was taken into account by my raising things to the power of the number of particles in the universe. Each different arrangement would be different and I picked out the one that it actually is.

The other properties, again, add a few to the exponent but won't get it up to a googol.

PS: We *do* see events with probabilities lower than 1 in a googol every instant. But a googolplex is much, much, much larger than a googol.

Yes, you can beat a 1 in a googol if you have $3.33 in pennies and that's doing it the lazy way. Or a couple well-shuffled decks of cards.

Again, like I said, we see odds of 1 in a googol all the time. We do NOT see odds of 1 in a googolplex all the time. Remember that a googolplex is 10^googol and a googol is a big number compared to the size of the universe or the number of particles in it.

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I thought about this a bit more and it is wrong. 4^^4 is about 10^10^154. A number with 50% more digits than a googolplex would be about
10^(1.5*10^100), so far less than 10^10^101 (which would have 10 times as many digits as a googolplex). The number 4^^4 is about a googolplex to the power of 10^50, so (googolplex)^\sqrt(googol)

By the way, 10^^10 is called Decker.

Why are people so impressed by the number of fingers on their hands? 9^^9 is so much more elegant.

If you want to be hateful you just take one complex irrational number and raise it to the power of another complex irrational. Or to the superpower if you want to get really nasty.

Or, instead of tetration, you can do higher order operations. Say, 4^^^^4, or 9^^^^^^^^^9. Those are both MUCH larger.

But the googology site still considers those to be *small*.

I'm not sure why you think a complex irrational to the power of a complex irrational has to be large at all. For example, e^(i\pi) =-1, famously.