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

[polymath257]

And if the probability is 1 in a googolplex, we don't *expect* to see actual examples in the current age of the universe (very far from it). But the probability is still non-zero.

If every fundamental particle in the universe was doing addition every Plank's time, the total computed since the Big Bang would be less than 10^135. This is *far, far, far* less than a googolplex.

I'll go further. If every fundamental particle added a total of a googol every Plank time, the total since the Big Bang would still be smaller than 10^235, which is still far, far, far less than a googolplex.

What do you think of ultimate reality being akin to Cantor's infinities within infinities, perhaps, our time and space being countably infinite sets within a infinitude of other countably infinite sets, across both time and space?

Well, we usually model things in quantum theory as operators on a Hilbert space of countably infinite dimension. The space itself is uncountable.

A countably infinite number of countably infinite sets only means countably many points. But the real line is an uncountable set, so it is much larger.

Generally speaking the cardinality of a set is far less important than the other structures put on that set (say, a metric, or a vector space structure). Cardinality is a very crude measure of the size of a set in most cases.

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In practice, people tend to be exceptionally bad at properly examining the probability space. Probabilities lower than 1 in a googolplex happen every instant of our lives but we fail to recognize them because of the stochastic nature of the universe that we inhabit. In the instance of the egg the rational course of action is not to start worshipping Gawd AllMighty Mender Of The Yolk but rather to look for the Gallifreyan pankster who has been unscrambling your omelettes.

On a side note, googol and googolplex have always failed to impress me. They're stunt numbers based on the number of fingers on your hands. If really big numbers is all you want then 4^^4 is a bit better than 50% more digits than a googolplex and 9^^9 should be more than sufficient to tie up any computer from now until the end of time.

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, 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.

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.