The infinitely large and the infinitely small

In physics we cannot extrapolate Planck's length to even smaller quantities, but in mathematics there are no limits to our imagination. Just as there is no known quantity smaller than Planck's length, so it is not known if there are more than 10100 atoms in our observable universe. In our imagination it's okay to conceive numbers like 101000 and 10-1000. What's more, the product of these two is equal to 1! This is the interesting part. There is a remarkable multiplicative structure that we will analyze in more detail now.

Symmetry in Mathematics V

There are astronomical mathematical models that predict the existence of a galaxy at a distance of about meters from here. It is very difficult to imagine this "reality". Even more difficult is to imagine that in this galaxy there are specimens of the homo sapiens just like us!

The authors of a February 2004 article in the Brazilian Scientific American say that the truth of these mathematical theories about the universe derives only from the elementary laws of probability. If the universe is "infinitely large," then there is a place where the most unlikely event occurs. Any combination of choices for our lives is experienced by others like us somewhere in the "infinitely large" universe.

No one will be able to meet your copies, as we can only see points no more than 42 billion light years away. Thus the farthest visible objects are at a distance of 4'1026 meters. It is our observable universe, also known as the hubble volume.

Our copies are also on planets located on Hubble balls, that is, balls that have the Hubble volume, or balls with a radius of 4 '10.26 meters.

The above imagination is only part of a variant of multiverse theories. What interests us here is the numbers with which these theorists work. Multiverse theories have four levels. At level 1, there are conclusions as follows:

"about meters from here there is a Hubble ball exactly like ours… ”

How does this number come up? Possible quantum states for a Hubble ball whose temperature does not exceed 108 degrees Kelvin. Question: How many protons can fit in a Hubble ball at a temperature of 108 degrees Kelvin? Answer: 10118 protons. As such a particle may or may not be present, the total proton array is . Therefore, this amount would exhaust all possibilities of Hubble balls. These balls would fill a distance of approximately meters. Beyond this distance the universes, that is, Hubble's balls, would necessarily begin to repeat themselves!

Level 1 of multiverse theories suggests that the universe (that is, all Hubble balls) is “infinitely large”…

What if we think in the opposite direction? What if we think symmetrically of the possible distances and ask what is the shortest possible distance? Well, the symmetry seems to break: we will not go less than Planck's distance of 10-33 cm!

What if we ask the same question about time? Time also seems to have a lower bound, at least as revealed by the mathematical theory of Loop Quantum Gravity, or Quantum Space-Time Theory, which proposes that space and time cannot be smaller than certain values. For time there is the lower limit of 10-43 seconds

So the lower bounds are 10-33 cm for distances, so 10-66 cm2 for areas, 10-99 cm3 for volumes and 10-43 seconds to time.

Thus, the Loop Quantum Gravity Theory brings an asymmetry between the big and the small in physics. However, we saw in the previous columns that a beautiful symmetry between large and small real numbers is natural. It seems that the "infinitely large" in physics has no limitations while the "infinitely small" has the above limitations.

Perhaps it was a case of speculating a little further on the assumption that if physical reality were symmetrical in terms of quantities, analogous to positive real numbers, then what would exist would be 'nothing'. By breaking the perfect physical symmetry between big and small, Hubble's balls appear. As Georg Cantor said,

the essence of mathematics is its freedom… ”.

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