An intuition I had about infinite numbers -- Is it valid?

[mr3000]

There are an infinite number of numbers. You can take those infinite numbers and slice them into an infinite number of infinite slices each of which can be sliced the same way ad infinitum.

 

[anuttarasammyak]

Yes. For an example, let p, q, r be prime numbers. There are infinite number of prime numbers. Infinite number of sets $$\{p^1,p^2,p^3,...\}$$ are disjoint infinite subsets of $$\{1,2,3,...\}$$. Infinite number of sets
$$\{p^{q^1},p^{q^2},p^{q^3},....\}$$ are disjoint infinite subsets of $$\{p^1,p^2,p^3,...\}$$. Infinite number of sets $$\{p^{q^{r^1}},p^{q^{r^2}}.p^{q^{r^3}},...\}$$ are disjoint infinite subsets of $$\{p^{q^1},p^{q^2},p^{q^3},....\}$$.......
We can continue this way infinite times.

 

[FactChecker]

There are an infinite number of numbers. You can take those infinite numbers and slice them into an infinite number of infinite slices each of which can be sliced the same way ad infinitum.

Sure. Conceptually, you can divide the whole into 2 sets, assigning them alternately to one of the sets. Then you can pick one of those sets and divide it similarly into two infinite sets. Then take one of those ... etc.
There are simple ways to avoid getting hung up on any one infinite partitioning.