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ly3xqhl8g9 parent
Some light, coffee reading "Cardinality of the continuum" [1]: in short, the cardinality of real numbers (ℝ) is often called the cardinality of the continuum, and denoted by 𝔠 or 2^ℵ_0 or ℶ_1 (beth-one [2); whereas, interestingly [3], the cardinality of the integers (ℤ) is the same as the cardinality of the natural numbers (ℕ) and is ℵ_0 (aleph-null) [perhaps what was meant initially?].

Related: the Schröder–Bernstein theorem [4], "if there exist injective functions f : A → B and g : B → A between the sets A and B, then there exists a bijective function h : A → B.".

Not related, but great: Max Cooper (sound) and Martin Krzywinski (visuals) did a splendid job visualising "ℵ_2" [5].

[1] https://en.wikipedia.org/wiki/Cardinality_of_the_continuum

[2] https://en.wiktionary.org/wiki/%E2%84%B6

[3] "Cardinalities and Bijections - Showing the Natural Numbers and the Integers are the same size", https://www.youtube.com/watch?v=kuJwmvW96Zs

[4] https://en.wikipedia.org/wiki/Schr%C3%B6der%E2%80%93Bernstei...

[5] "Max Cooper - Aleph 2 (Official Video by Martin Krzywinski)", https://www.youtube.com/watch?v=tNYfqklRehM

hegzploit
adding upon this comment to why the two cardinalities are not equal, on one hand we have the set of integers {..., -2, -1, 0, 1, 2, ...} and they can be put into a bijection with the set of natural numbers {1, 2, 3, 4, ...}, this is done by rearranging the set of integers like {0, -1, 1, -2, 2, -3, 3, ...}. so this is a countably infinite set (one that has a cardinality of ℵ_0)

As for the set of real numbers, we have the subset of irrational numbers which are uncountably infinite (see cantors diagonalization argument) thus making the whole set of real numbers, a set whose cardinality is ℵ_1.

The annotated turing book goes into this pretty well in the first couple pages.

orra
Quite. Then there is the question, is the cardinality of the continuum the first cardinality bigger than the cardinality of the naturals?

It turns out the 'continuum hypothesis' can be true or it can be false. Neither contradicts standard ZFC set theory: the hypothesis is 'independent'.

ly3xqhl8g9 OP
One way to think about it would be to replace or with and: the continuum hypothesis can be true and false: it is a 'polycomputational object' [1].

[1] Using the concept of polycomputing from There’s Plenty of Room Right Here: Biological Systems as Evolved, Overloaded, Multi-Scale Machines: "Form and function are tightly entwined in nature, and in some cases, in robotics as well. Thus, efforts to re-shape living systems for biomedical or bioengineering purposes require prediction and control of their function at multiple scales. This is challenging for many reasons, one of which is that living systems perform multiple functions in the same place at the same time. We refer to this as 'polycomputing'—the ability of the same substrate to simultaneously compute different things, and make those computational results available to different observers.", https://www.mdpi.com/2313-7673/8/1/110

orra
Interesting, that's not a concept I have come across before. But to be honest, I wasn't sure which conjunction to use (and, or or).
ly3xqhl8g9 OP
Here is Michael Levin, one of the paper's author, speaking at length about the polycomputing concept and more: "Agency, Attractors, & Observer-Dependent Computation in Biology & Beyond" [1].

[1] https://www.youtube.com/watch?v=whZRH7IGAq0

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