What is the required amount of information to specify an element in \omega_1?

[tzimie]

To select an element from countably infinite set (list set of integers) you need to provide finite amount of information. To specify an element in continuum in general case you have to provide infinite amount of information: any real number is specified as countable-infinite number of digits. So here is a pattern:

\aleph_0 - finite information
\aleph_{continuum} - countably infinite number of digits,

So the amount of information is 1 degree less than the cardinality of set.

Now, let's deny continuum hypotesis and let's assume continuum = \aleph_2, so there is 1 cardinality between countable and continuum (I've heard that Goedel believed in it), and that cardinality is obviously \omega_1

What information is required to completely specify an element in \omega_1 ?

 

[fresh_42]

To select an element from countably infinite set (list set of integers) you need to provide finite amount of information. To specify an element in continuum in general case you have to provide infinite amount of information: any real number is specified as countable-infinite number of digits. So here is a pattern:

\aleph_0 - finite information
\aleph_{continuum} - countably infinite number of digits,

So the amount of information is 1 degree less than the cardinality of set.

Now, let's deny continuum hypotesis and let's assume continuum = \aleph_2, so there is 1 cardinality between countable and continuum (I've heard that Goedel believed in it), and that cardinality is obviously \omega_1

What information is required to completely specify an element in \omega_1 ?

Shouldn't it be $\aleph_1$ instead of $\omega_1$ by its definition? But anyway. Whether you put it in a pseudo information theory language, or leave it where it belongs to, namely ZFC, makes no difference. Since it is undecidable in ZFC, it stays as such in all other languages based upon ZFC.

 

[mathman]

Rational numbers are real. Not all real numbers require an infinite number of digits.

 

[tzimie]

Rational numbers are real. Not all real numbers require an infinite number of digits.

yes, but in general case you have to provide an infinite number of digits
most of reals are random numbers.

 

[tzimie]

Shouldn't it be $\aleph_1$ instead of $\omega_1$ by its definition? But anyway. Whether you put it in a pseudo information theory language, or leave it where it belongs to, namely ZFC, makes no difference. Since it is undecidable in ZFC, it stays as such in all other languages based upon ZFC.

Thank you for the correction.
But then question "what amount of information is required to specify a countable ordinal?" (which is an element of $\omega_1$ ) must be also undecidable, because answering it you also provide one or another "answer" to CH?

 

[fresh_42]

Thank you for the correction.
But then question "what amount of information is required to specify a countable ordinal?" (which is an element of $\omega_1$ ) must be also undecidable, because answering it you also provide one or another "answer" to CH?

Yes, only that it can't be another answer. Assuming CH were decidable in some axiomatic system, then it would have to be different from ZFC. However, information theory isn't.

 

[Zafa Pi]

To select an element from countably infinite set (list set of integers) you need to provide finite amount of information. To specify an element in continuum in general case you have to provide infinite amount of information: any real number is specified as countable-infinite number of digits. So here is a pattern:

\aleph_0 - finite information
\aleph_{continuum} - countably infinite number of digits,

So the amount of information is 1 degree less than the cardinality of set.

Now, let's deny continuum hypotesis and let's assume continuum = \aleph_2, so there is 1 cardinality between countable and continuum (I've heard that Goedel believed in it), and that cardinality is obviously \omega_1

What information is required to completely specify an element in \omega_1 ?

If the cardinality of R is \aleph_2, then there is a subset A of R with cardinality \aleph_1. The elements of A can't be specified with a finite amount of information for that would imply A is countable. But the elements of A can be specified by a subset of those that specify R, so the answer is countable.

Given what everyone else has said I must have stepped in some pile of logical poop.