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

In summary: I don't think so. You might not be aware of it, but CH is undecidable in ZFC, so any attempt to provide a finite amount of information for an element in ##\omega_1## would also imply that CH is either true or false.In summary, the amount of information required to specify an element in the countable ordinal ##\omega_1## is undecidable, as it would also provide an answer for the undecidable Continuum Hypothesis.
  • #1
tzimie
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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:

[itex]\aleph_0[/itex] - finite information
[itex]\aleph_{continuum}[/itex] - 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 = [itex]\aleph_2[/itex], so there is 1 cardinality between countable and continuum (I've heard that Goedel believed in it), and that cardinality is obviously [itex]\omega_1[/itex]

What information is required to completely specify an element in [itex]\omega_1[/itex] ?
 
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  • #2
tzimie said:
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:

[itex]\aleph_0[/itex] - finite information
[itex]\aleph_{continuum}[/itex] - 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 = [itex]\aleph_2[/itex], so there is 1 cardinality between countable and continuum (I've heard that Goedel believed in it), and that cardinality is obviously [itex]\omega_1[/itex]

What information is required to completely specify an element in [itex]\omega_1[/itex] ?
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.
 
  • #3
Rational numbers are real. Not all real numbers require an infinite number of digits.
 
  • #4
mathman said:
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.
 
  • #5
fresh_42 said:
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?
 
  • #6
tzimie said:
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.
 
  • #7
tzimie said:
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:

[itex]\aleph_0[/itex] - finite information
[itex]\aleph_{continuum}[/itex] - 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 = [itex]\aleph_2[/itex], so there is 1 cardinality between countable and continuum (I've heard that Goedel believed in it), and that cardinality is obviously [itex]\omega_1[/itex]

What information is required to completely specify an element in [itex]\omega_1[/itex] ?
If the cardinality of R is [itex]\aleph_2[/itex], then there is a subset A of R with cardinality [itex]\aleph_1[/itex]. 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.
 
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1. What is "Information and Cardinality"?

"Information and Cardinality" is a mathematical concept that focuses on the relationship between the amount of information available and the number of elements in a set or group.

2. How is "Information and Cardinality" used in science?

"Information and Cardinality" is used in a variety of scientific fields, including statistics, computer science, and biology. It is used to quantify and analyze the amount of information present in a given system or dataset.

3. What is the difference between information and cardinality?

Information refers to the knowledge or data that is available, while cardinality refers to the number of elements or objects in a given set or group. In other words, information is the content and cardinality is the quantity.

4. How does "Information and Cardinality" relate to data analysis?

"Information and Cardinality" is an important aspect of data analysis as it helps to measure and understand the amount of information contained within a dataset. This can aid in making informed decisions and drawing meaningful conclusions from the data.

5. Can "Information and Cardinality" be applied to real-life situations?

Yes, "Information and Cardinality" can be applied to real-life situations, such as analyzing customer data for a business, studying population trends, or understanding the genetic diversity in a species. It is a versatile concept that can be used in various contexts.

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