NON-well-orderability vs. NON-enumerability

[stevnewb]

[/size][size=medium](CONJECTURED?) THEOREM: The class of OMEGA-random binary sequences is not capable of being well-ordered. (1)

Putative proof:  Suppose the contrary.  Then let M(i, j) be the representation of such a well-ordering of all the OMEGA-random binary sequences in the open real interval, (0, 1), where  i,j  are indicial numerals, not necessarily belonging to any countable ordinal.  Consider the "Cantorian Diagonal", represented as M(k,k) where
k = i = j:  Assuming that M(k,k) is necessarily also a OMEGA-random binary sequence, the fact that the complement M'(k,k) differs from each element of M(i,k) in at least one component, hence, as defined, M'(k,k) is not included in M(i,j), and since the complement of any OMEGA-random binary sequence is another OMEGA-random binary sequence,  we have the sought-for contradiction which proves the theorem.  QED

Comment:  Since any consideration of the cardinal-magnitude of the index set was explicitly excluded in the statement of assumptions, it follows that the attribute of NON-well-orderability is LOGICALLY PRIOR to the attributes of countability/uncountability.

With that in mind, consider the contra-factual hypothesis under which the researches of Professors Gregory Chaitin, Per Martin-L\"ov, and Robert Solovay (establishing the existence and critical characteristics of OMEGA-random binary sequences) might have preceded Cantor's work: Then the (also contra-factual) hypothesis that the above given result might have also preceded Cantor's proof of the uncountability of the reals in (0, 1) would raise the (presently?) open question of whether there is any necessary correlation between the attributes of NON-well-orderability and NON-denumerability.

Quoting the last sentence of Henkin's doctoral dissertation, "But perhaps this is only Philosophy?"

Reference (1): G. J. Chaitin, "Algorithmic Information Theory", Cambridge University Press 1987.


Any comments?

[bo198214]

Hello Stev and wellcome on board,

before continuing can you just say what Omega-random binary sequences are? I could not find references in the web and didnt had time yet to read the reference given by you.

[bo198214]

stevnewb Wrote:(CONJECTURED?) THEOREM: The class of OMEGA-random binary sequences is not capable of being well-ordered. (1)

Putative proof:  Suppose the contrary.  Then let M(i, j) be the representation of such a well-ordering of all the OMEGA-random binary sequences in the open real interval, (0, 1), where  i,j  are indicial numerals, not necessarily belonging to any countable ordinal.  Consider the "Cantorian Diagonal", represented as M(k,k) where
k = i = j:  Assuming that M(k,k) is necessarily also a OMEGA-random binary sequence, the fact that the complement M'(k,k) differs from each element of M(i,k) in at least one component, hence, as defined, M'(k,k) is not included in M(i,j), and since the complement of any OMEGA-random binary sequence is another OMEGA-random binary sequence,  we have the sought-for contradiction which proves the theorem.  QED

The problem with the proof is that i and j in M(i,j) are possibly of different ordinals (j is of but i can be of a bigger ordinal). But M(k,k) can be taken for every k, only if i and j belonged to the same ordinal.

[stevnewb]

bo198214 Wrote:Hello Stev and wellcome on board,

before continuing can you just say what Omega-random binary sequences are? I could not find references in the web and didnt had time yet to read the reference given by you.

Google "Gregory Chaitin"

[stevnewb]

bo198214 Wrote:

stevnewb Wrote:(CONJECTURED?) THEOREM: The class of OMEGA-random binary sequences is not capable of being well-ordered. (1)

Putative proof:  Suppose the contrary.  Then let M(i, j) be the representation of such a well-ordering of all the OMEGA-random binary sequences in the open real interval, (0, 1), where  i,j  are indicial numerals, not necessarily belonging to any countable ordinal.  Consider the "Cantorian Diagonal", represented as M(k,k) where
k = i = j:  Assuming that M(k,k) is necessarily also a OMEGA-random binary sequence, the fact that the complement M'(k,k) differs from each element of M(i,k) in at least one component, hence, as defined, M'(k,k) is not included in M(i,j), and since the complement of any OMEGA-random binary sequence is another OMEGA-random binary sequence,  we have the sought-for contradiction which proves the theorem.  QED

The problem with the proof is that i and j in M(i,j) are possibly of different ordinals (j is of but i can be of a bigger ordinal). But M(k,k) can be taken for every k, only if i and j belonged to the same ordinal.

Yup.  
I should have specified that  i, j are elements of

Thanks for response.