Quick question

[ughaibu]

Anything that can be known can be named, so the most things that can be known is aleph null. But some unknown things can not be named, does this mean that the number of unknowable things is aleph one?

[Qfwfq]

Well it depends on the number of things. Let's call this number [imath]\cal{T}[/imath]. Now, first of all, I agree the number of known things can't exceed and could hardly reach [imath]\aleph_0[/imath] but that isn't the number of knowable things, which will most certainly be greater than the number of known things. Anyway if we call the number of knowable things [imath]\cal{K}[/imath] we can say there are [imath]\cal{T}-\cal{K}[/imath] unknowable things.

[ughaibu]

Qfwfq: Thanks and sorry about my inconsistent wording.

Anything that can be known can be named ie any knowable thing (once known) can be named, so the number of knowable things cant exceed aleph null.

Some unknowable things can not be named (due to them being unknown), so the number of unknowable things cant exceed aleph one.

Question 1) is the cardinality of the set of unnameable things aleph one, by convention, or does this need to be established?

Question 2) if the number of unnameable things is aleph one and if it can be shown that the number of both knowable and unknowable things is infinite, are we in a position to say that the number of unknowable things is uncountably infinitely greater than the number of knowable things or can we say no more than that the number of unknowable things is at least as great as the number of knowable things?

[Qfwfq]

First, you might gather from my previous post that I'm not sure of [imath]\aleph_0=\cal{K}[/imath], while it's trivial that [imath]\aleph_0\le\cal{K}[/imath]. Must it be possible to know all knowable things?

 

Question 1) is the cardinality of the set of unnameable things aleph one, by convention, or does this need to be established?

If it is possible to define a cardinality greater than [imath]\aleph_1[/imath] then surely it will be possible to construct a set having that cardinality?

 

Question 2) if the number of unnameable things is aleph one and if it can be shown that the number of both knowable and unknowable things is infinite, are we in a position to say that the number of unknowable things is uncountably infinitely greater than the number of knowable things or can we say no more than that the number of unknowable things is at least as great as the number of knowable things?

It depends on exactly which the two infinite cardinalities are.

[ughaibu]

Thanks for the reply. Interesting, I'll give it some more thought.