Mike's question....

Originally posted by iamatiger
That seems to me to be an assertion of some fact rather than an identification of a flaw in my argument. I'll try to put my argument more clearly.

If a set S(N) is the set of all combinations of N digits (from 0 to 10), excluding combinations with trailing zeros, then, if the allowed values of N are the natural numbers SS, which is the set of all S(N) ...[text shortened]... e reals). However SSS has been defined to be the Union of a countable number of countable sets.

The flaw is that your SSS does not map to the reals, I'm afraid. Reals do not necessarily have some number N of digits, but rather an actual denumeable infinity of digits. It may be clear to try and work out for yourself the proof of the following:

"Any countable union of countable sets is countable, or equivalently, no uncountable set is a countable union of countable sets."

If you just start going ahead in a general way, the specifics of why your above argument fails can be made apparent. I'd be glad to look at abything you say here...this is quite an interesting thing 🙂.

The reals are the union of an uncountable number of countable sets, or the union of a countable number of uncountable sets. They are the union of all possible subsets of the naturals (first case), or equivalently the union of all continuous intervals.

Originally posted by royalchicken


The reals are the union of an uncountable number of countable sets, or the union of a countable number of uncountable sets.

What a wonderful image that brings up... I had never thought of that equivelency! But there is a lot in math I have never thought of... The first instance (union of uncountable number of countable sets) ... I see "the DNA structure of all possible life forms"... The second I see <Edit... I didn't like what I first posted. Let me get back to you on the second, ie, 'the union of a countable number of uncountable sets'> My "vision" of this comes with a lot of poetic license of course. Very interesting. Keep it going as long as there is interest. I am learning more about set theory and unions than i would have thought possible. I still have to "abstract", or "draw forth from a form in order to envision a familiar". But that is better than nothing. Just the way my brain works.
😠 Or Doesn't😕

Originally posted by iamatiger
That seems to me to be an assertion of some fact rather than an identification of a flaw in my argument. I'll try to put my argument more clearly.

If a set S(N) is the set of all combinations of N digits (from 0 to 10), excluding combinations with trailing zeros, then, if the allowed values of N are the natural numbers SS, which is the set of all S(N) ...[text shortened]... e reals). However SSS has been defined to be the Union of a countable number of countable sets.

SSS corresponds to some of the rational numbers in [0,1] (eg there's no member corresponding to 1/3), and none of the irrational ones. You can let N range over the naturals, but there's no such thing as S(?).

Originally posted by royalchicken
The flaw is that your SSS does not map to the reals, I'm afraid. Reals do not necessarily have some number N of digits, but rather an actual denumeable infinity of digits. It may be clear to try and work out for yourself the proof of the following:

"Any countable union of countable sets is countable, or equivalently, no uncountable set is a count ...[text shortened]... ble subsets of the naturals (first case), or equivalently the union of all continuous intervals.

Thanks. In the light of day I see that you are saying my method only maps to reals with a finite number of digits followed by trailing zeros. I think the flaw is that there it leaves out an uncountable number of reals with an infinite number of digits and no trailing zeros.

Originally posted by iamatiger
Thanks. In the light of day I see that you are saying my method only maps to reals with a finite number of digits followed by trailing zeros. I think the flaw is that there it leaves out an uncountable number of reals with an infinite number of digits and no trailing zeros.

Yes; also read Acolyte's post carefully. I'm quite glad we dragged him in.

Originally posted by StarValleyWy
What a wonderful image that brings up... I had never thought of that equivelency! But there is a lot in math I have never thought of... The first instance (union of uncountable number of countable sets) ... I see "the DNA structure of all possible life forms"... The second I see <Edit... I didn't like what I first posted. Let me get back to you on t ...[text shortened]... n a familiar". But that is better than nothing. Just the way my brain works.
😠 Or Doesn't😕

Good good. Just keep thinking about stuff like this. Use whatever analogies you can devise to make it make sense.

Acolyte, maybe when you're doing the tutoring you mentioned, take a hint from StarValleyWy here, and think of some colorful analogies to illustrate your points.