Mike's question....

Originally posted by StarValleyWy
Very good. I think I understand that there are an infinite number of reals between 1-2, 8-9 etc. I just boggle at the idea of ever arriving at a "terminus" as we approach "the speed of light" for example... Seems weird that in real life we can "always get there", yet in Plancks small world, and Einsteins big world, we can never get there. I have o ...[text shortened]... t "all known types" from the remaining Reals, what is left at the end of all map-and-removals?

Subtract rationals from reals, you get irrationals. Irrationals are just as frequent as reals. Try to work out why this is so, in view of what was said before.

Originally posted by royalchicken
Subtract rationals from reals, you get irrationals. Irrationals are just as frequent as reals. Try to work out why this is so, in view of what was said before.

i was actually thinking not of a mathematical "subtraction" operation, but a set removal... though i sense that it might be equivelent. Map a real to a rational, remove both from further consideration in the aggragate set under discussion. It probably equates to an exact 1 to 1 map, but i'm not sure.

Originally posted by StarValleyWy
i was actually thinking not of a mathematical "subtraction" operation, but a set removal... though i sense that it might be equivelent. Map a real to a rational, remove both from further consideration in the aggragate set under discu ...[text shortened]... on. It probably equates to an exact 1 to 1 map, but i'm not sure.

If you remove all the rationals, which are a countable set, from the uncountable set of reals, you are left with the uncountable set of irrationals I think. You can take any number of different countable sets away and you are still left with an uncountable set. So it doesn't really get you anywhere much.

To get to any countable set of reals you would have to remove an uncountable set from them. (Or remove an infinite (countable) number of countable sets).

Perhaps royale chicken will comment on that bracketted statement! 🙂

Originally posted by StarValleyWy
i was actually thinking not of a mathematical "subtraction" operation, but a set removal... though i sense that it might be equivelent. Map a real to a rational, remove both from further consideration in the aggragate set under discu ...[text shortened]... on. It probably equates to an exact 1 to 1 map, but i'm not sure.

Yeah...I was thinking of the same thing....removing all of the rationals from the reals will result in another uncountable set of irrational reals, as iamatiger correctly said. So there is no 1-1 map, because the reals are uncountable and the rationals are countable.

Iamatiger is correct in his bracketed statement. A argument to this effect from either of you would be cool...😏....(expectant smiley).

Could you explain how a set of rationals is countable? I thought we had agreed that there were infinitely many rationals. :😕

Originally posted by royalchicken
Yeah...I was thinking of the same thing....removing all of the rationals from the reals will result in another uncountable set of irrational reals, as iamatiger correctly said. So there is no 1-1 map, because the reals are uncountable ...[text shortened]... ffect from either of you would be cool...😏....(expectant smiley).

😕 Looks more "smug" than "expectant"! 😉
Let me think about it for a while... as you know, i am not much for higher states of math... Higher being "Check Book Balancing"...

Expectant woud be more like seeing a truck approaching you and your driver's ed teacher at 80 miles per hour, your car stalled in the middle of an intersection... like this... 😞😲😲😲

Originally posted by jot
Could you explain how a set of rationals is countable? I thought we had agreed that there were infinitely many rationals. :😕

Yes! Jot ! You have discovered the meaning of life and death and afterlife. You and me together with all other of the damned... Counting for 19.6 billion years... That is about what i would expect Hell to be like... And God Said... "You gotta start over if you lose your place!"... And the Damned rebelled and Burnt Hell To the Ground. And God was Pissed! "You Destroyed One Of My Favorite Toys, you buggers!" he said. Oh Oh! The Damned tried to hide, but haveing burnt the joint down, they had to just stand in a corner (which was hard to find in the unbounded universe in which they dwelled) for eternity. Without supper. And "No TV, Damnit"

Actually, I think "Count" would be better said "Account for Mathematically in a systematic matching"?... Nobody could "literally count", but could 'fell swoop' the buggers using the magic of math.

Originally posted by iamatiger
To get to any countable set of reals you would have to remove an uncountable set from them. (Or remove an infinite (countable) number of countable sets).

'A countable union of countable sets is countable.' I seem to remember having to prove that in an exam a couple of months ago. So if you remove countably many countable sets, what you're left with is still uncountable.

Originally posted by Acolyte
'A countable union of countable sets is countable.' I seem to remember having to prove that in an exam a couple of months ago. So if you remove countably many countable sets, what you're left with is still uncountable.

How did you do in the exams, if it is not a presumptuous question?

Originally posted by royalchicken
How did you do in the exams, if it is not a presumptuous question?

Well, I got a first, but I don't know much more than that.

Congratulations 🙂.

Originally posted by Acolyte
'A countable union of countable sets is countable.' I seem to remember having to prove that in an exam a couple of months ago. So if you remove countably many countable sets, what you're left with is still uncountable.

I think that perhaps if you express all the real numbers between 0 and 1 as decimals then you can cancel them out with a countable number of countable sets as follows:

The Nth countable set is the set of all possible N digit numbers. Any such set is obviously countable. To remove a countable set we put a decimal point before each number and remove all equivalent real numbers from the 0..1 grouping.

It seems to me that if we repeat this removal for N=1 to infinity then we will have removed all the reals in the range 0..1. The set of all the countable sets we used is also obviously countable as N is an integer.

I expect someone clever will immediately point out the flaw...

Originally posted by jot
Could you explain how a set of rationals is countable? I thought we had agreed that there were infinitely many rationals. :😕

By "countable set" I think we mean you can step through the elements of the set in some sequential manner and you get proportionally closer to the end. However I suspect that if the time for getting "1 notch" closer to the end increases exponentially then you have an uncountable set.

Originally posted by iamatiger
I think that perhaps if you express all the real numbers between 0 and 1 as decimals then you can cancel them out with a countable number of countable sets as follows:

The Nth countable set is the set of all possible N digit numbers. Any such set is obviously countable. To remove a countable set we put a decimal point before each number and remove all e ...[text shortened]... ntable as N is an integer.

I expect someone clever will immediately point out the flaw...

The reals cannot be counted with a countable number of countable sets. Each real consists of a string of digits (all 0's after a certain point if necessary). Thus each real is associated with a finite or countably infinite set of natural numbers. The reals can thus be put into 1-1 correspondence with the set of all subsets of the naturals. The set of all subsets of the naturals is the 'power set of N'. The power set of N is uncountable (not hard, bu interesting, to prove). Therefore, the reals can be expressed as uncountably many countable sets of natural numbers.

For more on this, look up the "Continuum Hypothesis" and the "Schroeder-Bernstein Theorem".

Originally posted by royalchicken
The reals cannot be counted with a countable number of countable sets. Each real consists of a string of digits (all 0's after a certain point if necessary). Thus each real is associated with a finite or countably infinite set of natural numbers. The reals can thus be put into 1-1 correspondence with the set of all subsets of the naturals. The set ...[text shortened]... For more on this, look up the "Continuum Hypothesis" and the "Schroeder-Bernstein Theorem".

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) is countable - as its members are directly mapped to the natural numbers.

Each S(N) is also countable, as it is easy to see that its entries can be arranged in numerical order. For instance S(2) is {01,02,03 .. 09, 11, 12, 13, ... 99) Each has 10^^N members.

The set SSS which is the union of all S(N) in SS can be shown to map to the real numbers between 0 and 1. The members of each S(N) map to the reals with N digits after the decimal point (not including trailing zeros).

Now, it may well be that SSS is uncountable (it better had be I suppose, as it maps to the reals). However SSS has been defined to be the Union of a countable number of countable sets.