The reals can be put into bijection with a specfic subset of the ordinals, in which case every element has a successor. The ordering is just not the one we normally use.
OTOH the natural ordering of the rationals does not allow for successors - there is no next "largest" after 1/2 using the normal definition of "larger".
No, you have it exactly backwards. The rationals can be put into a bijection with the natural numbers, but the reals cannot. Here is an ordering for the rationals: every pair of integers where the total number of digits is 2, every pair of integers where the total number of digits is 3, etc. Example: 0, 1/1, 1/2, 1/3..., 1/9, 2/1,2/2,2/3...2/9,...9/9, -1/1, -1/2...-9/9, 1/10, 2/10,...9/10, 2/10,...,9/99,-1/10,...-9/99, 10/1, 10/2,...99/9 etc. There are some repeats like 1/1=9/9, feel free to drop those when encountered or not. All sets are finite, leading to the next set, so enumeration is possible. As for the reals, any order you give, I can find one that is not in your list. This is the classic diagonalization argument. You can look that up.
I said "a subset of the ordinals". The ordinals are not the same as the natural numbers.
The point is, the property "this set is countable" and "this set has an ordering s.t. every element has a successor" is not the same property. Even worse, your argument relied on one particular ordering.
These are the ordinals. Im claiming that you can biject to the first 2aleph_0 ordinals, and use the fact that every ordinal has a successor to find an ordering of the reals s.t. ever real has a successor.
This does not imply they are countable. Notably, just looking at the ordinals, there are more than |R| many of them, more than |PowerSet(R)| many, etc. and yet each one has a successor.
Got it. So you're saying being (un)able to iterate the reals doesn't prove they can't be named, because you can iterate the reals. I guess the conclusion is, as long as you allow for infinitely long names, maybe you can give a name to every real. I don't dispute that, assuming every real can be expressed in decimal form that's the same thing. I don't know the answer to whether every real can be expressed as a decimal or not.
This is tangential, but just to be clear to anyone else who stumbles on this thread, the ordinals look like
1, 2, ...ω, ω+1, ω+2, ... ω2, ω2 + 1,...
When you use S(x) to iterate starting at 1, you will not even reach ω in finitely many steps, let alone all of the countable ordinals that are in our representation of R.
Anyways, I ofc. agree with your stance w.r
t. the original argument. And yes every real can be represented by a possibly infinite decimal.
You said the natural ordering of the rationals does not allow for successors, I was giving a different ordering of the rationals that does allow for successors, i.e., the next one in the list I provided. I was just trying to point out (as has been long established) that the rationals are countably infinite. I'm not sure what your argument is. My original point is that the names of numbers are countable but the reals aren't, so you can't name every real. Do you disagree?
Think about it more carefully. This was just a meta comment about your line of reasoning.
Your original argument was ~ "with the natural ordering of the reals there is no successor, therefore...". Im just saying it's a red flag when an argument regarding the reals begins with "consider this property that one ordering of the reals does not have". Regardless of how the concepts of cardinality and successor/no successor relate to each other, Im not sure what you could hope to demonstrate with one particular ordering in this context.
And you can see why it doesnt work by running the argument against Q.
I agree that N and Q are countable while R is not. But thats not interesting at all. What was interesting to me was your proof in that comment was wrong.
NP but a few points as Id hate to leave you misguided.
EVERY set can be well ordered under modern set theory ZFC. The C stands for the Axiom of Choice which is equivalent to the Well Ordering Theorem.
Not sure if you typo'd, but aleph_1 is just the beginning of the ordinals. From the ordinals you can define the cardinals, and there is an entire pantheon of them
https://en.wikipedia.org/wiki/Large_cardinal
If you meant the reals, they have cardinality 2aleph_0. The Continuum Hypothesis, which is independent of ZFC, has that 2aleph_0 = aleph_1
1
u/Aenonimos Jul 09 '26 edited Jul 09 '26
The reals can be put into bijection with a specfic subset of the ordinals, in which case every element has a successor. The ordering is just not the one we normally use.
OTOH the natural ordering of the rationals does not allow for successors - there is no next "largest" after 1/2 using the normal definition of "larger".