It is not true that according to set theory all logically possible (consistent) collections exist. First, it's not even clear how that would be stated...
If you think the continuum hypothesis is false, then you think there is a set with cardinality strictly greater than the cardinality of N and strictly...
No. The way you wrote it is wrong. The continuum hypothesis is that the cardinality of the set of reals is aleph_1. This point keeps getting lost. Don...
thm: n is a natural number <-> (n is finite & n is an ordinal) dfn: card(x) = the least ordinal k such that x is 1-1 with k dfn: c is a cardinal <-> t...
I addressed that about half a dozen times in posts above. The statement "aleph_1 is the cardinality of the set of real numbers" is the continuum hypot...
Yes, x is a natural number iff x is a finite cardinal. And aleph_1 is not a finite cardinal. And the poster is asking about finding a certain natural ...
aleph_1 is the least cardinal greater than aleph_0. That is the case by definition. "aleph-1. The the infinite cardinal of the real numbers" That is t...
When you use reductio ad absurdum, you construct a denumerable binary sequence not in the range of f, which contradicts the assumption that f is a bij...
Here's the argument, which is not reductio ad absurdum: Let f be a function from the set of natural numbers to the set of denumerable binary sequences...
That is not the axiom of infinity. The statement that for any natural number n there is a set with n members is a trivial theorem, not even an axiom. ...
Not merely a lack of rigor. Rather, your statement "the continuum hypothesis is that from aleph_0 the next is aleph_1" is plainly false. There is no q...
By the way, I did not say that the cardinality of the set of real numbers is aleph_1. I said that "The cardinality of the set of real numbers is aleph...
It seems that you wish not to recognize that: is the continuum hypothesis and that is incorrect, since "from aleph_0 the next aleph is alelp_1" is not...
As to the other poster, the current question of the continuum hypothesis does not stem from the definition of 'countable'. Rather, the current questio...
Yes, the continuum hypothesis is about the first two infinite cardinals. Meanwhile, what I said stands: That is the continuum hypothesis. In other wor...
It is correct that the next aleph after aleph_0 is aleph 1. That follows trivially from the definition of the alephs. Since the alephs are indexed by ...
card(reals) = aleph_1 is the continuum hypothesis. It is not provable in ZFC. It is thought to be true by some mathematicians and false by other mathe...
It's not about going into the meanings of "appearance", "filtered way", "unfiltered way", or "our action". Rather, it's about the logical form, no mat...
What does that mean? Do you mean that there is no sentence that is true in all models? But there are sentences that are true in all models. No, there ...
You haven't shown that my answer is incorrect. Nor has anyone said what other answer is "the" correct answer. The problem that deserved an answer (in ...
And your criticism is belied by the fact that the poster himself explicitly said that my answer was clear and helpful, and his followup questions do s...
That's your informal understanding. I can't comment with real definiteness, because your informality doesn't provide a clear, definite meaning of 'qua...
(1) I don't know exactly which post(s) and passage(s) he is saying "no" to. (2) I understood everything in that thread prior to his post. And, modulo ...
And my rigorous, mathematical and standard use and explanations are not refuted (or whatever your disagreement is supposed to be) by your own informal...
Why are you exclaiming that to me? Of course I agree. There is no class of all groups in set theory. Set theory has only classes that are sets. In cla...
For example, in set theory, we have a defined predicate 'is an natural number' (I'll use 'B'). So theorems such as: Ax(Bx -> (x is even or x is odd)) ...
The question was: "all inclusive in one way or another" is not definite. My choice was to give a mathematically definite framework for the the questio...
There also is the notion of proper classes as models, or more specifically, inner models. However, I think (I am rusty on this) that when we state thi...
Right. But even with those theories, the domain of discourse for a model for the language of the theory is a set. Even a class theory such as NBG has ...
Not formally. Formally, any model of set theory has as a set, not a proper class, as its domain of discourse. For any model, the universal quantifier ...
Your question is insightful. You're thinking along the right lines. But there is no set of all things not in the domain of discourse. We take only rel...
Your neck of the woods is a fantasy place. People in everyday life don't take 'countable' to mean "one could, in principle only, count an infinite ser...
No one doubts that you are a grown up person who can do numerical reckoning just fine. The point is that you don't know anything about the mathematica...
It should not still be needed to say: The notion that some infinite sets are countable is a special mathematical notion. It is technical and has a rig...
Oh come on! How captious can a person get? 'countably infinite' and 'countable infinity' are tantamount to each other. 'countable infinity' though is ...
No, I clearly see what you actually posted. In earlier posts, you mentioned that there is a layman's notion of countably infinite. Then later you aske...
It doesn't invite mathematical proof because 'actual' and 'potential' are not mathematically defined terms. Meanwhile, formal set theory does not use ...
That is such an inpenetrable mess that it would be a task to unsort it all. But a couple of points: If we said that there are half as many even number...
I don't have "a magical view of how mathematics is physically done", so you can leave me out. But I thought you might have some particular mathematici...
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