The discussion revolves around the possibility of establishing a 1-to-1 correspondence between any two real numbers within the interval [a,b]. Participants explore the implications of such a correspondence, its mathematical definitions, and the nature of the elements involved.
Participants do not reach a consensus on the definitions and implications of the proposed 1-to-1 correspondence. Multiple competing views remain regarding the nature of x and the validity of the arguments presented.
The discussion highlights limitations in the definitions and assumptions made by participants, particularly regarding the nature of elements that are not classified as normal real numbers and the implications of such classifications on mathematical reasoning.
Which of them (axioms)?
Is the section of General physics>Theory Development math section? I thought, that it is a Physics theory development section... If I've misunderstood, then forgive me
For Zermelo-Frenkel system axioms (AUTHOR's DENOTATIONS, not Ito and somebody else) see link below
No, I don't.
Why do you need another axiom? Don't you think that we could do with existing ones?
In the case of the idea of the set, x is only an uncertain set, nothing else.
Of course! Bingo! And I think so, too! Maybe I was not clear enough. My point is that ZF exclude an axiom of the empty set because the empty set concept is a theorem within the framework of ZF, not an axiom. But I asked Organic to give the empty set definition of his own, because the ZF empty set definition doesn't allow to make the deductions he has made.Originally posted by Hurkyl
I've been keeping tabs on this thread... but I can't seem to figure out what your point is, Anton, can you enlighten me?
Some comments on the empty set that haven't been mentioned:
Given the existence of any set S, the ZF axiom of subsets can be used to construct the empty set:
ExAy:(y in x) <=> (y in S and ~(y in S))
It's easy enough to prove that a set constructed this way is unique and doesn't depend on the identity of S. I'll call the empty set 0 because I don't know the code to give me the empty set symbol.
It is easy enough to show that 0, constructed this way, satisfies, for any set T, 0 U T = T, 0 x T = 0, and it is also easy enough to show that 0 is the unique set satisfying these two properties. In fact, in ZF, the four statements:
Why include the axiom of the empty set if you can use the axiom of subsets to construct it? Minimalism. In order to get started, the existence of at least one set must be postulated. Because no matter what set gets postulated one can construct the empty set, the most minimal postulate you can state is that the empty set exists.
Could you specify what you mean? And where are the contradictions with my definitions?
Once more, which of my definitions (formulas) exactly redundant and complicated? The emty set is only a formula, not an axiom (prove me that my definition of FORMULA of empty set is an axiom; the formulas !x=Æ and z=Æ (in Z7 and Z8) can be substituted with full definition of the empty set... am I worng? Just prove it!).
Well... one and the last time. The set theory operates sets by using operations such as x - direct product of sets, Å - direct addition of sets, U - join of sets, Ï - intersection of sets. The main requirement is that the results of all the above operations with sets are sets as well. Only this requrement makes the set theory use the empty set concept, therefore the empty set can only be defined by non-empty sets and operations. NOTHING ELSE! BECAUSE THE EQUALITY OR UNEQUALITY TO THE EMPTY SET (=Æ OR !=Æ) ARE ONLY FORMULAS. THE ICON Æ WITHOUT "=", "!=", "Î" or "Ï" MAKES NO SENSE. And this is really for the last time.