zylo's Axiomatic Set Theory

Mar 2015
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Set: Things for which = and \(\displaystyle \epsilon\) are defined.
Axiom 1: A = B and A \(\displaystyle \epsilon\) B are mutually exclusive.

Theorem 1: A set cannot consist of one thing.
Proof: Definition of set and Axiom 1.

Theorem 2: A is a set iff B exists st B \(\displaystyle \epsilon\) A.
Proof: Definition of set, Axiom 1, and Theorem 1.
 
Mar 2015
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Meanwhile have you heard of Zermelo-Fraenkel Set Axioms?
https://en.wikipedia.org/wiki/Zermelo–Fraenkel_set_theory
I have googled ZFC and followed all the links to Bernays, Von Neumann, etc and found them not worth the effort of wading through because they are all rife with undefined terms, and so incapable of being rationally defined. For example, what does "all sets" mean?

Personally, I find the core concepts of set theory to be =, and \(\displaystyle \epsilon\), and they are mutually exclusive. Anything that doesn't start off explicitly with that doesn't interest me.

My system may be imperfect, but I prefer an imperfect intelligible system to an imperfect unintelligible system.

Things like "the class of all sets" and "proper class satisfies ZFC" I find to be simply word games.

But whatever floats your boat. Only I find the notion that these various set theories, including mine, are the "foundation" of mathematics ludicrous, but fun politics- I know something you don't know so I'm smarter (better) than you- I'll play to try and bolster my self-confidence. I know the meaning of that abstract painting, do you?
 
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zylo's Axiomatic Set Theory Rev 01

Revison 01

Set: Things for which = and \(\displaystyle \epsilon\) are defined.
Axiom 1: A = B and A \(\displaystyle \epsilon\) B are mutually exclusive.
Axiom 2: A is a set iff B exists st B \(\displaystyle \epsilon\) A.


"The forks on the table" refers to the individual forks on the table. The forks on the table are salad forks.
"The set of forks on the table" refers to the forks as a group. The forks on the table are members of the set of forks on the table.
The set of forks on the table is the set my grandmother gave me.

Studiot. Sorry I didn't answer a post of yours. Which one?
 
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Jun 2015
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Studiot. Sorry I didn't answer a post of yours. Which one?
Post#23 here

http://mymathforum.com/topology/314638-well-ordering-theorem-2.html

There, as here, I was offering something extra in the way of information that I thought might help.

In another forum they are also discussing set theory and I particularly like this comment

Once again we see the principle than math is not based on the axioms; rather, our choice of axioms is based on what math we want to do!
 

skipjack

Forum Staff
Dec 2006
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Set: Things for which = and \(\displaystyle \epsilon\) are defined.
Is there an empty set, and how is a definition of ∈ checked for validity?
 
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In another forum they are also discussing set theory and I particularly like this comment
You're very good at taking things out of context. And in this case, by failing to provide a link, you make it impossible for anyone to determine the context for themselves. Clever rhetoric, but not math.
 
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Is there an empty set, and how is a definition of ∈ checked for validity?
There is no empty set. Empty set is a contradiction in terms. Empty set is a definition of a set.
A herd of elephants is a set. An empty herd of elephants is a definition of a set.
An empty herd of elephants is not the same as an empty herd of cows. A herd of elephants can not contain cows.

\(\displaystyle \epsilon\) is membership, it's axiomatic. Any definition that makes sense and can be understood and satisfies a simple intelligible axiom system qualifies. The essence of \(\displaystyle \epsilon\) is that it is different than =. Halmos, Suppe, ZFC, etc don't make the distinction.

Beyond that it gets too abstract, virtually meaningless symbolism:
What is the mathematics of two different symbols which can be placed between any two things other than the two symbols? This leads to a set of rules about symbols. As you get more and more rules to cover every eventuality you have to question consistency.

On the positive side, if you can get a group of people to start coming up with and agreeing on a set of rules (axioms), you can call them the foundations of mathematics and invoke them in response to questions about mathematics, implying that anyone who doesn't "know" them doesn't know mathematics. In a way it's like abstract painting and the implication that if you don't understand abstract paintings you don't understand art.
However, I can understand that some people might enjoy the APPEARANCE of abstract art, just as some people might enjoy the APPEARANCE of arranged symbols. But I have no quarrel with such people, as long as they don't intrude on Art and Science.

It may happen that a symbolism system whose elements are given real interpretations may actually come up with a verifiable prediction. In that case it could be considered A methodology of science, but not a foundation of mathematics.

Finally, to protect against obvious criticism, my views are not absolute.

Does that explain \(\displaystyle \epsilon\)?



EDIT: Empty set may have a relation to sets like 0 to the natural numbers 1,2,3... ; 0 is not a natural number. But I have to think about it.
 
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You're very good at taking things out of context. And in this case, by failing to provide a link, you make it impossible for anyone to determine the context for themselves. Clever rhetoric, but not math.
Again you make statements without support as though they were fact.

In this case the statement is a personal one, not a mathematical one.

Are such allowed here?

So far as I can see ( and my question to the OP indicates I cannot see very far, certainly not far enough) this thread is about mathematical axioms.

So how is comment about mathematical axioms and their use, 'out of context'?
 
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skipjack

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. . . it's axiomatic. Any definition that makes sense and can be understood and satisfies a simple intelligible axiom system qualifies.
How does that imply that members of the set have any specific common characteristic, such as being an elephant? Wouldn't it suffice (at least for a finite set) that I specify the elements of the set by listing them?