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Cont: Deeper than primes - Continuation 2

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doronshadmi said:
You and little 10 toas actually do not understand what is the meaning of "a ZF version", so let me help you.
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You are still not answering the question. Telling us what something would be like is not telling us what it is.

Little 10 Toes asked you to provide the axioms for your particular ZFU set theory.
 
jsfisher said:
You are still not answering the question. Telling us what something would be like is not telling us what it is.

Little 10 Toes asked you to provide the axioms for your particular ZFU set theory.
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Let's start by ZF axioms with the modification of the Axiom of extensionality:

"Given any set A and any set B, if A is a nonempty set (that is, if there exists a member X of A), then if A and B have precisely the same members, then they are equal"

Now please show that this ZF version is inconsistent, or does not allow non-classical collections, as explained in http://www.internationalskeptics.com/forums/showpost.php?p=11006288&postcount=960 and http://www.internationalskeptics.com/forums/showpost.php?p=11002986&postcount=951.
 
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As written in http://www.internationalskeptics.com/forums/showpost.php?p=11006288&postcount=960:
doronshadmi said:
In order to conclude that set N is not the one and only one set that is not included in the infinite square matrix, one has to construct some proper infinite subset of set N, which is different than all of the proper infinite subsets of set N in the given infinite |N|*|N| square matrix.
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Currently I do not see how one enables to construct such proper infinite subset of set N, and it will be nice to prove if such proper infinite subset exits (in the abstract sense), does not exist, or can't be proved or disproved within any ZF version.
 
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I have a general question about identity.

Eech finite collection can be based on the building block | , for example:

|
<|>
<||>
<|||>
<||||>
<|||||>
...

etc. ad infinitum.

If the notations above are understood in terms of collection of sets, then the building block of such collection is the abstract existence of the empty set exactly as | is the building block of each collections above, as follows:

{}
{{}}
{{}{}}
{{}{}{}}
{{}{}{}{}}
{{}{}{}{}{}}
...
etc. as infinitum.

So the size of collections is derived from some building block, where being a building block means that the considered object is a non-composed existing (at least in the abstract sense) thing.

Moreover, the identity of the collected building blocks is insignificant for the determination of the size of a given collection.

It means that the identity of the size of a given collection is basically derived from things that do not have identity.

If ZF is an axiomatic mathematical system that is based on identity as its fundamental property, it actually misses the more fundamental notion of collections of things that do not have identity.

So my question is: Do ZF or ZFC are not based on identity as their fundamental property?

If the answer is YES then please demonstrate how ZF or ZFC are not based on identity as their fundamental property.
 
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doronshadmi said:
Let's start by https://en.wikipedia.org/wiki/Zermelo–Fraenkel_set_theory where the Axiom of Extensionality is modified, as seen in http://www.internationalskeptics.com/forums/showpost.php?p=11006351&postcount=962.
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That's ZFC, so I will assume you exclude the final axiom.

I asked you, too, if there were other modifications to ZF, but said no. At the very least you will need to extend "set" and "membership" as primitive concepts to include "ur-element", but I'll assume you meant to provide that extension.

Then there is this:
If A has no members and B has no members, is A = B?
If A and B are sets and A has no members and B has no members, is A = B?

Your axioms appear insufficient to decide either question.

You will need some additional axioms.
 
jsfisher said:
That's ZFC, so I will assume you exclude the final axiom.

I asked you, too, if there were other modifications to ZF, but said no. At the very least you will need to extend "set" and "membership" as primitive concepts to include "ur-element", but I'll assume you meant to provide that extension.

Then there is this:
If A has no members and B has no members, is A = B?
If A and B are sets and A has no members and B has no members, is A = B?

Your axioms appear insufficient to decide either question.

You will need some additional axioms.
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The null set axiom:

There is a set such that no set or ur-element is a member of it.
 
doronshadmi said:
The null set axiom:

There is a set such that no set or ur-element is a member of it.
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So, when you said your ZF axiom set was as described in https://en.wikipedia.org/wiki/Zermelo–Fraenkel_set_theory, you meant something entirely different.

You'd like to add the (unnecessary) Axiom of the Empty Set to your list, is that right?

Ok, even with the Axiom of the Empty Set, how does that answer these questions:

If A has no members and B has no members, is A = B?
If A and B are sets and A has no members and B has no members, is A = B?
 
jsfisher said:
If A has no members and B has no members, is A = B?
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If A AND B ur-elements, then A = B OR A ≠ B.

jsfisher said:
If A and B are sets and A has no members and B has no members, is A = B?
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Yes (I change "The null set axiom:" to "a null set axiom:"), and please also reply to http://www.internationalskeptics.com/forums/showpost.php?p=11008041&postcount=967.

jsfisher said:
You'd like to add the (unnecessary) Axiom of the Empty Set
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Why it is unnecessary?
 
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doronshadmi said:
If A AND B ur-elements, then A = B OR A ≠ B.
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How do you tell? What test within your ZFU theory can you apply?

Yes (I change "The null set axiom:" to "a null set axiom:")
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The Axiom of the Empty Set, despite your fixation on an irrelevant semantic point, asserts the existence of an empty set. It does not guarantee uniqueness.

Be that as it may, since your response was hollow, I ask again the question:
If A and B are sets and A has no members and B has no members, is A = B?

(The answer needs to be a test from within your axiom set.)
 
jsfisher said:
you finally exposed what you advertised as the full axiom set
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There no "finally" at this preliminary stage so the actual meaning that you are still missing is that "let's start by ..." is an axiomatic framework under construction, or in other words, nothing is full or final at this stage.

As long as you do not understand this simple fact, there is not communication between us.
 
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