Cont: Deeper than primes - Continuation 2

[doronshadmi]

How do you tell? What test within your ZFU theory can you apply?

If A is indistinguishable from B, then A=B, otherwise A ≠ B.

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.

Exactly, identity is not a fundamental property of the considered axiomatic framework under construction, (you still ignore http://www.internationalskeptics.com/forums/showpost.php?p=11008041&postcount=967, why is that?)

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?

All you are doing is forcing identity as a fundamental property of the considered axiomatic framework under construction.

Again, you continue to ignore http://www.internationalskeptics.com/forums/showpost.php?p=11008041&postcount=967 as an important part of the the considered axiomatic framework under construction.

(The answer needs to be a test from within your axiom set.)

This is exactly what you get within an axiomatic framework where identity is not a fundamental property of it.

For example an empty set is not unique, for example, in {{}{}{}{}{}} but it can be unique in {{}}, or {{} 1 @ ? ...}.

In other words, in the considered axiomatic framework under construction objects have OR do not have unique identity, which enables richer mathematical framework, simply because its objects and their relations, are not limited by uniqueness, as probably done by the, so called, standard view of the foundation of Mathematics.

If you disagree with me, then please demonstrate within any ZF or ZFC version that you like, that identity is not a fundamental property of it.

 

[jsfisher]

If A is indistinguishable from B, then A=B, otherwise A ≠ B.

It was a straightforward question I asked. Are you unable to answer it? How, within your axiom set, would you distinguish A from B?

 

[jsfisher]

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.

So when I asked this:

Any other additions or alterations to ZF for your ZFU?

and you responded by saying this:

No. Only the modification of the Axiom of Extensionality of as seen in http://www.internationalskeptics.com/forums/showpost.php?p=11006351&postcount=962.

You were lying. There were additions and alternations yet to be revealed.

Let me know when you have settled on a final version, ok? I am not interested in your wild claims founded on fluid, ever-changing base.

 

[doronshadmi]

You were lying.

No, you are still missing the meaning of "Let's start by ..." as appears in http://www.internationalskeptics.com/forums/showpost.php?p=11006351&postcount=962.

 

[doronshadmi]

It was a straightforward question I asked. Are you unable to answer it? How, within your axiom set, would you distinguish A from B?

It seems that http://www.internationalskeptics.com/forums/showpost.php?p=11010394&postcount=981 is beyond your identity-only reasoning.

 

[doronshadmi]

Let me know when you have settled on a final version, ok? I am not interested in your wild claims founded on fluid, ever-changing base.

For the last time, if you disagree with me, then please demonstrate within any ZF or ZFC version that you like, that identity is not a fundamental property of it.

jsfisher, if no answer will be given by you about this fine subject, it will be clear that you are not interested in any discussion that re-examining identity as a property of Mathematics.

In that case, I am not interested in your dogmatic approach of this fine subject.

 

[jsfisher]

jsfisher, now I am sure that you are not interested in any discussion that re-examining identity as a property of Mathematics.

Is that the topic now? It is so hard to tell with you continually changing it.

You've abandoned your ZFU set theory, then. Probably just as well, since your formulation was, well, let's just say troubled, your formulation was troubled.

 

[doronshadmi]

Is that the topic now?

No, it is my topic all along this thread.

Dealing with identity is deeper than primes.

Once again, it seems that your identity-only reasoning is simply blind to posts like http://www.internationalskeptics.com/forums/showpost.php?p=11010394&postcount=981 (including the links there).

 

[doronshadmi]

I find http://skullsinthestars.com/2014/01/18/infinite-series-not-quite-as-weird-as-some-would-say/ interesting.

 

[doronshadmi]

If building-block means that the considered existing thing (at least in the abstract sense) is non-composed, then an empty set or a ur-element are indistinguishable of each other.

If being distinct in terms of non-empty collections is done by building-blocks, then these collections are distinct of each other by the amount of their biding-blocks.

Form this point of view, the natural numbers in their simplest form, are collections of indistinguishable building-blocks that are distinguished of each other by the amount of their indistinguishable biding-blocks, as observed, for example, in http://www.internationalskeptics.com/forums/showpost.php?p=11008041&postcount=967.

Order is possible only among distinguished things (at least in the abstract sense), which means that amount is more fundamental than order, and indistinguishable building-blocks are more fundamental than amount.

From this observation, a fundamental mathematical theory has to deal with indistinguishability and its transformation into distinguishability, and such theory "is deeper than primes".

----------------

When a think about the notions above, I realize that existence (at least in its abstract sense) is more fundamental than indistinguishability or distinguishability, for example:

{} ==> {{}} means that there is a function from at least one existing set to another existing set, no matter if these sets are composed, or non-composed.

On the basis of existence, there is, for example, the following bijection:

{{},{},{}} ==> {} (this is a bijection of existence that does not distinguish between the composed and the non-composed).

On the basis of existence, there can be a bijection of a thing to itself (for example {} ==> {}) or not to itself (for example {{},{}} ==> {}).

So the bijection by existence ignores the contents of the mapped things, and it is more fundamental than a given function between the contents of things, which is possible only among composed things.

The composed things can be based on distinct or non-distinct things, and here is some example of a bijective function between distinct and non-distinct things:

{ {}  , {}  ,  {}  }
  |      |      |
  V      V      V
{ {}  ,{{}}, {{},{{}}} }

So cardinality > 1, measures only the content of composed things, by ignore the property of their contents.

Also cardinality = 1, has at least two types:

Function by existence, which ignores the content of things.

Function between contents of things.

So cardinality is more fundamental than indistinguishability or distinguishability, where order "gets on stage" only among collections of distinct things that have cardinality > 1.

 

[doronshadmi]

Again, here is some simple diagram of Neutral Monism's reasoning:

                      Absolute

                        YESx

                          |
                          |

          Relative     Neutral     Relative

           SOMEx  ---     x    ---  EVERYx

                          |
                          |

                         NOx

                      Absolute

The cardinality of YESx is the result of external observation of an existing (at least in the abstract sense) mathematical object, where such absolute existence is notated by the outer "{" and "}" that enables the function by existence, which is beyond the scope of all functions under the outer "{" and "}".

The cardinality of NOx is the result of internal observation of an existing (at least in the abstract sense) mathematical object, such that no parts are found inside it (it is a building-block, where a building-block can't be but a non-composed mathematical object).

SOMEx and EVERYx are the relative domain that its cardinality (by internal observation) > NOx AND < YESx, such that given any cardinality (by internal observation) > 1, the internal domain is a transformation between the indistinguishable and the distinguishable, as observed, for example under the relative cardinality 2:

etc. ... ad (potential) infinitum.

 

[doronshadmi]

I find https://www.youtube.com/watch?v=gNmRw7V8cbI , https://www.youtube.com/watch?v=5d2ZFQ8LMYw and https://en.wikipedia.org/wiki/Fuzzy_number interesting.

 

[doronshadmi]

Here is an interesting paper on "don't care" in Boolean Algebra https://courses.engr.illinois.edu/ece199/fa2012/notes/dontcare.pdf

 

[doronshadmi]

YESx or NOx has absolute cardinality, since no absolute is "one of many" thing.

SOMEx and EVERYx have relative cardinality, since any relative is "one of many" thing.

 

[doronshadmi]

Whoa! Stop. You are claiming the set of rows of the matrix includes all of the proper infinite subsets of the set N?

You will need to prove that.

I agree with you, it has to be proven, but you also say that

It cannot be proven under ZFC,...

and you also say that

There are several ways to formulate the ZF axiom set.

So according to your own words:

1) From one hand ZFC can't be used in order to prove that the matrix includes all of the proper infinite subsets of the set N.

2) On the other hand there are several ways to formulate the ZF axiom set.

So my question to you are:

1) Is there some particular version of ZF or ZFC, which enables to prove that the matrix includes all of the proper infinite subsets of the set N?

2) If the answer to (1) is NO, then what is the basis of your claim that ZF or ZFC (in, at least some version of it) can be considered as the fundamental axiom set for all Mathematics?

3) Also if the answer to (1) is NO, then please air your view about the layouts of an axiomatic system that, according to your view, enables to prove or disprove that the matrix includes all of the proper infinite subsets of the set N.

 

[jsfisher]

I agree with you, it has to be proven, but you also say that

and you also say that

So according to your own words:

1) From one hand ZFC can't be used in order to prove that the matrix includes all of the proper infinite subsets of the set N.

2) On the other hand there are several ways to formulate the ZF axiom set.

So my question to you are:

1) Is there some particular version of ZF or ZFC, which enables to prove that the matrix includes all of the proper infinite subsets of the set N?

All of the different axiom set variations for either ZF or ZFC yield the same set theory. The same theorems would exist in each variation. It is unlikely you will find one variation in which your statement is true since other versions prove it false.

 

[doronshadmi]

All of the different axiom set variations for either ZF or ZFC yield the same set theory. The same theorems would exist in each variation. It is unlikely you will find one variation in which your statement is true since other versions prove it false.

In that case please answer to the following question:

Is it possible to prove or disprove by ZF(C) that the matrix includes all of the proper infinite subsets of the set N?

 

[jsfisher]

All of the different axiom set variations for either ZF or ZFC yield the same set theory. The same theorems would exist in each variation. It is unlikely you will find one variation in which your statement is true since other versions prove it false.

In that case please answer to the following question:

Is it possible to prove or disprove by ZF(C) that the matrix includes all of the proper infinite subsets of the set N?

Please note the part I highlighted for the answer.

 

[doronshadmi]

Please note the part I highlighted for the answer.

You wrote:

All of the different axiom set variations for either ZF or ZFC yield the same set theory. The same theorems would exist in each variation.

So, I'll ask you again, is it possible to prove OR disprove by ZF(C) set theory that the matrix includes all of the proper infinite subsets of the set N?

You actually already gave your answer by saying the following:

It cannot be proven under ZFC,...

In other words, by your own words, ZF(C) can't prove OR disprove that the matrix includes all of the proper infinite subsets of the set N, which means that ZF(C) can't be considered as The foundation of Mathematics.

Moreover

since other versions prove it false.

has no basis whatsoever since ZF(C) can't determine if the statement "the matrix includes all of the proper infinite subsets of the set N" is True OR False (or, in other words, it can't prove OR disprove(= prove it false) it).

------------------------

If you disagree with my interpretation of your quotes, then please explicitly demonstrate exactly how the statement "the matrix includes all of the proper infinite subsets of the set N" is True OR False statement, by using ZF(C) set theory.

 

[jsfisher]

You wrote:

So, I'll ask you again, is it possible to prove OR disprove by ZF(C) set theory that the matrix includes all of the proper infinite subsets of the set N?

The reason ZF and ZFC cannot prove that your so-called matrix includes all of the proper infinite subsets of your reference set is because your baseless assertion is false. Set theory cannot prove a false theorem.

The highlighted part in my previous message said that ZF and ZFC prove that your baseless assertion is false.