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Cont: Deeper than primes - Continuation 2

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doronshadmi said:
Have you heard about the disagreement among mathematicians whether {} is used to define a natural number, or not?
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Given that the topic is and has been the Axiom of Infinity, your attempt to derail is rejected.

The Axiom of Infinity -- that would be the thing that postulates an infinite set which includes {} among its members, where the members are neither old nor new, just members, and members that do not require being written down to come into existence. That is to say, the axiom you reject because of properties you attribute to it but it does not have.
 
jsfisher said:
Given that the topic is and has been the Axiom of Infinity, your attempt to derail is rejected.

The Axiom of Infinity -- that would be the thing that postulates an infinite set which includes {} among its members, where the members are neither old nor new, just members, and members that do not require being written down to come into existence. That is to say, the axiom you reject because of properties you attribute to it but it does not have.
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26203067354_6e6c090e66.jpg
is not silent about {y} as a successor of y, and this is the right notion about inductive sets, which prevent their completeness.

Moreover, y is not necessarily a finite set.
 
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jsfisher said:
It is not even a well-formed formula in predicate calculus.
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tex2img.php
says:

"There is a set x (the set which is postulated to be infinite), such that the empty set is a member of x AND such that for any y that is a member of x, the set formed by taking the union of y with its singleton {y}, is also a member of x.


26203067354_6e6c090e66.jpg
says:

"There is a set x (the set which is postulated to be infinite), such that the empty set is a member of x AND such that for any y that is a member of x there is singleton {y} (as its successor) AND such that for any y that is a member of x, the set formed by taking the union of y with its singleton {y}, is also a member of x."

Both versions are wff, but the standard version is silent about {y} as a successor of y even if it is used by it, and therefore one can't understand that an inductive set is actually inherently incomplete.
 
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doronshadmi said:
...such that for any y that is a member of x there is singleton {y}...
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No, it doesn't say that, and that would be the part that is not well formed.

Also, what you seem to be trying to say (i.e., that if y is a set, then {y} is a set) is already a certainty by virtue of the Axiom of Pairing. Why are you trying to explicitly say what is already known to be trivially true?
 
jsfisher said:
You failed to do either.
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Well, maybe by using the formal notations as I did, but it does not change the notion of what I wish to express about {y} as a successor of any y in x (and the conclusion that is derived from it about the inherent incompleteness of inductive sets).

I'll try to search among the currently used formal notations in order to formally express the notion above, and if they do not exist yet, I'll define my own notations in order to express it formally (without using, so called, natural language).
 
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doronshadmi said:
I'll try to search among the currently used formal notations in order to formally express the notion above, and if they do not exist yet, I'll define my own notations in order to express it formally (without using, so called, natural language).
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Won't that make a change from the usual.
 
Nonpareil said:
Won't that make a change from the usual.
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No matter what formal notations are used, the notions (currently expressed by natural language) are as follows:


The standard version of the axiom of infinity:

"There is a set x (the set which is postulated to be infinite), such that the empty set is a member of x AND such that for any y that is a member of x, the set formed by taking the union of y with its singleton {y}, is also a member of x.


My non-standard version of the axiom of infinity:

"There is a set x (the set which is postulated to be infinite), such that the empty set is a member of x AND such that for any y that is a member of x there is singleton {y} (as its successor) AND such that for any y that is a member of x, the set formed by taking the union of y with its singleton {y}, is also a member of x."

The standard version is silent about {y} as a successor of y even if it is used by it, and therefore one can't understand that an inductive set is actually inherently incomplete.
 
doronshadmi said:
The standard version is silent about {y} as a successor of y even if it is used by it, and therefore one can't understand that an inductive set is actually inherently incomplete.
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You've said this quite a bit.

You have yet to make any coherent case for it being true.
 
Nonpareil said:
You have yet to make any coherent case for it being true.
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If one understands the notion of a singleton as a successor, one understands that an inductive set is inherently incomplete.

This notion is coherent and straightforward.
 
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(The dollar sign is defined above as "is a successor of")

The string above in natural language is:

"There is a set x (the set which is postulated to be infinite), such that the empty set is a member of x AND such that for any y that is a member of x, {y} is a successor of y AND such that for any y that is a member of x, the set formed by taking the union of y with its singleton {y}, is also a member of x."

If one understands the notion of a singleton as a successor, one understands that an inductive set is inherently incomplete.

This notion is coherent and straightforward.

EDIT:

My version of the axiom of infinity is exactly guarantees the incompleteness of any inductive set from within (as its inherent property).

As for natural numbers (which are particular case of an inductive set) there are infinitely many sets of natural numbers, whether they have finite or infinite number of members.

For example, there is an infinite set of natural numbers with 1,000,000,000,... members, and there is a bigger infinite set of natural numbers with 1+1,000,000,000,... members, etc. ad infinitum
 
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doronshadmi said:
If one understands the notion of a singleton as a successor....
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Why would one do that other than to appeal to ones misunderstanding of the Axiom of Infinity?

By the way, "singleton" is associated with cardinality, and that doesn't come along until long after the axioms. It has not place here, and other than being a space-filling adjective, it serves no purpose.

"Successor" is just something you've imagined to be important from your closet of misused, misdefined terms. It has no significance to the Axiom of Infinity.
 
doronshadmi said:
If one understands the notion of a singleton as a successor, one understands that an inductive set is inherently incomplete.

This notion is coherent and straightforward.
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It is neither of those things. Even if one were to use that definition, the set defined by the axiom of infinity continues to include all of the natural numbers.
 
jsfisher said:
By the way, "singleton" is associated with cardinality.
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EDIT:
If y is a set {y} must be a singleton set, but if you dislike the mention of singleton at this stage, you do not have to use it, the notion {y} as a successor of y holds.
 
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