Cont: Deeper than primes - Continuation 1/3*9

[jsfisher]

I take the property (x ∪ {x}) right from the axiom above, as follows:

Definition 1: A set K is a base set iff K property is ~(x∪{x}).

How can you determine whether K is of the form x ∪ {x} or not?

(By the way, (x ∪ {x}) is not a property and ~(x∪{x}) is not a valid formula. You don't negate sets.)

 

[jsfisher]

Each one of the given definitions simply declares the existence of some set that satisfies a given property.

No, they don't. They give names to things. Neither "declares the existence" of anything.

 

[doronshadmi]

How can you determine whether K is of the form x ∪ {x} or not?

No, the formula (x ∪ {x}) appears without "the need to first define it as a successor" because it is unnecessary to give it a name.

Is (x∪{x}) a wff?

 

[jsfisher]

Is (x∪{x}) a wff?

Not in propositional calculus, no. "Cup" is a set-valued binary operator, and x and {x} are sets. That makes x ∪ {x} a set and not a true/false ;proposition.

 

[doronshadmi]

Not in propositional calculus, no. "Cup" is a set-valued binary operator, and x and {x} are sets. That makes x ∪ {x} a set and not a true/false ;proposition.

A∪B = {x : x∈A OR x∈B}

A=x
B={x}

and we get A∪B = {x : x∈x OR x∈{x}}

Definition 1: A set K is a base set iff K ≠ {x : x∈x OR x∈{x}}.

Definition 2: A set K is a successor set iff K = {x : x∈x OR x∈{x}}.

 

[jsfisher]

Definition 1: A set K is a base set iff K ≠ {x : x∈x OR x∈{x}}

So, what is "x" thing (the non-bold, non-italic version)?

 

[doronshadmi]

So, what is "x" thing (the non-bold, non-italic version)?

Whet are the iff yes/no logical foundations of "x" and "x ∪ {x}" things in the following expression? :

∃I (∅∈I ∧ ∀x∈I [(x ∪ {x})∈I])

 

[jsfisher]

Whet are the iff yes/no logical foundations of "x" and "x ∪ {x}" things in the following expression? :

∃I (∅∈I ∧ ∀x∈I [(x ∪ {x})∈I])

Yes/no logical foundations? Where did that come from? I asked what was x in your definition. You cannot just introduce x ∪ {x} (in whatever form you care to represent set union) without telling us what x is. If you leave x unknown, as you have, then your definition isn't.

The Axiom of Infinity does include x ∪ {x}, but if first tells us about x. That's the ∀x∈I part.

So, what's your x?

 

[Little 10 Toes]

Whet are the iff yes/no logical foundations of "x" and "x ∪ {x}" things in the following expression? :

∃I (∅∈I ∧ ∀x∈I [(x ∪ {x})∈I])

That makes no sense.

"What are the 'if and only if' yes/no foundations of 'x' and 'x ∪ {x}" things in the following expression"?

There are none.

Axiom_of_infinity^WP: "∃I (∅ ∈ I ∧ ∀ x ∈ I [(x ∪ {x}) ∈ I]). In words, there is a set I (the set which is postulated to be infinite), such that the empty set is in I, and such that whenever any x is a member of I, the set formed by taking the union of x with its singleton {x} is also a member of I. Such a set is sometimes called an inductive set." (I use square brackets in the for readability).

No iff or yes/no there.

 

[doronshadmi]

x is a placeholder for any given set (without any information about its members).

x∪{x} = {x : x∈x AND {x∈x}}

Definition 1: A set K is a base set iff K ≠ {x : x∈x AND {x∈x}}.

Definition 2: A set K is a successor set iff K = {x : x∈x AND {x∈x}}.

 

[Little 10 Toes]

x is a placeholder for a set.

x∪{x} = {x : x∈x AND {x∈x}}

Definition 1: A set K is a base set iff K ≠ {x : x∈x AND {x∈x}}.

Definition 2: A set K is a successor set iff K = {x : x∈x AND {x∈x}}.

Define x.
Define x.
What is the difference between x and x?

 

[jsfisher]

x is a placeholder of any given set.

You need to quantify that in some way. Otherwise, x is just an unknown. The "right hand side" of your definitions are not allowed to contain unknowns.

You can have unknowns on the left, after all those are the things you intend to define, but by the time you get to the right, it all needs to be fixed.

 

[doronshadmi]

You need to quantify that in some way. Otherwise, x is just an unknown. The "right hand side" of your definitions are not allowed to contain unknowns.

You can have unknowns on the left, after all those are the things you intend to define, but by the time you get to the right, it all needs to be fixed.

x is a placeholder for any given set (without any information about its members).

The Axiom of Infinity does include x ∪ {x}, but if first tells us about x. That's the ∀x∈I part.

Once again you force "for all" (universal quantification).

 

[jsfisher]

Define x.
Define x.
What is the difference between x and x?

In the set-builder notation Doronshadmi is using, x has a defined meaning. It is simply a variable representing each possible member of the set being defined; everything following the colon provides the conditions all members of the set must meet (and which all non-members fail to meet).

E.g.,

{ w : w is an even whole number } = { 0, 2, 4, 6, ... }​

 

[jsfisher]

x is a placeholder for any given set (without any information about its members).

Then your definition fails to define anything. Whether K is equal to some set involving x cannot be decided since x is not known. You'd need to quantify x in some way, formally, right there in the definition.

(This is the second time I have provided that hint.)

 

[doronshadmi]

{ w : w is an even whole number } = { 0, 2, 4, 6, ... }​

In this example w meaning can also be "no members".

 

[Little 10 Toes]

In this example w meaning can also be "no members".

How? Are you saying that even whole numbers don't exist?

 

[doronshadmi]

Then your definition fails to define anything. Whether K is equal to some set involving x cannot be decided since x is not known. You'd need to quantify x in some way, formally, right there in the definition.

(This is the second time I have provided that hint.)

Here we come again to ∀.

 

[jsfisher]

In this example w meaning can also be "no members".

It is correct exactly as presented. You didn't understand the example at all, did you?

Stop pretending.

 

[doronshadmi]

How? Are you saying that even whole numbers don't exist?

My mistake of writing a misleading post , I mean that generally w meaning can also be "no members", where w in jsfishr's example and x in my post are used for the same purpose, to define a set according its structure and not according to its members.