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

[doronshadmi]

So N members are of the form (x∪{x}) (successors) OR not of the form (x∪{x}) (bases), which is a tautology but not a trivial one, since ~(x∪{x}) is actually a proper subset of forms in N, where (x∪{x}) is not one of these forms in N, where N is closed under succession since at least one of the (x∪{x}) forms does not have its largest (x∪{x}) form in N (the term "all" is not satisfied in N if N is an infinite set).

More details are given in http://www.internationalskeptics.com/forums/showpost.php?p=13197966&postcount=239 and http://www.internationalskeptics.com/forums/showpost.php?p=13197438&postcount=235.

 

[jsfisher]

The two properties are:

1) One property establishes sets, which are not of the form (x∪{x}) (where ∅ is an example of such set).

2) The other property establishes sets, which are of the form (x∪{x}).

No, and no. The Axiom of Infinity establishes that there exists a set satisfying two properties. The two properties are:

1. The set contains the empty set as a member.
2. For all members, x, of the set, x U {x} is also a member.

These two properties do not in any way establish sets.

 

[jsfisher]

It is a membership function that does not need a gate (yes/no) keeper.

Then it is not a membership function.

Your repeated attempts to redefine things demonstrate no failures in Mathematics, only your lack of understanding of Mathematics. Mathematics is not wrong just because you imagine it to be different from what it is.

 

[doronshadmi]

The Axiom of Infinity establishes that there exists a set satisfying two properties.

These two properties do not in any way establish sets.

Well, no consistency in your previous post.

 

[doronshadmi]

Then it is not a membership function.

Your repeated attempts to redefine things demonstrate no failures in Mathematics, only your lack of understanding of Mathematics. Mathematics is not wrong just because you imagine it to be different from what it is.

jsfisher, you are stacked with your gate keeper and don't aware of the amazing N paradise that you established.

This paradise, if used carefully, may help us to understand how complexity (successors) emerges from simplicity (bases).

 

[Hevneren]

First Axiom of Doronity

There exists a set whose members are unknown.

 

[jsfisher]

Well, no consistency in your previous post.

The two statements are completely consistent. The Axiom itself (not the two properties) asserts some set exists.

The properties are two characteristics the set (asserted by the Axiom) must have.

Neither Axiom itself nor the two properties give a precise definition for the set asserted to exist.

 

[jsfisher]

jsfisher, you are stacked with your gate keeper and don't aware of the amazing N paradise that you established.

This paradise, if used carefully, may help us to understand how complexity (successors) emerges from simplicity (bases).

Paradise? The person at the adjacent table wants what you're having.


If you want to use this set, N, of yours (and it looks like you have settled on N for its name), then you need to define it. Please do.

 

[doronshadmi]

There exists a set whose members are unknown.

N members are well-defined, as given in http://www.internationalskeptics.com/forums/showpost.php?p=13198055&postcount=241 and its links.

 

[doronshadmi]

Please do.

Done at http://www.internationalskeptics.com/forums/showpost.php?p=13198055&postcount=241 and its links.

 

[doronshadmi]

jsfisher and Hevneren,

N members are well-defined by the following rule:

Any given N member is of the form (x∪{x}) OR not of the form (x∪{x}), which is a tautology.

Please define some well-defined set that can't be N member according this tautology.

 

[jsfisher]

jsfisher and Hevneren,

N members are well-defined by the following rule:

Any given N member is of the form (x∪{x}) OR not of the form (x∪{x}), which is a tautology.

I see. So you would like your set N to be a set of everything. All sets satisfy your rule so every set must be a member of N.

Alas, though, while that is surely not the set you had in mind, it is also not a set in any axiomatic set theory.

This shouldn't be that hard, Doronshadmi. You just need a logic function M(x) such that M(x) is true if x is in N, and M(x) is false if x is not in N. Then, you can define N in a simple way:

N = { x : M(x) }​

The only constraint is that M(x) must be consistent with the Axiom of Restricted Comprehension to insure that N is a set.

 

[doronshadmi]

I see. So you would like your set N to be a set of everything. All sets satisfy your rule so every set must be a member of N.

No, you don't see that N can't be a set of everything, since if N is an infinite set at least one of the (x∪{x}) forms does not have its largest (x∪{x}) form in N (the term "all" is not satisfied in N if N is an infinite set, according to the following definition):

Definition: (x∪{x}) is called the largest successor in N iff (x∪{x}) does not have its successor in N.

jsfisher, you still do not capture http://www.internationalskeptics.com/forums/showpost.php?p=13198055&postcount=241 and its links.

Paradise? The person at the adjacent table wants what you're having.

jsfisher, you are the person at the adjacent table that is unaware of the paradise that he established on his table.

jsfisher, you are the founder of set N by using base member and successor member as its inherent properties.

 

[jsfisher]

No, you don't see that N can't be a set of everything

Of course it can't, but that's how you described it. You need to fix the description.

...the term "all" is not satisfied...

Set theory says otherwise. You don't get to redefine things to your liking. You do, however, get to define this set N you continually reference.

jsfisher, you are the person at the adjacent table that is unaware of the paradise that he established on his table.

You completely missed the movie reference, didn't you? Next time you hear a loud swooshing sound, look up.

 

[doronshadmi]

Of course it can't, but that's how you described it.

Well that's how you wrongly interpret it exactly because you ignore the following definition:

Definition 1: (x∪{x}) is called the largest successor in N iff (x∪{x}) does not have its successor in N.

I use the term "in N" and not "of N", which enables to define the term "the largest successor" not with respect to "all" the possible successors in N, but only with respect to the successors that emerge from a given base member in N, as seen in the following example:

Here is a bijective map between given infinite proper subsets in N, where for any such given infinte proper subset there is a distinct base member, and its other distinct members are defined according to (x∪{x}), such that no (x∪{x}) is the largest successor in any such given proper subset:

N = {
∅,                    ---> {{∅}}, ---> ...   <-- base members
{∅},                  ---> {{∅}, {{∅}} }, ---> ...
{∅, {∅}},             ---> {{∅}, {{∅}}, {{∅}, {{∅}}}}, ---> ...
{∅, {∅}, {∅, {∅}}},  ---> {{∅}, {{∅}}, {{∅}, {{∅}}}, {{∅}, {{∅}}, {{∅}, {{∅}}}}}, ---> ...
...                         ...
}

In other words, set N has at least one such proper subset that does not have its largest successor, and by definition 1 universal quantification (the term "For All") can't be used on N.

Set theory says otherwise.

ZF was defined by persons that their aim was to establish definitions of order and cardinality that hold both for finite and infinite sets (in the name of generalization).

It was done by using universal quantification (the term "For All") both for finite and infinite sets, where in the case of infinite sets universal quantification was arbitrarily forced on them without any reasoning.

It can be seen right at the ZF Axiom of infinity, universal quantification (the term "For All") is arbitrarily plugged into this axiom even though a set, named, for example, as N by jsfisher (where any other name can be used) clearly does not have all of its successors, as shown above.

It does not matter if the Axiom of Restricted Comprehension or other ZF axioms are used or not, the fundamental error was done by using universal quantification as a part of ZF Axiom of infinity.

You don't get to redefine things to your liking.

You can force the term "For All" as much as you like on an infinite set, a fundamental error it will stay.

You completely missed the movie reference, didn't you?

You are completely missing what you actually established in http://www.internationalskeptics.com/forums/showpost.php?p=13163976&postcount=92 and http://www.internationalskeptics.com/forums/showpost.php?p=13191454&postcount=202 , don't you?

Please read your http://www.internationalskeptics.com/forums/showpost.php?p=13163976&postcount=92 post very carefully to actually understand your own words about bases and their successors.

You still have no idea what a beautiful mathematical paradise you, jsfisher, established.

 

[jsfisher]

Well that's how you wrongly interpret it exactly because you ignore the following definition:

Definition 1: (x∪{x}) is called the largest successor in N iff (x∪{x}) does not have its successor in N.

No, that definition is irrelevant to my point. You described the members of set N as either successors (of some non-specific something) or not successors. You even admitted it was a tautology.

Your description admits all sets as members of your set N.

If that isn't the description you want, then you need a better description for your set N.

 

[doronshadmi]

No, that definition is irrelevant to my point. You described the members of set N as either successors (of some non-specific something) or not successors. You even admitted it was a tautology.

My description: Any given N member is of the form (x∪{x}) OR not of the form (x∪{x}), which is a tautology.

Definition 1: (x∪{x}) is called the largest successor in N iff (x∪{x}) does not have its successor in N.

Your description admits all sets as members of your set N.

There is no all in my description exactly because for any given proper subset in N that its base member is, by definition, not a successor ( defined as ~(x∪{x}) ), does not have its largest successor in N by Definition 1, which means that the term "For All" (also called universal quantification that is notated by ∀) is not satisfied in N.

So Definition 1 is relevant for N.

Details are already given in http://www.internationalskeptics.com/forums/showpost.php?p=13199949&postcount=255 but you chose to ignore them.

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

B.T.W, ∀ is the cause of Russell's Paradox, whether a given collection of distinct objects is finite, or not.

For example:

U is a set of two distinct members, such that one of the members, called u, shaves ∀ the members of set U that do not shave themselves and only these members of set U (this is supposed to be his property in order to be a member of set U).

Who shaves u?

If u shaves himself, then he must not shave himself (shaves AND ~shaves himself, which is a contradiction) exactly because of the term ∀.

If u does not shave himself, then he must shave himself (~shaves AND shaves himself, which is a contradiction) exactly because of the term ∀.

So, because the term ∀ is used as a part of the terms that define u as a member U, u must be referred to himself, and we get the contradictions that actually prevents to well-define ∀ the members of set U (the term ∀ itself is actually not well-defined in case of U).

The same problem holds also among infinite sets that the term ∀ is one of their properties, therefore the Axiom of Restricted Comprehension was add to ZF in order to avoid Russell's Paradox, but it is done without being aware of the fact that the term ∀ is the cause of any given contradictory self reference, whether it is used among finite or infinite sets.

 

[jsfisher]

My description: Any given N member is of the form (x∪{x}) OR not of the form (x∪{x}), which is a tautology.

So which is it: You are telling us that everything is a member of your set N, or you are not telling us what things are not members? Your choice, but neither describe a specific set.

How's that membership function coming along?

 

[doronshadmi]

So which is it: You are telling us that everything is a member of your set N, or you are not telling us what things are not members? Your choice, but neither describe a specific set.

How's that membership function coming along?

By using definition 1, no largest successor ( where successors are of the form (x∪{x}) ) is emerged from at least one base member of N ( where base members are of the form ~(x∪{x}) ), yet any given member of N is of the form (x∪{x}) OR ~(x∪{x}), which is tautology.

Please look at set N:

N = {
∅,                    ---> {{∅}}, ---> ...   <-- base members
{∅},                  ---> {{∅}, {{∅}} }, ---> ...
{∅, {∅}},             ---> {{∅}, {{∅}}, {{∅}, {{∅}}}}, ---> ...
{∅, {∅}, {∅, {∅}}},  ---> {{∅}, {{∅}}, {{∅}, {{∅}}}, {{∅}, {{∅}}, {{∅}, {{∅}}}}}, ---> ...
...                         ...
}

Given any base member (where a base member is of the form ~(x∪{x}) ), no largest successor (where a successor is of the form (x∪{x}) ) is emerged from it.

The lack of largest successors in N is exactly the property of an infinite set, therefore the term "For All" is not satisfied in N, and yet given any member of N, it is ~(x∪{x}) OR (x∪{x}), which a tautology.

Moreover, you still do not capture the link between ∀ and Russell's Paradox, whether a given set is infinite or finite, in terms of ∀.

Look how gross is ZF.

From one hand ∃N (∅∈N ∧ ∀x∈N [(x∪{x})∈N])

But on the other hand the axiom schema of comprehension (https://en.wikipedia.org/wiki/Axiom_schema_of_specification) is needed in order to avoid Russell's Paradox, exactly because ∀ is a part of the axiom of infinity.

 

[jsfisher]

By using definition 1....

You still have not defined your set N. A membership function would be an important step towards that goal.