Cont: Deeper than primes - Continuation 2

[doronshadmi]

You seemed to have missed this part in what I wrote: If the set of natural numbers doesn't contain all the natural numbers, then it isn't the set of natural numbers.

You are missing the notion that no inductive set is complete, no matter what name is given to it, because of the "next" mechanism of a singleton as a successor.

The axiom of infinity is exactly this notion, which guarantees the incompleteness of any inductive set from within (as its inherent property).

As for natural numbers, 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

 

[jsfisher]

You are missing the notion that no inductive set is complete, no matter what name is given to it, because of the "next" mechanism of a singleton as a successor.

And this circles us back to you rejecting the Axiom of Infinity, which clearly postulates a complete set in all its infinite-ness.

 

[doronshadmi]

And this circles us back to you rejecting the Axiom of Infinity, which clearly postulates a complete set in all its infinite-ness.

EDIT:

You are missing the rest of http://www.internationalskeptics.com/forums/showpost.php?p=11259373&postcount=1181 and also the previous http://www.internationalskeptics.com/forums/showpost.php?p=11257846&postcount=1172 post.

I reject your notion of what is considered as a successor, no less, no more.

 

[zooterkin]

I do reject the existence of the non-finite set of all natural numbers.

Which ones are missing? In the set of all natural numbers, which ones are not there?

 

[doronshadmi]

Let's understand better the possible difference between infinite numbers, by some example:

If 1... is an infinite number and base-10 is used, then number 10... is 10 times infinitely bigger than 1... , where the relations between these infinite numbers is expressed as follows:

10...
 1...

Also there can be a finite difference between two infinite numbers, for example:

1+1... > 1... by 1 finite number.

Infinite and/or finite numbers are used to determine the sizes of sets.

 

[Nonpareil]

I reject your notion of what is considered as a successor, no less, no more.

And, in doing so, you reject the axiom of infinity.

This is not complicated.

 

[doronshadmi]

Which ones are missing? In the set of all natural numbers, which ones are not there?

Those that are not written down.

EDIT:
In order to understand my answer, you first have to understand the notion of a singleton as a successor (for example http://www.internationalskeptics.com/forums/showpost.php?p=11259142&postcount=1179 , http://www.internationalskeptics.com/forums/showpost.php?p=11259373&postcount=1181 and http://www.internationalskeptics.com/forums/showpost.php?p=11259463&postcount=1185).

 

[doronshadmi]

And, in doing so, you reject the axiom of infinity.

No, I reject its standard notion, as explained in http://www.internationalskeptics.com/forums/showpost.php?p=11259412&postcount=1183.

 

[Nonpareil]

If 1... is an infinite number and base-10 is used, then number 10... is 10 times infinitely bigger than 1... , where the relations between these infinite numbers is expressed as follows:

10...
  1...

Also there can be a finite difference between two infinite numbers, for example:

1+1... > 1... by 1 finite number.

Infinity is not a finite number. Infinity does not behave in the same way that finite numbers do. Adding a finite number to infinity gives you infinity, no greater or lesser than before (at least in the case of infinite cardinals like Aleph-0, which is what we are talking about here).

I wonder about the possibility of bringing up the surreal numbers here, but I don't want to become an enabler.

 

[Nonpareil]

No, I reject its standard notion

...which means you reject the axiom.

This is not complicated.

 

[zooterkin]

Those that are not written down.

They are in the set whether you write them down or not. You don't need to list every member of a set for them to be in the set, just define which ones are in it.

 

[doronshadmi]

They are in the set whether you write them down or not. You don't need to list every member of a set for them to be in the set, just define which ones are in it.

Again, in order to understand my answer, you first have to understand the notion of a singleton as a successor (for example http://www.internationalskeptics.com/forums/showpost.php?p=11259142&postcount=1179 , http://www.internationalskeptics.com/forums/showpost.php?p=11259373&postcount=1181 and http://www.internationalskeptics.com/forums/showpost.php?p=11259463&postcount=1185).

 

[doronshadmi]

...which means you reject the axiom.

This is not complicated.

..which means that I reject the standard notion of this axiom.

This is not complicated.

 

[doronshadmi]

Infinity is not a finite number.

You are right, therefore I am talking about infinite numbers and not about Infinity.

 

[doronshadmi]

They are in the set whether you write them down or not.

This notion is derived from the determination that a successor is a natural number (in case that inductive set is taken as a set of natural numbers).

I do not share with you this determination, because my notion is derived from the determination that a successor is a singleton.

 

[Nonpareil]

You are right, therefore I am talking about infinite numbers and not about Infinity.

So we're just down to outright ignoring what is actually said, then. Wonderful.

 

[doronshadmi]

So we're just down to outright ignoring what is actually said, then. Wonderful.

You wrote:

Infinity does not behave in the same way that finite numbers do

Well, Infinity is not a number and the transfinite numbers system is not my infinite numbers system (that some of its properties are shown in http://www.internationalskeptics.com/forums/showpost.php?p=11259463&postcount=1185).

Also, do you remember this?

A hint: infinity is not a number.

 

[jsfisher]

The Axiom of Infinity I'm using is this:

Perhaps you, Doron, have something else in mind?

In my version the axiom postulates the existence of an infinite set. There is nothing incomplete about this set.

The set of natural numbers is a subset of the set postulated, and with the assistance of a few other of set theory's axioms, the set of natural numbers can be identified. The set of natural numbers is complete in every regard. No numbers are omitted; they are all there.

In fact, the set of natural numbers defines what numbers are the natural numbers. Were any missing (that is, not in the set of natural numbers), then they wouldn't be natural numbers, by definition.

Or, Doron, are you suggesting that there are objects that are simultaneously members and not members of the set of natural numbers? That would be bizarre, don't you think, but is that what you have in mind?

 

[Nonpareil]

You wrote:

Well, Infinity is not a number and the transfinite numbers system is not my infinite numbers system (that some of its properties are shown in http://www.internationalskeptics.com/forums/showpost.php?p=11259463&postcount=1185).

Also, do you remember this?

Yes. And precisely none of this does anything to change the fact that adding any finite amount to infinity leaves you with infinity.

 

[doronshadmi]

Yes. And precisely none of this does anything to change the fact that adding any finite amount to infinity leaves you with infinity.

The infinite number 1... < 1+1... by 1, where in both cases an infinite number is involved (or as you put it, it "leaves you with infinity").

Once again, the transfinite numbers system is not my infinite numbers system.