Cont: Deeper than primes - Continuation 2

[jsfisher]

You are the one focused on the reciprocal of aleph₀, not I.

You wrote:

1 is infinitesimal with respect to aleph₀.

Yes, I did. Notice I made no mention of the reciprocal of aleph₀.

So please show where what you wrote is used in the mathematical literature.

It follows directly from the meaning of infinitesimal with respect to the Archimedean Property. But since you don't understand any of that, you don't see the obvious. If you did, not only would you understand why 1 is infinitesimal with respect to aleph₀ in the algebraic structure I proposed, you would also understand that this whole tangent upon which you've embarked on the Archimedean Property is absolutely irrelevant to Cantor's diagonalization proof method.

 

[doronshadmi]

That isn't what you wrote.

Once again you simply ignore my use of the diagram of diagonalization (this time shown in http://www.internationalskeptics.com/forums/showpost.php?p=11255999&postcount=1137) that was taken from wikipedia.

So please stop doing it, if you really wish to discuss about the infinite set of natural numbers as represented (by _i) in the diagonalization taken from wikipedia.

If you repeat once again about the values of s_i and not only on _i, it will be clear to me that you simply do not wish to discuss about the infinite set of natural numbers.

So, what is your question?

After I was clear about my using of _i in the diagonalization taken from from wikipedia, please tell me what is the _i value of the aleph₀-1 member in that diagonalization?

 

[doronshadmi]

Notice I made no mention of the reciprocal of aleph₀.

Before we continue, please explain exactly what do mean by "the reciprocal of aleph₀" (is it 1/aleph₀)?

It follows directly from the meaning of infinitesimal with respect to the Archimedean Property.

Great, so please show that "1 is infinitesimal with respect to aleph₀" is found in the mathematical literature.

 

[jsfisher]

That isn't what you wrote.

Once again you simply ignore my use of the diagram of diagonalization (this time shown in http://www.internationalskeptics.com/forums/showpost.php?p=11255999&postcount=1137) that was taken from wikipedia.

No, I did not ignore your use of the Wikipedia diagram. What is obvious, though, is that you did not read and understand the Wikipedia article from which you cribbed the diagram.

Instead, you simply posted the diagram without any of the context or details and started proclaiming, "See?? Cantor's wrong."

The details are important. The context is important.

The diagram you have been posting is part of a discussion of the proof that set of infinite sequences of 0s and 1s is non-countable.

But you keep assuming it was for something else, and you keep assuming it means exactly what you want it to mean.

It doesn't.

So please stop doing it, if you really wish to discuss about the infinite set of natural numbers as represented (by _i) in the diagonalization taken from wikipedia.

Case in point.

 

[jsfisher]

Before we continue, please explain exactly what do mean by "the reciprocal of aleph₀"?

https://en.wikipedia.org/wiki/Multiplicative_inverse

Great, so please show that "1 is infinitesimal with respect to aleph₀" is found in the mathematical literature.

I've already responded to that. Since you didn't read and understand it the first time, I'll give you another chance: It follows directly from the meaning of infinitesimal with respect to the Archimedean Property. But since you don't understand any of that, you don't see the obvious. If you did, not only would you understand why 1 is infinitesimal with respect to aleph0 in the algebraic structure I proposed, you would also understand that this whole tangent upon which you've embarked on the Archimedean Property is absolutely irrelevant to Cantor's diagonalization proof method.

 

[doronshadmi]

The diagram you have been posting is part of a discussion of the proof that set of infinite sequences of 0s and 1s is non-countable.

Once again, In my last discussion about diagonalaization I do not reject Cantor's proof, that there are more members in the set of real numbers than the set of natural numbers.

What I reject is the notion that there is a complete set of infinitely many elements.

 

[doronshadmi]

I've already responded to that.

Without any concrete example, so I'll give one, once again:

h of non-standard analysis is smaller than any reciprocal of some natural number.

EDIT:
Now by not using the reciprocal of aleph₀, please support your "1 is infinitesimal with respect to aleph₀ in the algebraic structure I proposed".

... you would also understand that this whole tangent upon which you've embarked on the Archimedean Property is absolutely irrelevant to Cantor's diagonalization proof method.

I am not talking about Cantor's diagonalization proof method, but about the incompleteness of any infinite set, where aleph₀ is claimed to be the exact size of a complete infinite set.

 

[jsfisher]

Once again, In my last discussion about diagonalaization I do not reject Cantor's proof, that there are more members in the set of real numbers than the set of natural numbers.

What I reject is the notion that there is a complete set of infinitely many elements.

So, once again, this boils down to you rejecting the Axiom of Infinity. What would you like in its place?

 

[doronshadmi]

So, once again, this boils down to you rejecting the Axiom of Infinity. What would you like in its place?

You have missed http://www.internationalskeptics.com/forums/showpost.php?p=11255234&postcount=1124.

Also please reply to http://www.internationalskeptics.com/forums/showpost.php?p=11256133&postcount=1147.

 

[jsfisher]

You have missed http://www.internationalskeptics.com/forums/showpost.php?p=11255234&postcount=1124.

Didn't ignore. Just dismissed it. If you want an alternate set theory, go for it...but you'll need a substitute for the Axiom of Infinity.

 

[doronshadmi]

Also please, this time, answer to the following question:

After I was clear about my using of _i in the diagonalization taken from from wikipedia, please tell me what is the _i value of the aleph₀-1 member in that diagonalization?

 

[doronshadmi]

Didn't ignore. Just dismissed it. If you want an alternate set theory, go for it...but you'll need a substitute for the Axiom of Infinity.

Here is the axiom of infinity, as written in wikipedia:

"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."

It does not follow from this axiom that there is a complete set of infinitely many elements.

EDIT:

On the contrary, this axiom actually determines that whenever a given element is a member of a given set, the union of this member with its singleton is the next member this set (or in other words, this addition (being next) of members is an inherent property of being an infinite set).

Make no mistake, there is no process of any kind here, but simply an infinitely long addition (of being next) that is done simultaneously (in parallel) as an inherent property of being an infinite set.

No finite set has this inherent property.

 

[Nonpareil]

Here is the axiom of infinity, as written in wikipedia:

"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."

It does not follow from this axiom that there is a complete set of infinitely many elements.

On the contrary, this axiom actually determines that whenever a given element is a member of a given set, the union of this member with its singleton is added to this set (or in other words, this addition of members is an inherent property of being an infinite set).

Oh, for heaven's sake. You could at least take the time necessary to actually read a Wikipedia article on the subject before trying to act as though you understand it.

"...that is, for each element of I, the successor of that element is also in I.

Thus the essence of the axiom is:

There is a set, I, that includes all the natural numbers."

No elements are being added. The set I is simply defined as containing all successors of its elements. It is exactly a "complete set of infinitely many elements" - assuming that you haven't tried to make that phrase mean something else when our backs were turned, anyway. You seem quite fond of trying to do that.

Make no mistake, there is no process of any kind here, but simply an infinitely long addition that is done simultaneously (in parallel) as an inherent property of being being an infinite set.

Oh, wonderful. We're back to the parallel nonsense.

 

[doronshadmi]

Thus the essence of the axiom is:

There is a set, I, that includes all the natural numbers.

EDIT:
No, the set of natural numbers is some particular case of an inductive set, as determined by the axiom of infinity.

The essence of the axiom is that whenever a given set is a member of some set, the union of this member with its singleton is added to this set.

In other words, given some member of a given set, there is always the next member (the successor) as an inherent property of such set.

 

[Nonpareil]

No, the set of natural numbers is some partial case of an inductive set, as determined by the axiom of infinity.

The essence of the axiom is that whenever a given set is a member of some set, the union of this member with its singleton is added to this set.

In other words, given some member to a given set, there is always the next member (the successor) as an inherent property of such set.

Doron, the bit that you are arguing with is from the article. It is the literal definition of the axiom of infinity. The definition that you are ostensibly using.

Please, at least pretend to put some time and effort into reading these things.

 

[doronshadmi]

Doron, the bit that you are arguing with is from the article. It is the literal definition of the axiom of infinity. The definition that you are ostensibly using.

I am talking about the interpretation of an inductive set, and the existence of the next element of an inductive set, is an inherent property of being an inductive set, in the first place.

Once again, given some member of a given set, there is always the next member (the successor) as an inherent property of such set.

No finite set has this inherent property.

 

[Nonpareil]

I am talking about the interpretation of an inductive set

This is not a matter of interpretation. This is a matter of definition. There is, by definition, no "addition" going on in the set posited by the axiom of infinity. It is defined as a set which contains all of its members' successors, not a set which instantly adds those successors to itself whenever you ask.

Your idea of "parallel addition", or whatever it is that you are calling it this time around, remains completely incoherent and useless. Even if it weren't, you still wouldn't be talking about the set defined in the axiom of infinity, because that set explicitly does not behave that way.

 

[jsfisher]

Also please, this time, answer to the following question:

After I was clear about my using of _i in the diagonalization taken from from wikipedia, please tell me what is the _i value of the aleph₀-1 member in that diagonalization?

You can say you use the subscripts like that all you like, but that doesn't mean the diagonalization method uses them in the manner you claim.

The diagram you cribbed without understanding it was in Wikipedia to assist in showing the set of all infinite sequences of 0s and 1s to be non-countable. The approach is to observe that if the set were countable (the opposite of what is to be proved), then each member of the set could be mapped to a unique natural number. In effect, they could be listed in some order such that element s_j of the set of infinite sequences of 0s and 1s is mapped to j, a natural number.

The diagonalization then is used to show that at least one element of the set (of sequences) is unmapped. Period. Full stop. This contradicts the assumption the set be countable. It must be non-countable. The proof is over.

There is no waiting-to-be-added natural number that appears in this anywhere. Please stop pretending there is.


Also, there is no "aleph₀-1 member" of the set. Please stop pretending there is. I can, however, tell you "what is the _i value of the 107th member in that diagonalization";it is 107. And "the _i value of the 9,144,853th member in that diagonalization" is 9,144,853. Were aleph₀-1 a natural number, then the answer to your question would be trivial, but since it isn't, your question is meaningless.

 

[doronshadmi]

The approach is to observe that if the set were countable (the opposite of what is to be proved)

I am not talking about terms like countable or non-countable, but about the inherent incompleteness of an inductive set.

Also, there is no "aleph₀-1 member" of the set. Please stop pretending there is.

EDIT:
I am not claiming such thing, exactly because the number of infinite _i's values is greater than any given _i value.

 

[Nonpareil]

I am not talking about terms like countable or non-countable, but about the inherent incompleteness of an inductive set.

Which is an incoherent idea based on your failure to understand set theory.