Deeper than primes

[doronshadmi]

As is your technique, you blame others for your failure. You again assert as fact your conclusion, yet you offer no basis for that assertion. The failure is and has always been yours.

Wrong again jsfisher.

Here is your assertion and conclusion:

If 0-dimensional space exists, then given any dimensional space greater than 0-dimensional space, it is defined as a collection of all distinct and uncountably infinite 0-dimensional spaces.

 

[HatRack]

HatRack, support in details your argument, which asserts that the following is not formulated within a clear, precise framework of logical deduction:

Theorem: The collection of all distinct points of [0,1] can't completely cover [0,1].

I'm going to request at this time that you word your theorem more precisely. In particular, and most importantly, give a precise definition of what you think it means for a collection of points to completely cover an interval.

This definition must be readily reducible to nothing more than familiar mathematical concepts such as sets and functions, but it need not be worded in formal logic.

For example, here is a definition that is readily reducible to a statement about sets and functions:

A set S is said to be countably infinite if there exists a bijection of N onto S, where N is the set of natural numbers.

This definition is very clear. If we can construct a bijection from N onto S, then S is countably infinite. If we can prove that no such bijection exists, then S is not countably infinite. It uses nothing more than familiar set theoretic concepts.

You are proposing a theorem that potentially upsets centuries of mathematical knowledge. Your theorem, and every definition it uses, must be readily reducible into familiar concepts so that it may be verified precisely. If that's not possible, due to a "notion" or some type of undefined concept, then you must give an appropriate axiom.

So, let's get this theorem worded precisely, using only sets and functions if possible, before proceeding.

 

[doronshadmi]

You are proposing a theorem that potentially upsets centuries of mathematical knowledge. Your theorem, and every definition it uses, must be readily reducible into familiar concepts so that it may be verified precisely. If that's not possible, due to a "notion" or some type of undefined concept, then you must give an appropriate axiom.

The axiom of the continuum:

If 0-dimensional space exists, then given any dimensional space greater than 0-dimensional space, it can't be defined as a collection of all distinct and uncountably infinite 0-dimensional spaces.

 

[laca]

laca, we have here jsfisher.

He is a professional mathematician, and he uses "adjacent" in the following post http://www.internationalskeptics.com/forums/showpost.php?p=6585515&postcount=12497 .

Is you wish to get an answer from a professional mathematician about "adjacent" , then just ask him.

I'm asking you doron, because it seems like such a central point in your "argument". It's coming from the same delusion that somehow makes you think that we would need to find two distinct points with no gap between them to prove that a line is completely covered with points. I'm thinking by adjacent you mean no gap. But that's utterly ridiculous, because if there's no gap then they are one and the same point and discussing adjacency becomes moot.

This is why I'm asking you to define adjacent in the context of points.

After all he is the one that first used "adjacent" in this thread.

Not that I care, but I would think he was, you know, playing along, trying to communicate in your "special" language. If he wishes, he can elaborate on that.

Please not that even without defining the term, he said that no two points are adjacent. Which is correct, since the term is meaningless for points.

Let's get back to your first attempt at defining adjacent. You said adjacent means distance > 0. Do you still think this is an accurate definition or would you like to refine it?

 

[laca]

The axiom of the continuum:

If 0-dimensional space exists, then given any dimensional space greater than 0-dimensional space, it can't be defined as a collection of all distinct and uncountably infinite 0-dimensional spaces.

You're hilarious. Do you even know what an axiom is? Are you even aware what is it you're trying to do? You're trying to dismiss a known and proven fact by negating it and attempting to smuggle it in as an axiom. That ain't going to happen, sorry.

 

[HatRack]

The axiom of the continuum:

If 0-dimensional space exists, then given any dimensional space greater than 0-dimensional space, it can't be defined as a collection of all distinct and uncountably infinite 0-dimensional spaces.

Awesome, we can stop right there then. I don't accept your axiom, and neither will any professional mathematician, ever.

The ZFC axioms are widely accepted because they state things that are relatively simple and obvious. Your axiom makes an assertion about what infinity can and cannot be, something that should be a theorem which follows from much simpler axioms.

But, the true problem with your axiom is that it makes ZFC inconsistent. The negation of your axiom can in fact be derived from ZFC. That is because ZFC implies the existence of the real number line (1-dimensional space), which is in fact an uncountably infinite collection of distinct points (0-dimensional spaces).

Well, that wraps it up. Unless you annihilate one or more ZFC axioms, you have been disproven.

 

[jsfisher]

Wrong again jsfisher.

Here is your assertion and conclusion:

If 0-dimensional space exists, then given any dimensional space greater than 0-dimensional space, it is defined as a collection of all distinct and uncountably infinite 0-dimensional spaces.

Nope. Again you misrepresent, misunderstand, and fantasize. That is not my assertion nor my conclusion.

 

[doronshadmi]

he said that no two points are adjacent. Which is correct, since the term is meaningless for points.

Laca, how exactly jsfisher's proposition "no two points are adjacent" is a correct proposition, if the term "adjacent" (which is a part of jsfisher's proposition) is (as you claim) "meaningless for points"?

 

[zooterkin]

Laca, how exactly jsfisher's proposition "no two points are adjacent" is a correct proposition, if the term "adjacent" (which is a part of jsfisher's proposition) is (as you claim) "meaningless for points"?

Because the term cannot apply to points, as I said earlier. We know what adjacent means, in general. Because points don't occupy any space themselves, if there is no distance between them (the normal definition of adjacent), then they are the same point. Therefore, adjacent does not apply as a meaningful term for points.

 

[doronshadmi]

But, the true problem with your axiom is that it makes ZFC inconsistent.

No HatRack, this axiom is strictly not ZFC, because ZFC is closed under the concept of Collection.

Any one that gets infinity only in terms of the concept of Collection, can't comprehend infinity in terms of this axiom.

 

[doronshadmi]

Because the term cannot apply to points, as I said earlier. We know what adjacent means, in general. Because points don't occupy any space themselves, if there is no distance between them (the normal definition of adjacent), then they are the same point. Therefore, adjacent does not apply as a meaningful term for points.

You are wrong, zooterkin.

"Not adjacent" (distance>0) is related to more than one 0-dimensional space along 1-dimensional space.

"Adjacent" (distance=0) is related to exactly one 0-dimensional space along 1-dimensional space.

 

[doronshadmi]

Nope. Again you misrepresent, misunderstand, and fantasize. That is not my assertion nor my conclusion.

Please answer only by yes or no:

k=1 to ∞

Does any given k-dimensional space is distinct from the collection of all distinct and uncountably infinite 0-dimensional spaces?

 

[doronshadmi]

I don't accept your axiom, and neither will any professional mathematician, ever.

Since when you are the voice of any professional mathematician?

Again,

Any one (professional mathematician or not) that gets infinity only in terms of the concept of Collection, can't comprehend infinity in terms of this axiom.

 

[HatRack]

No HatRack, this axiom is strictly not ZFC, because ZFC is closed under the concept of Collection.

Any one that gets infinity only in terms of the concept of Collection, can't comprehend infinity in terms of this axiom.

Oh, so you choose to annihilate ALL of the ZFC axioms then. Don't you hate it when the pesky mathematics that has been developed over the centuries by all those cranks such as Newton and Cauchy and Weierstrass gets in the way of your grand new theory? Well, to each his own. Good luck deducing some useful theorems from your theory and its lone axiom.

Hey, Doron really did come up with a good idea. Next time I run into a mathematical theorem that is above my understanding, I'll just introduce an axiom which asserts its falsity.

 

[HatRack]

Since when you are the voice of any professional mathematician?

Again,

Any one (professional mathematician or not) that gets infinity only in terms of the concept of Collection, can't comprehend infinity in terms of this axiom.

The fact that you've been going on about this nonsense for how many years now and you've yet to produce a peer-reviewed paper that has gained acceptance in the mathematical community (let alone an Internet forum) is a pretty good indication that my assertion is true.

 

[zooterkin]

You are wrong, zooterkin.

No, I'm not. See my signature.

"Not adjacent" (distance>0) is related to more than one 0-dimensional space along 1-dimensional space.

"Adjacent" (distance=0) is related to exactly one 0-dimensional space along 1-dimensional space.

Well, yes, if two points are adjacent (distance=0), then we are, in fact, talking about just one point; but I don't think you've quite grasped that yet.

 

[laca]

Laca, how exactly jsfisher's proposition "no two points are adjacent" is a correct proposition, if the term "adjacent" (which is a part of jsfisher's proposition) is (as you claim) "meaningless for points"?

Well, think about it real hard... I know you can do it! Oh, wait...

 

[laca]

"Not adjacent" (distance>0) is related to more than one 0-dimensional space along 1-dimensional space.

"Adjacent" (distance=0) is related to exactly one 0-dimensional space along 1-dimensional space.

In that case, adjacent is equivalent to "the same". And you just made a fool out of yourself by arguing essentially for "no two distinct points are the same". Congrats. Quite an achievement.

 

[doronshadmi]

The fact that you've been going on about this nonsense for how many years now and you've yet to produce a peer-reviewed paper that has gained acceptance in the mathematical community (let alone an Internet forum) is a pretty good indication that my assertion is true.

Your assertion is true as long as the mathematical community gets infinity only in terms of the concept of Collection (it can't comprehend infinity in terms of this axiom).

 

[laca]

Please answer only by yes or no:

k=1 to ∞

Does any given k-dimensional space is distinct from the collection of all distinct and uncountably infinite 0-dimensional spaces?

You're asking whether one space is distinct from a collection of spaces. I'm almost sure you're not as stupid as that question leads us to believe.