New Paradox for the "Principles of Physics".

1234567890
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21 years 7 months ago #5642 by 1234567890

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<BLOCKQUOTE id=quote>quote:<hr height=1 noshade id=quote>
<BLOCKQUOTE id=quote>quote:<hr height=1 noshade id=quote>[123...]: I disagree with your statement that the number of integers is infinite if infinity isn't a finite number.<hr height=1 noshade id=quote></BLOCKQUOTE id=quote>

I repeat the definition: Infinity -- concept of being always unlimited: the concept of being unlimited by always being larger than any imposed value or boundary.

In what way does the number of integers fail to fulfill this definition? In what way is it limited?

<BLOCKQUOTE id=quote>quote:<hr height=1 noshade id=quote>Why should this relationship, that every sum of integers leads to another integer ever break down no matter how many integers I added?<hr height=1 noshade id=quote></BLOCKQUOTE id=quote>

It doesn't ever break down. No one suggested that it did. When we say that the number of integers is infinity, that certainly does not make infinity an integer or the sum of any integers or groups of integers, however large. The whole point is that the number of integers goes beyond any integer you can name, write, or even imagine. The number of integers is "unlimited". The same cannot be said for sums of integers or counts of integers in any group.

The trick here is for you to familiarize with the use of infinity in math and physics. First, study 1/0. Examine its properties. Then learn the difference between "infinite" and "indeterminate". Then learn the math of infinities (where <img src=icon_infty.gif border=0 align=middle> == infinity; xdt == indeterminate): <img src=icon_infty.gif border=0 align=middle>+<img src=icon_infty.gif border=0 align=middle>=<img src=icon_infty.gif border=0 align=middle>; <img src=icon_infty.gif border=0 align=middle>-<img src=icon_infty.gif border=0 align=middle>=xdt; <img src=icon_infty.gif border=0 align=middle>*<img src=icon_infty.gif border=0 align=middle>=<img src=icon_infty.gif border=0 align=middle>; <img src=icon_infty.gif border=0 align=middle>/<img src=icon_infty.gif border=0 align=middle>=xdt; <img src=icon_infty.gif border=0 align=middle>^<img src=icon_infty.gif border=0 align=middle>=<img src=icon_infty.gif border=0 align=middle>2, a higher order of infinity; etc. Then learn how these concepts and infinities and infiniteesimals in general arise in derivitives and integrals in calculus, and how useful they are. Finally, study singularities and physical phenomena such as the "ultraviolet xatastrophe" to see how this applies to physics.

It's a tall order. But if you really want to understand how concepts such as dimensions and the universe can be infinite, this is all essential background. Wouldn't those of you you have this background agree?

<BLOCKQUOTE id=quote>quote:<hr height=1 noshade id=quote>When you call the number line infinite, you are invoking a number to represent that quality. This number is the sum of all the numbers in the set. But if every sum leads to another number, it has to be a member of its own set.<hr height=1 noshade id=quote></BLOCKQUOTE id=quote>

I don't agree with any of these three statements. Infinity is not a member of the set of numbers. -|Tom|-

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The count of the numberline, which you state to be infinite, is just 2 times the largest integer on the number line (since the integers and the count are basically generated by the same method ), adding 1 for the number zero. But since there is no largest integer- you can keep counting forever- it is redundant to call the count infinite and a contradiction to say infinity isn't itself an element of a set that by definition has no bounds to the elements.

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Jeremy
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21 years 7 months ago #5316 by Jeremy

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<BLOCKQUOTE id=quote>quote:<hr height=1 noshade id=quote>
george:
Because the number line is imaginary, it doesn't exist. The number line is an concept and there is no use in argueing concepts.
<hr height=1 noshade id=quote></BLOCKQUOTE id=quote>

What is going on here except argueing concepts? Aren't existence and infinity concepts? The postulate is that the universe contains an infinite number of entities. Conceptual numbers can be put in one to one correspondence with these entities and therefore have analogous properties when discussing sets.

<BLOCKQUOTE id=quote>quote:<hr height=1 noshade id=quote>
[123...]
1. If one defines as EXISTENCE the set of all things that exist, then EXISTENCE must be a member of that set, simply because it also exists. This is an obvious, common sense definition.
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One does NOT define existence as the set of all things that exist, you did. The fallacy occurs from your circular definition. This most assuredly is NOT an obvious definition. Existence is a philosophical concept that probably is not well defined at all and is at the root of the confusion over what is meant by it. There is certainly a difference between when I say love exists and if I say a car exists. When we say that something "exists" all we are really saying is that we believe we perceive something as being real at the moment we are considering. The line of what is accepted as being real is mutable and is going to differ for everyone here. Perhaps the best answer to existence is that something doesn't "exist" until we perceive that it does. "coming into existence" is simply perceiving a form that we didn't perceive previously. Eternal existence of substance cannot be proven except by logical inference. If the universe at some time in the past did not exist then one has to explain where the present substance disappeared to. None of the critics has supplied any answer to this fundamental problem. How can perception arrive from nothing?

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tvanflandern
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21 years 7 months ago #5317 by tvanflandern

Replied by tvanflandern on topic Reply from Tom Van Flandern

<BLOCKQUOTE id=quote>quote:<hr height=1 noshade id=quote>[123...]: The count of the numberline, which you state to be infinite, is just 2 times the largest integer on the number line<hr height=1 noshade id=quote></BLOCKQUOTE id=quote>

I repeat my urging that you study the subject matter before going on.

For example, the above statement is again false. Lines are infinite, line segments are not. Guess which I was talking about, and which you were talking about.

<BLOCKQUOTE id=quote>quote:<hr height=1 noshade id=quote>But since there is no largest integer- you can keep counting forever- it is redundant to call the count infinite ...<hr height=1 noshade id=quote></BLOCKQUOTE id=quote>

The number of integers is infinite by definition of "infinite".

<BLOCKQUOTE id=quote>quote:<hr height=1 noshade id=quote>... and a contradiction to say infinity isn't itself an element of a set that by definition has no bounds to the elements.<hr height=1 noshade id=quote></BLOCKQUOTE id=quote>

You are no longer disputing anything in the Meta Model or any opinion of mine, or even my application of concepts. Rather, you are disputing the basic meaning of infinite. I am using the concept in the standard way, presently accepted worldwide. If you wish to set up a different set of definitions, relationships, and usages, you are free to try. But unless others can see the point, they probably will ignore your efforts.

I recommend you learn the standard definitions, relationships, and usages of "infinite" and "infinitesimal". Then we can get back to talking about whether or not I or others here are applying them correctly. -|Tom|-

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1234567890
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21 years 7 months ago #5318 by 1234567890

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tvanflandern
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21 years 7 months ago #5319 by tvanflandern

Replied by tvanflandern on topic Reply from Tom Van Flandern

<BLOCKQUOTE id=quote>quote:<hr height=1 noshade id=quote>If there is no limit to the size of integers then integers can be infinite sized by definition.<hr height=1 noshade id=quote></BLOCKQUOTE id=quote>

Incorrect. All integers are finite. The number of integers is infinite. The former are real and specific. The latter is a concept. No integer is infinite. But there are an infinite number of them.

But I've already said this in one form or another several times. It is one thing to reject reasoned conclusions, and another to reject definitions.

<BLOCKQUOTE id=quote>quote:<hr height=1 noshade id=quote>When you consider all the integer elements in the number set, this count itself is a quantity, which is represented by a number.<hr height=1 noshade id=quote></BLOCKQUOTE id=quote>

If the set is all integers, then the count is not represented by a number, but by a concept -- infinity. It is not a part of the set. If the set were ten apples, the number ten would not be a part of the set.

This point has also been made repeatedly. Please accept this as a true premise or show why it is unacceptable.

<BLOCKQUOTE id=quote>quote:<hr height=1 noshade id=quote>Why else do we have numbers if not to use it to represent a quantity? The count is an infinite number which is itself an element of the number set.<hr height=1 noshade id=quote></BLOCKQUOTE id=quote>

Infinity is not a member of the set of integers. It is a concept. As such, it serves to represent, not a quantity, but a lack of limits to the quantity.

Let's not go around this circle again, shall we?

<BLOCKQUOTE id=quote>quote:<hr height=1 noshade id=quote>OTOH, if we placed a limit to numbers , we can easily define the number set as a quantity that is not an element of the number set. For example, if we only have -10 to 10 as the number set, we can sensibly say that the number set is 21 in quantity and hence is not an element inside its own set.<hr height=1 noshade id=quote></BLOCKQUOTE id=quote>

The same holds true if we place no limit. -|Tom|-

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1234567890
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21 years 7 months ago #5644 by 1234567890

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<BLOCKQUOTE id=quote>quote:<hr height=1 noshade id=quote>
<BLOCKQUOTE id=quote>quote:<hr height=1 noshade id=quote>If there is no limit to the size of integers then integers can be infinite sized by definition.<hr height=1 noshade id=quote></BLOCKQUOTE id=quote>

Incorrect. All integers are finite. The number of integers is infinite. The former are real and specific. The latter is a concept. No integer is infinite. But there are an infinite number of them.

But I've already said this in one form or another several times. It is one thing to reject reasoned conclusions, and another to reject definitions.

<BLOCKQUOTE id=quote>quote:<hr height=1 noshade id=quote>When you consider all the integer elements in the number set, this count itself is a quantity, which is represented by a number.<hr height=1 noshade id=quote></BLOCKQUOTE id=quote>

If the set is all integers, then the count is not represented by a number, but by a concept -- infinity. It is not a part of the set. If the set were ten apples, the number ten would not be a part of the set.

This point has also been made repeatedly. Please accept this as a true premise or show why it is unacceptable.

<BLOCKQUOTE id=quote>quote:<hr height=1 noshade id=quote>Why else do we have numbers if not to use it to represent a quantity? The count is an infinite number which is itself an element of the number set.<hr height=1 noshade id=quote></BLOCKQUOTE id=quote>

Infinity is not a member of the set of integers. It is a concept. As such, it serves to represent, not a quantity, but a lack of limits to the quantity.

Let's not go around this circle again, shall we?

<BLOCKQUOTE id=quote>quote:<hr height=1 noshade id=quote>OTOH, if we placed a limit to numbers , we can easily define the number set as a quantity that is not an element of the number set. For example, if we only have -10 to 10 as the number set, we can sensibly say that the number set is 21 in quantity and hence is not an element inside its own set.<hr height=1 noshade id=quote></BLOCKQUOTE id=quote>

The same holds true if we place no limit. -|Tom|-

<hr height=1 noshade id=quote></BLOCKQUOTE id=quote>

The number of integers in the integer set is 2 times the largest integer in the set plus 1. If you also stipulate that there are an infinite number of integers, this contradicts the definition of an integer as being finite and discrete in size since they would also become infinitely large. So the concept of infinity becomes an element in the set of integers under these assumptions.

Only if you limited the number of integers can the number of elements in the integer set be a finite quantity of which is outside the integer set. This number would be 2 times the largest positive integer plus 1, but would now be represented by a number that is not an integer- an imaginary integer.

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