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Cont: Deeper than primes - Continuation 2

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Please pay attention that according to post http://www.internationalskeptics.com/forums/showpost.php?p=11243802&postcount=1040, there are different sizes of infinite sets and yet any given infinite size can be changed by adding to it finite or infinite amount of members.

Maybe I am wrong but I think that the notion that, for example, aleph0=aleph0+1 is based on the fact that a proper subset of a given set with infinitely many members is in bijection with its infinite set (for example: the set of odd numbers is in bijection with the set of natural numbers), but being in bijection between a given infinite set and its proper subset does not avoid the conclusion that the number of members of that set+1 > the number of members of that set by 1 more member (as shown in http://www.internationalskeptics.com/forums/showpost.php?p=11243802&postcount=1040).
 
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doronshadmi said:
Ok.
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"Ok" as in "I accept what you said and won't simply continue to use meaningless notation", or "OK" as in "I'll continue without regard to anything you said"?

Let's accept that there is no bijection between the real numbers and the natural numbers, where both of them are represented by the place value method.
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No, let's not simple "accept". The non-existence of a bijection between the set of natural numbers and the set of reals is a proven mathematical fact. Your acceptance is not required. Moreover, that fact is not influenced in any way by your choice in numerical representation.

For example:
...
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You cannot just steal an image from Wikipedia and say "Tada!". The image is quite a few steps away from establishing the cardinality relationship between the set of natural numbers and the set of reals.

All you have done is pull the image out of context and claimed it shows something it does not directly show.

...if we are able to add real numbers to the list (including their matched s numbers) we actually can't conclude that the s numbers can represent the set of all natural numbers, in the fist place.
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Maybe if you actually followed Cantor's proof rather than substituting all the details with the imaginary, you'd understand why the proof is valid and your objections to it nonsense.

...
This is not the case about, for example 1,000,000,... < 1+1,000,000,... by 1, which clearly shows this mathematical fact
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And now we have the answer to the my question, above. And since your notation is meaningless, your conclusions derived from it are equally so.
 
jsfisher said:
Maybe if you actually followed Cantor's proof ...
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Cantor's proof about the non-bijection between a set and its power set (whether it is finite or infinite set) uses the diagonalisation argument.

In the case of an infinite set, we must assume that this set is complete (no one of its members is missing).

But since we are able to add any given diagonal number to the list, our assumption that the "left" side is a complete infinite set, is not satisfied, and aleph0+1=aleph0 simply can't show it.

On the contrary by using the place value method, for example, 1,000,000,... < 1+1,000,000,... by 1, so unlike the aleph0 case, it can show the incompleteness of the "left" side.
 
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jsfisher said:
Maybe if you actually followed Cantor's proof rather than substituting all the details with the imaginary, you'd understand why the proof is valid and your objections to it nonsense.
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You are still missing my argument.

Even if I have no objection about Cantor's proof, it does not mean that the list he uses has the all natural numbers, exactly because more natural numbers can be added to that list (as shown in http://www.internationalskeptics.com/forums/showpost.php?p=11243802&postcount=1040).

aleph0+1=aleph0 does not deal with this addition.

On the contrary, for example, 1+1,000,000,... > 1,000,000,... deals with this addition and therefore it has what you call: "superiority over aleph0".
 
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jsfisher said:
A diagonalization proof does not involve adding things to a list.
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We are not talking about a diagonalization proof, but about the superiority of the place value over aleph0, such that by using place value, one enables to refine the difference between the sizes of infinite sets (as explained and shown in http://www.internationalskeptics.com/forums/showpost.php?p=11243802&postcount=1040), where this difference can't be shown by using aleph0, exactly because, for example, aleph0+1=aleph0.

If you are still missing it, by using place value an infinite set with 1+1,000,000,... members does not have a bijection with set with 1,000,000,... members, and aleph0 can't be used in order to show this non-bijection.
 
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doronshadmi said:
We are not talking about a diagonalization proof...
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You were talking about a misrepresentation of diagonalization and then spring-boarding from that to a justification for a meaningless notation.

If your starting point is bogus, then your conclusions are as well. Diagonalization does not involve "adding to the list" and has no dependence on "+1".
 
jsfisher said:
Diagonalization does not involve "adding to the list" and has no dependence on "+1".
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This is the view of aleph0 number (aleph0+1=aleph0) about the size of an infinite set.

EDIT: It seems that you simply do not wish to reconsider the size of an infinite set by using the place value method, that if properly developed, may refine our understanding of the sizes of infinite sets.
 
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jsfisher said:
What is?

And why are you implying aleph0 is a matter of opinions? Are you once again intending to disprove a definition?
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You are using definitions, which according to them the size of the set of natural numbers is called aleph0, such that, for example, aleph0+1=aleph0 (with all the consequences that are derived from such equality about the measurement of the size infinite sets).

I start with the notion that the number that measures the size of the set of natural numbers, is greater than any natural number, which means that by using the place value method, such number has infinitely many digits, for example: 1,000,000,... (more details are found in http://www.internationalskeptics.com/forums/showpost.php?p=11242445&postcount=1036).

If you disagree about this notion (as explained in http://www.internationalskeptics.com/forums/showpost.php?p=11242445&postcount=1036) than please explain why it is fundamentally mathematically\logically invalid?
 
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doronshadmi said:
You are using definitions, which according to them the size of the set of natural numbers is called aleph0
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Yes, the cardinality of the set of natural numbers is represented by the notation, aleph0.

...such that, for example, aleph0+1=aleph0
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No. There is no "such that" condition on the representation.
 
doronshadmi said:
You have missed the meaning of ", for example," in this sentence.
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No, didn't miss that at all. Your "for example" would be just one example of presumably many conditions you imagine for aleph0. Since there are no "such that" conditions, your "for example" would be an example of one that isn't.

Please answer to the rest of post http://www.internationalskeptics.com/forums/showpost.php?p=11244226&postcount=1052.
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Until such time as you establish that your claimed positional notation (which is not positional notation) has some meaning that makes it worthy of consideration, there is no point moving beyond that issue.
 
This current arc started with this:

doronshadmi said:
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if we use the place value method in order to represent transfinite number like aleph0, it has to be represented by infinitely many digits, for example: 1,000,000,000,...
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Doronshadmi, you need to focus on this part before you can advance.

You clearly said, "place value method." What, then, is the place value for the 1 in 1,000,000,000,...? And for each of the zeros?

Without place values, yours is not a place-value-method representation for the cardinality of the set of integers. It is just a string of 17 symbols. You may as well use aleph0. It is widely accepted so unlikely to confuse, and you won't mistakenly treat it as if it where a natural number.

Or was that your intent? To confuse, and to mistreat is as if it were a natural number?
 
jsfisher said:
No, didn't miss that at all. Your "for example" would be just one example of presumably many conditions you imagine for aleph0. Since there are no "such that" conditions, your "for example" would be an example of one that isn't.
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It is accepted among traditional mathematicians that by using the notion of aleph0 an equality like aleph0+1=aleph0 holds.

If you claim otherwise (for example, that aleph0+1>aleph0 by 1) than please support your claim by using traditional mathematics.
 
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doronshadmi said:
Worng, it represents a number with infinitely many symbols such that 1+1,000,000,000,... > 1,000,000,000,... by 1, where 1,000,000,000,... is greater than any natural number.
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You claimed it was "place value method". Does each digit position have a place value or not?

If not, than this isn't "place value method".

Moreover, since you are also claiming behavior different from aleph0, your sequence of 17 symbols doesn't represent the same thing as aleph0. It does not represent the cardinality of the set of natural numbers.
 
doronshadmi said:
It is accepted among traditional mathematicians that by using the notion of aleph0 an equality like aleph0+1=aleph0 holds.
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This has nothing to do with what I wrote. I took issue with, and continue to take issue with, your abuse of the phrase, "such that".
 
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